AN INTRODUCTION TO
GAME THEORY
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Contents
Preface
1
xi
Introduction 1
1.1
What is game theory? 1
An outlíne of the history of game theory 2
John von Neumann 3
1.2丁he theory of rational choìcε4
1.3
Coming attraζtions: interacting deζision-makers
Notes 9
Games with Perfect Information
7
11
2
Nash Equilibrium: Theory 13
2.1
Strategic games 13
2.2
Example: the Prisoner主 Dilemma 14
2.3
Example: Bαch or 5travinsky? 18
2.4
Example: Matcfling Pennies 19
2.5
Example: the 5tag Hunt 20
2.6
Nash equilibrium 21
John 在 NashJr. 23
5tudying Nash equilibrium experimental非 24
2.7
Examples of Nash equilibrium 26
Experimental evidenceon the Prisoner's Dilemma 28
Focal points 32
2.8
Best response funζtions 35
2.9
扫ominated actions 45
2.10 Equilibrium in a single population: symmetrìc games and
symmetric equilibria 50
Notes 53
3
Nash 在quilibrium: lII ustrations
55
3.1ζournot's 阳odel of oligopoly 55
3.2
Bertrand's model of oligopoly 63
Courno乙 Bertrancl and Nash: some historical notes
3 .3到ectoral competition 70
3.4
The War of Attrition 77
69
vi
Contents
3.5
Auctions 80
Auctions from Baby/onia to eBay
3βAccident law 91
Notes 97
4
81
Mixed Strategy Equilibrium 99
4.1
Introduζtion 99
Somεεvidence on expected payoff functions 104
4.2
Strategic games in which players may randomize 106
4.3
Mixed strategy Nash equilibrium 107
4.4
Dominated actions 120
4.5
PlJ re equilibria when randomization is allowed 122
4.6
IlI ustratíon: expert diagnosís 123
4.7
Equilibrium in a single population 128
4.8
IIlustration: reporting a crime 131
Reporting 0 críme: socia/ psych%gy and game theory 133
4.9
丁he form在tion of players' belìefs 134
4.1 0 Extensíon: finding all mixed strategy Nash equilibria 137
4.11 Extension: games in which each player has aζontinuum of actions
4.12 Appendix: representing preferenζes by expected payoffs 146
Notes 150
5
Extensive Games wi白 Perfect Information: Theory 153
5.1
Extensive games with perfect information 153
5.2
Strategies and outcomes 159
5.3
Nash equilibrium 161
5.4
5ubgame perfect equilibrium 164
5.5
Finding subgame perfect equilibria of nnite horizon games:
backward induction 169
Ti cktacktoe, ches5, and related games 178
Notes 179
6
Extensive Games with Perfect Information: lII ustrations 181
6.1
The ultimatum game, the holdup game, and agenda control
Experiments on the ultimatum game 183
6.2
Stackelberg's model of duopoly 187
6.3
Buying votes 192
6.4
A race 197
Notes 203
7
142
181
Extensive Games with Perfect Information: Extensions 我nd Discussion 205
7.1
Allowing for simultaneous moves 205
Mor!ε experimenta/ evidence on 5ubgamε perfect equifibrium 211
7.2
IIlustration: 台ntry into a monopolized industry 213
7.3
IlI ustration: electoralζompetition with strategic voters 215
Contents
7.4
7.5
7.6
7.7
vii
Il! ustration: committee decision-making 217
IlI ustration: exit from a declining industry 221
Allowing for exogenous un优 rtainty 225
Discussion: subgame perfect equilibrium and baζkward induction
231
Experimental evidence on the centipεdegamε234
Notes
236
8ζoalitional Games and the Core
239
8.1ζoalitional gamε5
8.2
8.3
3.4
8.5
8.6
8.7
239
The core 243
IlI ustration: ownership and the distribution of wealth 247
IlI ustration: exζhanging homogeneous horses 251
IlI ustration: exζhanging heterogeneous houses 256
IlI ustration: voting 260
1I 1ustration: matching 263
Matching doctors with hospital5 268
8.8
Discussion: other solution concepts
Notes 270
269
11
Games with Imperfect Information
271
9
Bayesian Games 273
9.1
Motìvational examples 273
9.2
General definitions 278
9.3
Two examplesζoncerning information 282
9.4
'"ustration: Cournot's duopoly game with imperfect information 285
9.5
IlI ustration: providing a public good 289
9.6
IlI ustration: auζtions 291
Auctions of the radio spectrum 300
9.7
lII ustration: juries 301
9.8
Appendix: auctions with an arbitrary distribution of valuations 307
Notes 311
10
Extensive Games with Imperfect Information 313
10.1 Extensive games with imperfect information 313
10.2 Strategies 317
10.3 Nash equilibrium 318
10.4 Beliefs and sequential equilibrium 323
10.5 Signaling games 331
10.6 lII ustration:ζonspicuous expenditure as a signal of quality 336
10.7 IIlustratìon: eduζation as a signal of ability 340
10.8 IIlustration: strategic information transmission 343
10.9 IlI ustratìon: agenda control with imperfect information 351
Notes 357
Conter王军S
viii
111
Variants and Extensions 359
11
Strictly Competitive Games and Maxminimization 361
11.1 Maxminimization 361
11.2 Mδxminimization and Nash equilibríum 364
11 .3 5trictly competitive games 365
11 .4 Maxminimization and Nash equilibrìum in strictly competitìve games
Maxminimization: some history 370
Empiricaf tes位 experiments， tennis, and soccer 373
Notes 375
12
Rationalizability 377
12.1 Rationalizability 377
12. 2 Iterated elimination of strictly dominated actions 385
12.3 Iterated elimination of weakiy dominated actions 388
12.4 Domina 门ζe solvability 391
Notes 392
13
Evolutionary Equilibrium 393
13.1 Monomorphic pure strategy equilibrium 394
Evofutionary game theory: some history 399
13.2 Mixed strategies and polymorphic equilibrium 400
13.3 Asymmetricζontests 406
Side-bfotched Iizards 407
Expfaining the outcomes of contests in nature 409
13 .4 Variation on a theme: sibling behavior 411
13 .5 Variation on a theme: the nesting behavior of wasps 414
13.6 Variation on a theme: the evolution ofthe sex ratio 416
Notes 417
367
14 Repeated Games: The Pr;soner结 Dilemma 419
14.1 The main idea 419
14.2 Preferenζes 421
14.3
只epeated games 423
14.4 Finitely repeated Prisoner 's Dilemma 424
14.5 Infinitely repeated Prìsoner's Dif，εmma 426
14.6 5trategies in an in位 nítely repeated Prisoner 's Difemma 426
14.7 50me Nash equilibria of an infìnitely 陀 peated Prisoner 's Difemma 428
14.8 Nash 号quilibrium payoffs of an in在nitely repeated Prisoner 's Difemma 431
Experimentaf evidence 436
14.9 Subgame perfect equilibria and the one-deviation property 437
Axefrod 's tournaments 439
Contents
1)(
14.10 Some subgame perfect equilibria of an in台 nitely repeated
Prisoner 's Dilemma 441
Rec年Jrocal altru也m among sticklebαcks 445
14. ì 1 Subgame perfect equilibrium payoffs of an infinitely repeated
Prisoner主 Dilemma 446
Medieval trade 句;rs 448
14.12ζonduding remarks 449
Notes 449
15
16
17
Repeated Games: General Results 4S 1
15.1 Nash equilibria of general infinitely repeated games 451
15.2 Subgame perfect equilibria of general 汩的 nitely repeated games
15 .3 Finitely repeatεdgamε5 460
15 .4 Variation on a themε: imperfe口 observability 461
Notes 463
Bargaining 465
16.1 Barg吕 ining as 吕 n extensiγegame 465
16.2 IlI ustration: trade in a market 477
16.3 Nash's axiomatic model 481
1 己.4 Relation between strategic and axiomatic models
Notes 491
Appendix: Mad将matics
17.1 Numbers 493
17.2 Sets 494
17 .3 Functìons 495
17.4 Profiles 498
17 .5 Sequences 499
17β
Probabìlity 499
17.7 Proofs 505
References
Index
525
507
493
489
455
Preface
- AME-THEORE Tl C REASONING pervades economic theory and is used widely
in other social and behavioral sciences. This book presents the main ideas
of game theory and shows how they can be used to understand economic, social ,
political, and biological phenomena. lt assumes no knowledge of economics , political scie!1ce, or any other social or behavioral science. 1t emphasizes the ideas behind the theory rather than their mathematical expression a!1 d assumes no specific
mathematical knowledge beyo口d that typicaily taught in U. 5. and Canadian high
schools. (Chapter 17 reviews the mathematical concepts used in the boo k.) 1n particular, calculus is not used , except in the appendix of Chapter 9 (Section 9.8). Nevertheless, a11 concepts are defined precisel予 and logical reasoning is used throughou t. My aim is to explain the main ideas of game theory as simply as possible
while maintaining complete precision; the more comfortable you are with tight
logical analysis, the easier you will find the arguments.
The only way to appreciate the theory is to see it in action, or better still to put
it into action. 50 the book includes a wide variety of illustrations from the social
and behavioral sciences, and over 280 exercises.
The structure of the book is illustrated in the figure on the next page. The
gray boxes indicate core chapters (the darker gray, the more important). A black
arrow from Chapter i to Chapter j means that Chapter f depends on Chapter i.
Thεgray arrow from Chapter 4 to Chapter 9 means that the latter depends weakly
on the former; for all but Section 9.7 only an understanding of expected payoffs
(5ection 4. 1. 3) is required， ηot a knowledge of mixed strategy Nash equilibrium.
(Two chapters are not included in this figure: Chapter 1 reviews the theory of a
single rational decision-maker, and Chapter 17 reviews the mathematical concepts
used 1.n the book.)
Each topic is presented with the aid of "Examples气 which highlight theoretical points , and "Illustrations" , which demonstrate how the theory may be used to
understand social , economic, political, and biological phe口omena. The μ Illustra­
tions" introduce no new theoretical points , and any or all of them may be skipped
without loss of con由mi可. The "Illustrations" for the key models of strategic and
extensive games are grouped in separate chapters (3 and 6)
The limited dependeηcies between chapters mean that several routes may be
taken through the book
-
VI
Preface
xii
Strategic games
3: IlI ustrations (55)
4: Mixed strategies (99)
Imperfect information 一
9: Bayesian games (273)
Topìcs
111: Maxminimization (361)
1 1 2 : Rationaliz在 bility (377)
!1 比olutionary 仰ilibrium (则|
Extensive game主
6: IlI ustrations (181)
7: Extensions (205)
Imperfect information
10: Signaling games (313)
Topics'
14， 15: 只epeated games (419 , 451)
比)
Coalitional games
16: Bargaining (465)
Th苦 structure of the book. Each chapter is represented 如 ya box whose area is proportional to the length
of the chapter; the number i只 parentheses is the page on which the chapter begins. Boxes shad在d
gray correspond to core chapters; dark gray chapters are more central than Jight gray ones. An arrow
纭。m Chapter í to Chapter j means that C加pter í is a PI院吨uísite for Chapter j. The gray arrow from
Chapter 4 to Chapter 9 means that the latter depends only weakly on the former.
Preface
XIII
• You may add to this plan any combination of Chapt时54 (Mix时 Strategy
Equi1ibrium) , 9 (Bayesian Games, except Section 9.7, which requires Chapter 钙， 7 (Extensive Games with Perfect Information: Extensions and Discussion) , 8 (Coalitional Games and the Core), and 16 (Bargaìning).
• If you read Chapter 4 (Mixed Strategy Equilibrium) , then you may in addition study any combination of the remaining chapters covering strategic
games, and if you study Chapter 7 (Extensive Games wìth Perfect Informa柱。n: Extensions and Discussion) , then you are ready to tackle Chapters 14
and 15 (Repeated Games).
Whichever route you take, you can choose the examples and illustrations to fit
one of several themes. You can, for instance, study all theeconomic εxamples， or
all the po 1itical or biological ones. Altemativel予 you can build a course around
all the examples on a more narrow topic; two possibilities are auction theory and
oligopoly theory.
All the material is intended to be accessible to undergraduate students. A onesemester co时se for tlùrd or fourth year North American εconomics majors (who
have been exposed to a few of the main ideas in first and second year courses)
could cover up to about half the material in the book in moderate detaìl粤
Osbome and Rubinstein (1994), a graduate text, offers a more advanced treatment of the field 建 With few exceptions, the two books use the same notation and
terminology.τhe exceptions are noted on the website for tlùs book (see the end of
the preface).
应xamples modeling economic, political , and biological phenomena
τhe main examples that involve economic, po1itical, or biological phenomena
are listed below. (Examples used to make a primarily game-theoretic point are
omitted.)
Economic phenomena
Accident law: Section 3.6
Adverse selection: Exercise 282.3
Auctions: Section 3.5 (perfect inforrnatio吟， Example 143.1 (all-pa抖， S仅tion 9.6
(imperfect information), Section 9.8 (imp时fect inforrn州on)
Bertrand's model of oligopoly: Section 3.2, Exercise 136.2, Exerdse 146.1 ,
Exercise 192.1 , Exercise 214上 Exercise 392.1 (dominance solvabi1ity) ,
Exerdse 454 .3 (repeated game), Exercise 459.1 , Exerdse 459.2, Section 15.4
(repeated game wi在 imperfectly observable actions)
Chain-store game: Example 23 1.1
Collective decision-making: Section 2.9.4
Common property: Section 3 .1.5
-
XIV
Preface
Competition in product characteristics: Exercise 76 .1
Cournot's model of o1ìgopoly: Section 3.1, Exercise 136.1, Section 9.4 (imperfect
informatio时， Exercise 388.2 (rationalizability)
Entry into an industry by a financially-constrained challenger: Exercise 174.2
Entry into a monopolized indust主y: Section 7.2
Exit from a dedi我ing industry: Secti∞ 7.5
Expert diagnosis: 如ction 4.6
Fi口n-unÍon bargaining: Exercise 177.1 , Exerdse 227.2 , Exercîse 489 .1
Holdup game: Section 6. 1. 2
Market games: Exercise 21 1. 2, Example 245 .5, Section 8.4, Section 8.5, Section 16.2
Matc趾ng: Example 242.2, Section 8.7
Ownership and the distribution of wealth: Example 240.2 , Example 244.2 ,
Section 8.3
Prìce competition between sellers: Exercise 128.t Exercise 212 .1
Provision of a public good: Eχercise 33.t Section 立8 .4， Exercise 44.1 ,
Exercise 132 .3, Section 9.5, Exercise 388 .1 (rationalizability)
Reporting a crime (private provision of a pu剖ìc good): Section 4.8
"Rotten kid theorem": Exercise 177.2
Signaling ability with education: Section 10.7
Signaling quality with conspìcuous expenditure: Section 10.6
Stackelberg's model of duopoly: Section 6.2
Strategìc information transmission: Section 10.8
Timíng product release: Exercise 80 .1
Po /i tical phenomena
Agenda contJτ01: Section 6. 1.3 (perfect ínformation) , Section 10.9 (imperfect
information)
Allocating resources in election campaigns: Exercise 141.3
Approval voting: Exercise 49.2
Buying votes in a legislature: Section 6.3
Cohesion of governing coa 1i tions in legislatures: Exercise 228 .1
Collective decision-making: Section 2.9.4
Committee dedsion-making: Section 7.4
Electoral competition between citizen而candidates: Exercîse 75.2
Electoral competition be tJw een policy-motivated candidates: Exercise 75.1
Hotelling's model of electoral competition: Section 3.3, Exercise 196.3,
Exercise 196.毡，Section 7.3 (strategic voters) , Exercise 387.5 (rationalizability) ,
Exercise 454.2 (repeated game)
Juries: Sεction 9.7
Lobbying as an auction: Exercise 9 1. 1
Majority game: Example 240.3 , Exercisε245 .1
Vote trading: Exercise 247.1
Voter partidpation: Exerdse 34.2, Exercise 118.2
Preface
xv
Voting: Section 2.9.3 (strategic game) , Section 8.6 (coalitional game) ,
Exercise 307.1 (swing voter's curse) , Exercìse 39 1. 3 (dominance solvability)
Voting by alternating veto: Exεrcise 173.3
Biological plJ enom衍10
Evolution of sex ratio: Section 13.6
Hawk-Dove: Example 398.2, Example 404.2, Example 409.1
过ermaphrodìtic fish: Exerdse 18 .1
N
叶es仗t州in咆
g behavior of was呼
ps
尔: Section 13.
Reci￥
严rocal altruism: Box on page 445
P
Siblìng competition: Section 13 .4
Signalìng hunger (Sir Philip Sydney game): Exercisε335.2
W盯 of Attritíon: Sectio川4
Personal pronouns
τhe Englísh language lacks a third person singular pronoun widely interpr!εted
to be sex neutral. 1n particular, many experiments have shown that "he" ìs not
neutral1, a fìnding consistent witl飞 the observation that whereas p巳ople may say
"when an airplane pilot is working, he needs to concentra栓"， they do not llsually
say 气vhen a flight attendant is working, he needs to concentrate" or "when a secretary is working, he needs to concentrate气 To quote the American Heritage Dictionary
(third edition, page 831) , '节lUS he is not really a gender-neutral pronoun; rather
it refers to a male who ìs to be taken as the representative member of the group
陀ferred to by its antecedent. The traditional usage, then, is not simply a grammatical conver飞tìon; it also suggests a pa时icular pattern of thought." Like many
writers, 1 regard as unacc心ptable the bias 泣nplicit in the use of "he" for indìvid伦
比 of unspecified sex. The New 0功rd Dictionary of Eηglísh states the case clearly:
"[the use of he to refer to a persoηof unspecifìed sex] has become . . . a hallmark of
old-fashioned language or sexism in language." Writers have becomεsensìtive to
thìs issue in the last 豆fty years , but the lack of a sex们eutral pronoun "has been felt
since at least as far back as Middle English" (Webster's Díctionnry of Englislz Usage ,
Merriam-Webster Inc. 1989, 499). A common solution has been to use "they",2 a
usage that the New Oxford Dictionary of Englísh en巾rses (and employs). 1n some
contexts this usage sounds natural, but in others it does not; it can also create ambiguity when the pronoun follows references to 1τlore 也an one person. 1 choose a
different solution: 1 use "she" exclusively. Obviously this usage, like that of "he" ,
lS伐， for example, Jaηice Moulton, Geor军e M. Robinson, and Cherin Elias, "Sex bias in language
use: 'neutral' pronouns that arenγ ， American Psychologist 33 (1舍78)， 1032-1036; Janet Shibley Hyde,
"Children's understanding of sξxist language", Development Psycholo，即 20 但她F 的工706; and John
Gastil, "Generic pronouns and sexist language: th哇。xymoronic character of masculine generics", Sex
Roles 23 (1990), 629-643.
2Por a discussion of the history of the use of "they飞 seeA挝1 Bodin哇， "Androcentrism ìn prescriptive
granunar: singular 'they', sex-indefinite 'he', and 'he or she' "，如nguage 仇 Society 4 (1975), 129-146.
Pref，在ce
xvi
is not sex neutral, but it may help to counterbalance the wides严ead use of "he",
and seems unlikely to do any harm.
References
η况 "Notes" section at the end of each chapter attempts to assign credit for the
ideas discussed. Several cases present difficulties. Jn some cases, ideas evolved
over a long period of time, with ζontributions by many pεople， making their origins hard to summarize in a sentence or t认咆In a few cases, my research has led
to a conclusion about the origins of an idea different from the standard one. ln all
cases, 1 cite the relevant papers without regard to their difficulty.
Over the years, I have taken exercises from many sources. 1 have attemptεd to
remember the origins of the ones 1 use in this book, and give credit appropriately.
ζonventions， numbering, and nomencla栩栩
1 use the term "dollar" for a unit of money because it is probably recognizable as
such to a majority of readers of this book, even if they pay their b il1 s in rupees,
yuan, or pa' anga.
In formal definitions, the terms being defined are set in boldface. Terms set in
italics are informal definitions.
Definitio邸， propositions, lemmas , examples , exercis时， and equations are numbered according to the page on which they appear. If the 在rst such object on
page n is an exercise , for example, 让
i t is called Exercise fη1.上
e
t出
ha
挝t page is a definition, it is called Definition n.L. For example , the definition
of a strategic game with ordinal prefereηces on page 13 is De自nition 13. 1. Figures are numbered separately, using the same page-based system. The scheme
allows number付 items to be found rapidly and enhances the precision of indεx
entrìes.
Symbollterm
Meaning
(
Exercise, with solution on website
<1D
Hard exercise , with solution on website
主
Exercise， with solution available only to instructors
苦
•
(see website for conditions)
Hard exercise, with solution available only to instructors
(see website for conditions)
Definition
Result
Ex在mple: a game that illustrates a g在me-theoretic point
I1lustration
A game , or family of games，也at shows how the theory can
ìlluminate observed phenomena
Prefaω
XVII
Acknowledgments
Iowe ahuge dεbt to Arìel Rubir飞stein. 1have learn驭ì， and continue to Iearn, vastly
from him about game theory. His influence ∞ this book will be clεar to anyone
familiar with our jointly authored book A course in gamf t险。 ry. Had we not written
that book and our previous book Bargaining and markets, 1 doubt that 1 would have
embarked on this project.
1 was privileged as a graduate student at Stanford University to learn game
tÌ况。ry from Robert Aumann, Sergiu Hart , Mordecai Kurz , Al Roth, and Robert
Wilson. Discussions over the years with Jean-Pierre Benoît, Haruo lmai , Vijay
Kris怡la， Michael Peters, and Carolyn Pitchìk have improved my understanding
ofmany game俨theoretic topics.
Many people have generously commented on a11 or parts of drafts of the book.
1 am partiωlarly grateful to Peter McCabe for many very thoughtful comments.
1 am very grateful also to Gian Luigi Albano, Jeffrey Banks, Nikolaos ßenos, Ted
Bergstrom, Tilman B怂rgers， Randy Calvert, Vu Cao, In叩Koo Cho, Rachel Cτ050n，
Eddie Dekel, Marina De Vos , Lauríe 口uke， Patrick Eli邸， Mukesh Eswaran, Xinhua
Gu , Costas Halatsis, JoεHarrington， Hiroyuki Ka飞Nakatsu， Lewis Kornhauser, Jack
Leach，丁ao Li, Simon Link , Bart Lip邵阳，Kin Chur飞gLo， M部simo Marinacci , Mas呻
simo Morelli, Nathan Nunn, Barry O'Neíll, Robin 口 .Osborr飞e， Marco Ottaviani,
C挝olyn Pitchik , Marie Rekkas, Bob Rosentha l, Al Roth , Matthew Shum, ß1笔anislav
L. Slantchev, Giora Slutzkì, Michael Smart, Nick Vriend , Charles A. Wìlson, Luís
Zemborain, and several a口onymous reviewers.
My editors at Oxford University Press, Kenneth MacLeod and Stephen McGroarty, offered much encouragement and advice. 丁heir enthusiasm for thεproject
was a great help in keeping the book on trac k. The many suggestìons of Oxford's
production editor Brian Kinsey greatly improved the appearance of the book.
τhe book has its origins in a coursε1 taught at Columbia Universìty in the
early 1980s. My experience in that course , and in courses at McMaster Universìty
and the University of Toronto, brought the book to its current forrr飞 The Kyoto
Institute of Economic Research at Kyoto 飞Jniversitγ 在nd the School of Economics
at the Australian 时ational University provided me with splendid envìronments ìn
whìch to work on the book during visits in 1999 and 200 1.
1 maintain a websitεfor the book. τhe current URL is
http://www.economics.utoronto.ca/osbo古ne/igt/
MAR Tl N J. OSBORNE
martin.osborne@utoron乞 o.ca
http://www.ecor飞omics.utoronto.ca/osborne/
ηepartment of Economics, University of Toronto
150 St. George Street， τoronto， Canada, M.5S 3G7
May 200:1
AN INTRODUCTION TO
GAME THEORY
Introduction
1.1
What is game theory? 1
1.2τhe theory of rational choice
13
1.1
4
Coming attractions: interacting decision-makers 7
What is game theory?
AMEτHEORY aims to help us understand situations in which decision-makers
interact. A game in the everyday sense-"a competitive actìvity ... in which
playεrs contend with each other according to a set of rules'二 in the words of a
dictionary-is an εxample of such a situation, but the scope of game theory ìs very
much larg时. lndeed , 1 devote very lit t1e spacεto games in the everyday sense;
my main focus is th号 use of game thεory to illuminate economic , politicat and
biological phenomena.
A list of some of the applications 1 discuss will give you an idea of the range
of situ拭ìons to whích game theory can be applied: firms competing for busi时部，
political candidates competing for votes, jury members deciding on a verdict, animals fighting over prey, bidders competing in an auction, the evolution of síblings'
behavior towards each other, competing exp但怒， incenti飞'es 如 correctly diagnosε
a problem, legislators' voting beh在飞rior under pressure from interest groups, and
the role of threats and punishment in long-term relationships.
Like other sciences, gam台 theoryωnsists of a collection of models. A model
ìs an abstraction we use to understand our observations and experiences. What
"understanding" entails is not clear-cu t. Partly, at least, it entails our perceiving
relationships between situations, isolating principles that apply to a range of problems, so that we can fit into our thinking new situations that we encounter. For
example, we may fit our oDservatiOI飞 of the path taken by a lobbed teru豆s ball into
a rnodel that assumes the ba l1 moves forward at a constant veloci咛 and is pul1 ed
towards the ground by the constant force of "gravity". This model enhances our
understanding bεcause ít 自ts well , no matter how hard or in which di陀ction the
ball is hit, and appliεs also to the paths taken by baseballs , cricket balls, and a wide
variety of other missiles , launched in any direction.
A model is unlikely to help us understand a phenomenon if its assumptions are
wildly at odds with our observations. At the same time, a model derives power
from its simplicity; the assumptions upon which it rests should capture the essence
of the situation, not irrelevant details. For example, when consideriì1g the path
Chapter 1. Introduction
2
AN OUTLINE OF THE HISTORY OF GAM芜丁 HEOl之Y
Some game-theoretic ideas εan be traced to the 18th century, but the major development of the theorγbegan in the 1920s with the work of the mathematician
主mile Borel (1871-1956) and the polymath John von Neumam飞 (1903-57). A de翩
cisive event ir飞出e development of the 位eory was the publication in 194圣 of the
book Theory of games and economic behavior by von Neumann and Oskar Morgenstern , which established the foundations of the 在eld. In the early 1950s, John F.
N出h (seεthe box 0饥 page 23) developed a 检y concept (Nash 叫uilibrium) and
initiated the game-theor时ic study of bargaining Soon aft，εr Nash's work , gametheoretic models began to be used in economic theory and po1itical science, and
psychologists began studying how human subjects behave in experimental games.
In the 1970s game theory was first used asa tool in evolutionary biology. Subsequently, game蝠theoretic methods have come to dominate microeconomic theory
and are used also in many other fields of eζonomics and a wide range of other
social and beha时oral sciences. The 1994 Nobel Prize in Economic Sciences was
awarded to the game theorists)o加C. Harsanyi (1920-2000) ， )ohn 巨时ash (1928寸，
and Reinhard Selten (1930一).
‘
taken by a lobbed tennis ball we should ignore the dependence of the force of
gravity on the distance of the ball from the surface of the earth.
民
Mo
叫
de
岱
i沁$ εu
创
缸nno
a
悦tb
挝
e JU
叫
dgedb
切
ya川b
捡享范
臼
S
olu
时t怆
们
e
时
C
ri技t怡加加
创灿加
e
时r‘ io
，υ'wroα飞飞喀
g". Whether a model is useful or not d叩ends， in part, on the purpose for
which it 1S used. For example, when 1 determine the shortest route from Florence
to Venice , 1 do not worry about the projection of the map 1 am using; 1 work under
the assumption that the earth is fla t. When 1 determine the shortest route from
Be中吨 to Havana, however, 1 pay close attention to the projection-I assumεthat
the earth is spherical. And were 1to climb thεMatterhorn 1 would assume that the
earth is neither flat nor spherical!
One reason for improving our understanding of the world is to enhance our
ability to mold it to our desires. The understanding that game-theoretic models
give is particularly relevant in the social, political, and economic arer飞as. Studyi我g
game-theoretic models (or othεr models that apply to human interaction) mayalso
suggest ways in which an individual's behavior may be modified to improve h时
own welfare. By analyzing the iηcentives faced by negotiators locked in battle, for
example, we may see the advantagεs and disadvantages of various strategies.
τhe models of gam台 theory are precise exprεssions of ideas 也at can be pre伽
sented verbally随 Verbal descriptions tend to be long and imprecise; in the interest of
conciseness and precision, I frequently employ mathematical symbols. Although
1 use the language of mathematics, 1 use few of its concepts; the ones 1 use are described in Chapter 17.1 aim to take advantage of the precision and conciseness of
a mathematical formulation without losing sight of the undεrlying ideas.
1.1 Whatis game theory?
3
JOHN VON NEUMANN
J000 von Neumann, the most important figurεin the
early development of game theory, was born in Budapest, Hungary，约 1903. 日e displayed exceptional
mathematical ability as a child (he had mastered calculus by the age of 8) , but his father, concerned about his
son's financial pro年时怡， did not want him to become a
mathεmatician. As a compromise he enrolled in mathematics at the University of Budapest in 1921 , but imm争
diately left to study chemistry, first at the Unîversity of
Berlin and subsεquently at the Swiss Federal Institute of
Technology in Zurich , from which he earned a degree in
chemical engineering in 1925. During his time in Ger而
many and Switzerland he returned to Budapest to write examinatio邸， and in 1926
obtained a Ph.D. in mathematics from the Uni飞!ersity of Budapest. He taugl飞t in
Berlin and Hamburg, and , from 1930 to 1933, at Princeton University. ln 1933 he
becamεthe youngest of the first six professors of the School of Matl飞ematics at the
lnstitute for Advanced Study in Princeton (Einstei口 was another).
VonNe飞lmam1' s first published scientifìc paper appeared in 1922, when he was
19 years old. ln 1928 he published a paper that establishes a key resu1t on strictly
competitive gam伍， a resu1t that had eluded Bore l. He made many major contrib法
tions in pure and applied mathematics and in physics-什enough， according to Hal伽
mos (1973) , "for about three ordinary careers , in pure mathematics alone". 队íhile
at the Institute for Advanced Study he collaborated with the Princeton economist
Oskar Morger飞stern in writing Theory of games and economic behavio了， the book that
establísh叫 gamεtheory as a field. 1n the 1940s he b心came ìncreasingly involved
in applìed work. In 1943 he became a consu1tant to the Manhattan Proje时， which
was developing an atomic bomb , and in 1944 he became involved with the development of the first electronic compute乙 to which he made major contributions.
He stayed at Princeton 山1til1954， when was appointεd to the US Atomic Energy
Commission. He died in 1957.
Game-theoretic modeling starts with an idea related to some aspect of the inter二
action of decisio扣makers. We express this idea precisel y in a model, incorporating
features of the situation that appear to be relevant.丁his step is an art.听'e wish to
put εnough ingredier飞ts into the model to obtaín nont主ivial insíghts, but not so
many that we are led into írrelevant complications; we wish to lay bare 吐lew飞der偏
lying structure of the situation as opposed to describing its every detaiL The r飞ext
step is to analyze the model to discover its ímplicatìons. At this stage we need to
adhere to the rígors of logic; we must not introduce extrar冶ous considerations abser飞t from the model. Our analysis may confirm our idea , or suggεst it is wrong. If
•
Chapter 1. Introduction
4
it is 飞!\Trong， the analysis should help us to understand why it is wrong. We may see
that an assumption is inappropriate, or that a口 important element is missing from
tl程 model; we may ∞口clude that our idea is invalid , or that we need to investigate
it furth在r by studying a different model.百1US， the interaction between our ideas
and models designed to shed light on them runs in two directions: the implications
of models help us determine whether our ideas ma立.e sense, and these ideas, in the
light of the implications of the models, may show Us how the assumpti∞s of our
models are inappropriat巳In either case, the process of formulating and anal严ing
a model should improve our understanding of the situation we are considering.
1.2τ!1 e theory of rational choice
The theory of rational choice is a component of many models in game theor予
Briefly, this theory is that a decision-maker chooses the best actíon according to
her preferences, among all the actions available to her. No qua 1itative restriction
is placed on the decision-maker's preferenc但; her "rationaHty" lies in the consistency of her dεcisions when faced with different sets of available actìons, not in the
nature of her likes and dislikes.
1.2. 1 Actions
τhe theory ìs based on a model with two components:
a set A consisting of all
the actions that, under some circumstances, are avaílable to the deεision-ma挝r，
and a specification of the decision-maker's preferences. In any given situation,
the decision-maker is faced with a subset1 of A , from which she must choose a
single elemen t. The decision-maker knows this subset of available choices , and
takes it as given; in particula 乙 the subset is not influenced by the decision-maker's
preferences. The set A could, for example, be the set of bundles of goods that
the decision-maker can possibly consume; given her income at any time , sh已 is
restricted to choose from the subset of A containing the bundles she can afford.
1.2.2
Preferences and payoff functions
As to preferences, we assume that the decision-make乙 when presented with any
pair of actions , knows which of the pair she prefers, or knows that she regards
both actions as equally desirable (in which case she is "indifferent between the
actions"). We assume further 出at these preferences are consistent in the sense that
if the decision栅maker prefers the action a to the action b, and the actíon b to the
actio口 c， then she prεfers the action a to the actíon c. No otheτrestriction is imposed
on p陀ferences. In particulaζwe allow a person's preferences to be altruistic in
the sense that how much she likes an outcome depends on some other person's
welfare. Theories that use the model of ratíonal choice a加l to derive implications
that do not depend on any qualitatíve characteristic of preferences.
J
See Chapter 17 for a description of mathematical terminology.
1.2 The th跑。ry of rational choice
5
How can we describe a decisiOI飞-maker's preferεnces? One way is to specify,
for each possìble pair of actions, the action the decisiOI干maker prefers, or to note
that the decision-mak盯 is indifferent between the actions. AJtematively we can
"represent" the preferences by a payoff function , which associates a number with
each action in such a way that actions with higher numbers are preferred. More
preciselμthe payoff function u represents a decision-maker's preferences it for
any actions a in A and b in A,
叭的>叫的 if and only if the decision-maker prefers a to b.
(5.1)
(A better name than payoff function might be "p陀fe陀nce indicator function".
1n economic theory a payoff function that represents a consumer's preferences is
often called a "utility function".)
EXAMPL巴 5.2 (Payoff function representing preferences)
A person is faced with
the choice of three vacation packages, to Havana , Paris, and Venice. She prefers
thεpackage to Havana to the other two , which she 陀gards as equivalent. Her
preferences bεtween the three packages ar.εrepresented by any payoff function
that assigns the samε number to Paris and Veni优 and a hi培骂h
怡
er num
引lbe
衍r 怡
t o I-日f妇
扣a
飞vana. For εxamp
抖le， we car口1 set 缸叫(Hava扫
na) = 1 and u(Pa
盯ris
吟) 口 lμI(Ve ηni文Cα
叫
叫j 工 0,
e
啊r 叫f江1:avana) 江 10 and u(Par垃is斗) 口 u(Ver时飞让i叫
O
cα
叫) 且 1, or 叫
e
u (江a飞vana
叫) 江 o a口
呻〉、
ar丘is苟) 二二 u(Venice) = … 2.
(j) EXERCISE 5.3 (A1truistic preferences) Person 1 cares about both h盯 income and
person 2's income. Precisely, the 飞ralue she attaches to each unit of her own income
is the same as the value she attaches to any two units of person 2's income. For
example , she is indifferent between a situation in which her income is 1 and person 2's is 0, and one in which her income is 0 and person 2、 is 2. How do her
prefere时es order the outcomes 口，的， (2 , 1) , and (3，时， where the first component
in each case is her income and the second component is person 2's ìncome? Give a
payoff function consistent with these preferences.
A àecisÌOIνmaker's preferences, in the sense used here , convey only ordinal
information.τhey may tell us that the decision-因此er prefers the action ({ to the
action b to the action c, for example , but they do not te Il us "how much" she prefers
a to b, or whether she prefers a to b "more" than she prefers b to c. Consequently
a payoff function that represents a decision-maker's preferences also conveys only
ordinal information. It may be tempting to think that the payoff numbers attached
to actions by a payoff function convey intensity of preference-…that if, for example,
a de杭ci必si始
on舶吁m飞ake
凹r's pr陀
efe
时rences are represent怡
ed by a payoff function μ for which1
叫(a
u
的) = O， 叫
u (b的)
1, a汩I叫 u(非仙
C叶) 口 10
∞)旧
O， 出
t始
h1冶e
臼
阳
阳I川 hede
仅ci必siωor仆
川
νH
ma
协
b
b飞时
1过t finds 恤
1i让tt由
l沁
ed
甜
ifference between a and 扛 A payofffunction contains no such information! The only condusion we ca刊raw from the fact thatμ(的= 0, u( 的 l ， and
u (c) = 100 is that the decision-maker prefers c to b to a. Her preferences are repre100， and
sented equally well by the payoff function v for which L巾) = 0, v(的
z牛) = 101 , for example, or any other function w for whichω (a) < 时 b) < 叫 c).
Chapter 1. Introduction
6
From this discussion we see that a decision-maker's preferences are represented
by rnany different payoff functìons. Looking at (5.1) , we see that if u represents a
decision-rnaker's preferences and the payoff function v assigns a higher munber
to the action a than to the action 告 if and only if the payoff function u does so,
then v also represents these preferences. Stated more cornpactly, if u represents a
decisio口-rnaker's preferences and v is another payoff function for which
v(a) > v(b) if an巾
then v also represents the decision-maker's preferences. Or, more succinctly, if u
represents a decision-maker's preferences, then any increasing function of u also
represents these preferences.
命 Ex照CISE 6.1 (Alter到ative reprεsentations of preference咛 Adecisionωmaker'spref的挝es over the set A
u(的 斗， 咐
{a , b, c} are represented by the payoff function u for w恒ch
= 1, and u(cμ 4. Are
陀 t问
怆h均飞随吨
e句y 巾 r即
e叩
pre叫ted
付
db
均
y t.加挝怆
efu
u瞅
曰
n1
which 纱tI(抖仅a叫例) 土 一 1 ， 叫
v (b的) 口 0, ar飞d v(c)
m 刨 z ω( 抖
Oandω (c) = 8?
2? How about the functionωfor which
Sometimes it is natural to formulate a rnodel in terms of preferences and then
find payoff functions that represent these preferences. In other cases it is natural
to start with payoff functions , even if the analys抬出pends only on the underlying
prefe纪时俑， not on the specific representatior飞 we choose.
1.2.3 The theory of rational choice
The thεory of rational choice rn在y be stated simply: in any given situation the
decision-maker chooses thernember of the available subset of A that is best according to her preferences. Allowing for the possibility that there are several equally
attractive best actions, the theoη， of rationaI choice is
the actíon chosen by a decisio作疗taker is at least as good , aà:ording to hcr
pr拉re町怒I as every other available action.
For any action , we can desi♂1 preferences with the property that no other ac位m
is preferred. Thus if we have no information about a decision-maker's preferences ,
and rnake no assump位。阳在bout their character, any single action is consistent with
the theory However, if we assume that a decision-m吕kεr who is indifferent between two actions sometimes choos部 one action and sometimes the other, not ev跚
ery collection of choices for diff，古ent sets of available actions is consistent wi由 the
theory. Suppose, for example, we observe 在at a decision-maker chooses a whenever she faces the set 初，时， but sometimes chooses b w边n facing the set {a , b, c}.
The fact that she always chooses a when faced with {a , b} rneans that she prefers
a to b (if she were indifferent, she would sometirnes choose b). But then when she
faces the set {a , b, c} , she must choose either a or c, never b. Thus her choìces are
‘
1.3 Coming attractions: interacting decision-makers
7
inconsister飞t with the theory.
(More concretely, if you choose the same dish from
the menu of your favorite bistro whenever t检re are no specials, then , regardless
of your prefe肥口ces， it is inconsistent for you to choose somεother item 卢om the
menu on a day when there is an off-menu specia l.)
If you have studied the standard economic theories of the consumer and the
firm, you have encountered the theory of rational choice before. In the economic
theory of the consumer, for example, the set of available actions is the set of all
bundles of goods that the consumer can afford. In the theory of the firm , the set of
available actions is tl飞e set of all input-output vectors, and the action a is preferred
抬出e action b if and only if a yields a higher profit than does b.
1.2.4
Discussion
The theory of rational choice is enormously successful; it is a component of countless models that el飞hance our understanding of social phenomena. It pervades
economic thωry to such an extent that arguments are classified as economic" as
much because they involvεrational choices as because they involve particularly
屯conomic" variables.
?、Jevertheless， under some circumstancεs its implications are at variance with
observations of human decision-making. To take a small example, adding an un
desirable action to a set of actions sometimes sig烈的cantly changes the action cho制
坐n (see Rabin 1998, 38). The sigr让豆cance of such discordance wìth the theory
depends upon the phenomenon being studied. If we are considering how the
markup of price over cost ìn an industry d叩ends on the number of firms , for
example, this sort of weakness in 也e theory may be unimportan t. But if we are
studying how advertising, designed speci自cally to influence peoples' preferences ,
affects consumers' choices, then the inadequacies of the model of rational choice
may be crudal
No general theory currently challenges the supremacy of rational choice theory.
But you should bear in mind as you read this book that the theory has its limits,
and some of the phenomena that you may think of explaining by using a game唰
theoretic model may lie beyond these limits. As always, the proof of the puddíng
ìs in the eating: íf a model er飞hances our understandìng of thεwor1d， then ìt serves
1ts purpose.
It
1.3
Cor￥，ing attractions: interacting decision-makers
In the model of 也e prevíous section, the dedsion-maker chooses an action from
a set A and cares on1y about this action. A dedsion-mak岱 in the world often
does not have the luxury of controlling all the variables that affect her. If some of
the variables that affect her are the actions of other decision附makers， her decisionmaking problem is aìtogether more challenging than that of an isolated dedsionmaker. The study of such situations, which we may model as games , occupies 也e
remainder of the book.
8
Chapter 1. Introduction
Consider, for example, firms cornpeting for bllsiness. Each 在rm controls its
price, bllt not the other firms' prices. Each firm cares, however, abollt all the firms'
prices , becallse these prices affect its sales. How ShOllld a fìrm choose its price in
these cìrcllmstances? Or consider a candidate for political office choosing a policy
platforrn. She is likely to care not only abollt her own platform, bllt those of her
rívals, which affect her chance of being elected. How ShOllld she choose her platform , k到 owing that every other candidate fò.ces the same problem? Or sllppose
yOll are negotiating a purchase. YOll care abollt the price, which depends on both
yOllr behavior and that of the seller. How ShOllld yOll decìde what price to offer?
Part 1 presents the rnain models of game theory: a strategic game, an extensive
game , and 没 coalitional game. These models díffer in two dimεnsions. A strategic
game and an 以tensive game fOCllS on the actions of individllals, whereas a coalitio口al game focllses on the Olltcomes that can be achieved by grollps of individ幡
II及ls; a strategic game and a coalitional game consider sitllations in which actions
are chosen once and for all, whεreas an 假如lS1飞'e game allows for the possibility
th放 plans may be revised as they are carried Ollt.
The model within which rational choice theory is cast is tailor-made for the
theory. If we want to devεlop another theory of a single decìsion-maker, we need
t。在 dd elements to the model in addítion to actions and preferences. Th e same is
not true of most models in game theory: strategic interaction is sllfficìently com搁
plex that even a relanvely simple model can admit more than one theo巧r of the
olltcome. 专Ve refer to a theory that specìfies a set of olltcomes for a model as a
"Sollltion". Chapter 2 describes the model of a strategic game and the Sollltion of
Nash 叫 llilibrillm for 四ch games. 在1e theory of Nash 吨llilibrillm in a strategic
game has been applied to a vast variety of sitllations; a few of the most significant
applicaticns are discllssed in Chapter 3.
Chapter 4 extends the notion of r、.Jash eqllilibrillm in a strategic game to allow
for the possibility that a decìsion-maker, when indifferent between actions, may
not always choose the same action.
The model of an extensive game , which adds a temporal dimension to the description of strategiζinteraction captured by a strat吨ic game, is studied in Chapters 5, 6, and 7. Part 1 concllldes with Ch apter 8, which discllsses the model
Notes
9
Notes
Von Neumann and Morgenstern (1944) established game theory as a field. The
information about John von l\ieumann 始 the box on page 3 is drawn from Ulam
(1958) , Halmos (1973) , Thompson (1987) , Poundstone (1992) , and Leonard (1995).
Aumann (1985) , on which 1 draw in the opening section, contains a very readable
discussion of the aims and achievements of game theor乒 τ'w o papers that discuss
the limitations of rational choice theory are 我abin (1998) and Elster (1998).
Games wÎth Perfect Information
2\lash"Equili 七 rium: Theory
13
3
剖 ash 王quìlibrium: IlI ustrations 55
4
Mixed 5trategy Equilibrium 99
5
Extensive Games with Perfect Information:
Theory 153
6
主xtensive Games with Perfect Information:
IIlustrations 181
7
Extensive Games with Perfect Informatìon:
Extensíons and Discussion 205
8ζoalitional Games and the ζore 239
Nash Equilibrium: Theory
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.1
Strategic games 13
Example: the Prisoner 's Dilemma 14
Example: Bach or St，阳 V阳ky? 18
Example: Matching Pennies 19
Exa.m ple: the Stag Hunt 20
Nash equilibrium 21
Examples of Nash equilibrium 26
Best response functions 35
Dominated actions 45
Equilibrium in a single population: symmetric games and
symmetric equilibria 50
Prerequisite: Chapter 1
Strategic games
STRATEGIC GAME is a mode1 of interacting decision-makers.
In recognition
of the interaction, we refer to the decision-makers as players. Each p1ayer
has a set of possible actio l1 s. The mode1 captures interaction between the p1ayers
by a110wing each p1ayer to be affected by the actions of all p1ayers , not on1y her
own action. 句
S pe
配
eα1
Cl凶
fic
臼a11
均
l协予 each 抖
p la
叮
ye
町r has 严
p r，吃h附
价
list of a11 the 抖
p 1a
叮
ye
町rs' actions. (See Section 17 .4, iE the mathematica1 appendix , for
a discussion of profi1es.)
More precise1y, a strategic game is defined as fo11ows. (ηle qualification "with
ordina1 preferences" distinguishes this notion of a strategic game from a more
genera1 notion studied in Chapter 4.)
DEFINITION 13.1 (Strategic ga111e with ordinal pr~作rences)
A strategic game (with
ordina1 preferences) consists of
• a set of players
• for each playe巳 a set of adions
• for each playe巳 preferences over the set of action profiles
A very wide range of situations may be modeled as strategic games. For example , the p1ayers may be 且rms， the actions prices , and the p配ferences a re f1ection of
the 且rms' pro且ts. Or the players may be candidates for political office, the actions
1 "J
Chapter 2. Nash Equilibrium: Theory
14
campaìgn expenditures , and the preferences a ri夺f1ection of the candìdates' probabilities of win时n各 Or the players may be anìmals fighting over some prey, the ac啕
tions concession times, and the prefereI飞ces a re f1 ection of whether an animal wins
or loses. In t刘白 chapter 1 descrìbe 50me simple games designed to capture fundamental con f1 icts present in a variety of situations. The next chapter is devoted to
rr飞ore detailed applications to speci在cphenomena
As in the model of rational choice by a single d时ision心laker (Sectio到1. 2)， ìt is
frequently convenient to specify the players' p陀ferences by giving payoffjU l1 ctions
that represent them. Suppose, for example , that a player prefers the action profile
fl to thεprofile b, and prefers b to c. We may specify these priε 坠 rences by assìgnìng
the payoffs 3 to fl , 2 to 0, and 1 to c. Or, altern挝ively， we may specify the preferences
by assigning thepayoffs 100 to a, 0 to b, and -2 to c. The two spεcifications are
equa11y good; in particula乙 the latter d优s 110t imply that the player's preferen优
between a and b ìs stronger than her preference between b and c. The point is that a
strategic game with ordinal preferences is defined by the players' preferences, not
by payoffs that represent these preferences.
丁imεis absent from the mode l. The id岱 is that each player chooses her action once and for a11 , and the players choose their actions "simultaneously" in the
sεnse that no player is informed , when she chooses her action , of the action chosen
by any other player. (For this reason, a strategic game is sometimes referred to
as a simultaneous-move game".) Nevertheles宫， an actìon may involve actìvities
that extend over time , and may take ìnto account an unlimited number of contingencies. An action might specify, for example , "if company X's stock faHs below
$10 , buy 100 shares; otherwis已 do not buy any shares". (For this reason, an action
is sometimes called a 飞 trategy".) H O\,vever, the fact that time is absent from the
model means that when analyzing a sìtuation as a strategic game, we abstract from
the complicatìons that rr飞ay arise if a player is allowed to change her plan as events
unfold: we assume that actions are chosen oncεand for a11.
If
2.2
应xample: the Prisoner's Dilemma
Its name
comes from a story involving suspects in a crime; its importance comes from the
huge vaI业ty of situations in which the participants face incentives simi!ar to those
faced by the suspects in tl陀 story.
01飞e of the most wε11-knm川ì strategic games is the Prisoner ￥ Dilemma.
EXA 剖 PLE 14 1 (Prisoncr's Díle l1lll1 fl) Two suspects in a major crìme are held in separate ce11s.τhere is enough e飞ridence to convict each of them of a minor offense ,
b时 noteno吨hevidεnce to convict either of them of the major crime unless one of
them acts as an informer against the other 币nk纱. Ifthey both stay quiet, each wìll
be convicted of the minor offense and spend one year in prison. If one and only
one of them finks , she w il\ be freed and used as a 飞4她lesS against the other, who
will spend four years in prison. If they both fink , each w i\l spend three years in
pnson.
,
2.2 Example: the Prisoner亏。ilemma
15
Thìs situation rnay be rnodeled as a strategic garne:
Players The two suspects.
Actions Each player's set of actìons is {Quiet， 时}
Preferences Suspect l's ordering of thεaction profiles, frorn best to worst, is
(Fì nk, Quiet) (she fìnks and suspect 2 rernaìns quì叽 so she is freed) , (Quiet ,
Quiet) (she gets one year in prison) , (Fink， Fir汰) (she gets thrεe years in
prison) , (Quiet，Fin主) (she gets four years in prison). Suspect 2's ordering is
(Quiet , Fink) , (Quiet , Quiet) , (Fì nk,Fink) , (Fí吮 QL时t).
We can represent the garne cornpactly in a table. First choose payoff functions
that represent the suspects' preferencεorderings. For suspect 1 we need a function
u} for whìch
Ul( Fí llk, Quiet) > u1(Quiet, Quiet) > L叭 Fink， fïn号 >u 迁台uiet， Fí nk).
A sirnple specification is !l} (Fí nk, Quiet) = 3, U1 (号uiet， Quiet) 2,!l 1( Fí nk, Fínk) =
1, and 问 (♀ uiet，Fil1k)
O. For s出pe仅ct
创 2 wε Cωar口飞 5豆i削larly ch飞O
∞
ose the function
U2 for 认w斗lhì
必创
甘ìch 1时 Ql肌
时
{α
iet， 旦
Fm
时
均) = 3, 衍
k
lt勺2(QU归t， Qr时t斗) = 2, U2丘(σ
山
旦i打I时k， 扫
F
F 11η伙
1k
均) 口 1, an
μ 2(Fiηn欢k， Qui花εd
叫
t斗
O. Usìng these representations, the garne is illustrated ín Figure 15 .1. In this figure the two rows correspond to the 机矿o possible actions of
player 1, tbe two colurnns correspond to the two possible actions of plaYE:r 2, and
the nurnbers in each box are the playεrs' payoffs to the action profile to which the
box corrξsponds， with player 1's payoff listed 自rst.
Suspect 2
Quiet Fink
suspectlQuiet21
Fink
3, 0
43
1, 1
Figure 15.1 The Prísoller's 0於mmn (Example 14.1)
The Prison衍'亏 Oìlemma models a situatior飞 in which there are gains frorn cooperation (each player prefers that both players choose Quiet than they both choose
Fink) but eaε且 player has an Ìncentive to "free ride" (choose Fíl1 k) what机'er the
other player does. The garne is irnportant not because we are interested in understanding the incentíves for prisoners to confess, but because rnany other situations
have sìrnilar structures. Whenever each of two players has two action日， say C
(corresponding to Quiet) and 0 (corresponding to Fínk) , player 1 prefers (0工) to
(C, C) to (0 0) to (C 0) , ar吐 player 2 prefers (C, 0) to (C,C) to (0 , 0) to (0 , C),
the Prisoner's Oilemma models the situation that the players face. Sorne exarnples
follow.
I
2.2.1
I
Working on a joint project
You are working with a friend on a joint projec t. Each of you can either work hard
or goof off. If your friend works hard, thεn you prefer to goof off (the outcorne of
Chapter 2. Na嚣h Equilibrium: Theory
16
the project would be better if you worked hard too, but the increment in its value
to you is not worth the extra effort). You prefer the outcome of your both working
hard to the outcome of your both goofing off (in which case nothing gets accon卜
plished) , and the worst outcome for you is that you work hard and your friend
goofs off (you hate to be "exploited"). If your friend has the same preferences,
then the game that models the situation you face is given in Figure 16.1, which, as
you can see, differs from the Prisoner's Dilemma only in the names of the actions.
Work hard
Goofoff
Work hard
Goofoff
2, 2
。， 3
3， 。
t1
Figure 16.1 Working 0盯在 loint project
1 am 110t cla江ning that a sìtuation in which two people pursue a joi时 project
necessaríly has the structure of the PrÎSOl1 er公 Dilemma， only that the players' pref伊
erences in such a situation may be the s没me as in the Prisoncr's Dilemmi军If， for
example， εach person prefers to work hard than to goof off when the other person
works hard , then the Pl台oner's Dilemma does 110t model the situation: the players'
preferences are dìfferent from those given in Figure 16.1
ø EXERCISE 16.1 (Workìng on a joint project) Formulate a strat吃ic game that models
a situation in which two people work on a joint project in the casεthat their pref
erences aDεthe same as those in the game in Figu陀 16 .1 except thatεach person
prefers to work hard than to goof off when the other person "works hard. Present
your gamε 扫1 a table like the one in Figure 16 .1.
2.2.2 Duopoly
ln a simple model of a duopol如 two firms produce the same good , for whích each
finn charges eíther a low price or a high price垂 Each fìrm wants to acmeve the
highest possible profit. If both firms choose High , then each earns a profit of $1000.
If one firm chooses High and the other chooses Loz/), then the firm choosing HiglI
obtains no customers and makes a loss of $200, wl飞ereas the firm choosing Low
earns a profit of $1200 (its unit profit is low, but íts volume is higl斗. If both firms
choose Low, then each earns a profit of $600. Each firm cares only abo飞lt Íts profit,
so we can represent its preferences by the profìt it obtains, yielding the game in
Fi gure 16.2、
High
H拉h
1000 , 1000
Low
1200， 一 200
Lo w
… 200, 1200
600, 600
Figure 16.2 A simple model of a price-优 tting duopol予
2.2 Example: the Prisoner亏。ilemma
17
Bearing in mind that what matters are the players' preferences, not the particular payoff functions that we use to represent them , we seεthat this game, like the
prevíous one, differs from the Prisoner's Dilemma only in the names of the actions.
Th e action High plays the τole of Quiet , ilnd the action Low plays the role of Fink;
firm 1 prefers (Low , H怡) to (High , High) to (Low , Low) to (H萨， Lozu) , and firm 2
prefers (High , Low) t。但拉h， High) to (Low , Low) to (Lo队 H兹的，
Q) EXERCISE 17 .1 (Games equivalent to the Prisoner's Dìlemma) Det但mine whether
each of the games in Figure 17 .1 differs from the Prìsoner's Dilemma only in the
names of the players' actions , or whether it differs also in one or both of the players'
preferences.
X
Y
X
3, 3
5, 1
Y
1, 5
X
。， 0
Y
X
2, 1
3,… 2
Y
。， 5
t
Figure 17.1 The strategic games for Exercise 17.1
人们n the previous example, 1 do not claim that the incentives in a duopoly are
l1 ecessarily those ìn the Prisoηer兰 Dilemma; differer飞t assumptions about the relatìve
sizes of the profìts in the four cases generate a different game. Furth毗 in this case
one of the abstractions incorporated into the model-that 位ch firm has only two
prices to choose bet飞veen-may not be harmless. If the firms may choose among
many prices , then the structure of the interaction may change. (A richer model is
studied in Section 3.2.)
2.2.3 The arms race
Under some assumptions about the countries' preferences, an arms race can be
modeled as the Prìsonel专 Dilemma. (The Prisoner's Dilemma was first studied in
the early 1950s, when the United States and the Soviet Union were involved in a
nuclear arms ra柜， so you rnight suspect that U.s. nuclear strategy was influenced
by game theo巧l; the evidεnce suggests that it was no t.) Assume thatεach country
can build an arsenal of nuclear bombs, or can refrain from doing so. Assume also
that each country's favorite outcome is that ít has bombs and the other conntry
does not; the next best outcome is that neìther country has any bombs; the next
best outcome is that both countries have bombs (what matters is relatìve strength,
and bombs are costly to build); and the worst outcome is that only the other country has bombs. 1n thi:弓 case the situation Îs modeled by the Prísoner's Dilemma , in
which the action Don't build bombs correspor飞ds to Quiet in Figure 15 .1 and the action Build bombs corresponds to Fi l1 k. However, once agaìn the assumptions about
pref在εnces necessary for the Prisoner's Dilemmαto model the situation may not be
satisfied: a country may prefer not to build bombs if the other country does not,
for example (b omb-building may be very costly)，如 which case the sìtuation is
modeled by a different game.
Chapter 2. Nash Equilibrium: Theory
18
2.2.4 Common property
fwo farmers are deciding how much to allow their sheep to graze on the village
con飞mon‘ Each farn飞ξr prefers that her sheep graze a lot rather than a little , re而
gardless of the other farmer's action, but prefers that both sets of sheep graze a
little rather than a lot (in which case thεcommon is ruined for future us的. Under
these assumptions the game is the Prisoner's Dilemma. (A richer model Îs studiεd
in Section 3.1 .5.)
2立5
Other situations modefed as the Pri50ner主 Difemma
A huge number of other situations have been modeled as the Priso /1 er's Dilemma ,
fromm在 ting hermaphroditic fish to tariff wars b时ween countries.
? EXERCISE 18 .1 (Hermaphroditic fish) M校mbers of some species of hermaphroditic
fish choose , in each mating encounter, whether to play the role of a male or a
femalε. Each fish has a preferred role , which uses up fewer resources and 1飞ence
allows more future mating. A 在sh obtains a payoff of H if it mates in its preferred
role and ζif it mates in the other role , \年hεre H > L. (Payoffs are measured in
terms of numb盯 of offspring, which fish are evol飞，ed to maximize.) Consìder an
encounter between two fish whose preferred roles are the same. Each fish has two
possible actions: mate in 的ther role or insist on its p陀 ferred role. If both fish offeτ
to mate in either role , the 古rol讼e岱s are assi毕gned 照
rζan
双ndom
丑11协
予; and each 负
y
f臼
is5由
h's pa
咛
yof丘f 必
i5
i(H 十 L引) (杠the 附Y臼age
吨们川叫
of H and ιL). 旦
1 f 巳ea阳C创Îs由
h insi
刽1S纣ts (ο
创∞)汀
m
川
l
do not m
丑飞at忧屹
吃; 础
e
e注 ch 伊
goe
岱s off in 时
s 已a
盯rch of another partner, and obtains the payoff 5.
The higher the chancεofm使 ting another partner, the larger is S. Formulate this
situation as a strategic game and determine the range of values of 5 , for any given
values of H and L, for \.vhich the game differs from the Pl' isoner's Dile111 11la oTÙy in
the n刑nes of the actions.
2.3ξxample: Bach or Stravinsky?
ln the Prisollεr's Dllemma the main Îssue is whεther the players will cooperate
(choose ♀uiet). 1n the following game the players agree that it is better to cooperate
than not to cooperate, but they disagree about the best outcome.
EXAMPLE 18.2 (Bach 0 1' 5travinsky?) Two pεople wish to go out togεther. Twoco阶
certs are available: one of music by Bach, al飞d one of music by Stravinsky. One person prefers Bach and the other prefers Stravinsky. If they go to different concerts,
each of them is 叫ually ur让lappy listening to the music of either composer.
We may model this situation as the two-player strategic game ìn Figure 19. 1,
in which the p在son who prefers Bach chooses a row and the person '^血 o prefers
Stravinsky chooses a column.
This game is also referred to as the "Battle of the Sexes" (though the conflict
it models surely occurs no more frequ在ltly between p芒。ple of the opposite sex
2 .4 Example: Matching Pennies
19
8ach
Stravinsk:y
Figur.程 19.1
8ach
Stravinsky
2, 1
0, 0
0, 0
1, 2
Bach or Stravil1沟户 (80S) (Example 18.2)
than it does b供weεn people of thεsame sex}. 1 call the game 80S , an acronym
that fíts both names. (l assume that each player is indifferent between listening
to Bach and listening to Stra飞rinsky when she is alone only for consistency with
the standard specifícati∞ of the game. As we shall see, the analysis of the game
remains the same in the absence of this indifference.)
Like the Prisoner's Dilemma, 80S models a widè variety of situations. Consider,
for example, two officials of a political 严 rty deciding the stand to take on an issue.
Suppose that they disagree about the best stand , but are both better off if they
take the same stand than if they take different stands; the cases in which they
take different stands, leading voters to be confused , are equally bad. Then 80S
captures the situation they face. Or consider two merging firms that currently use
different computer technologies. As two divisions of 法 single firm th咛 will both be
better off if they both use the same technology; each firm prefers that the common
technology be the one it used in the pas t. 80S models the choices the firms face.
2.4
Example: Matching Pennies
Aspects of both conflict and cooperation are present in both thε Prisol1er's Dilemma
and 80S. 丁he next game is purely conflictuaL
EXAMPLE 19.1 (Matching Pennics) Two people choose , simultaneousl予 whether
to show the head or the tail of a coin. If they show the same 革idε， perso口 2 pays
person 1 a dollar; if they show different sides, person 1 pays person 2 a dollar. Each
person cares only about the amount of money she receives, and (ηaturally!) prefers
to recei飞'e more than less. A strategic game that müdels this situation is shown
in Figure 19.2. (In this representation of the players' preferences, the payoffs arε
吨ual to the amounts of money involved. 科e could equallywell work with another
representation-for example, 2 could replace each 1, and 1 could replace each -1.)
ln this game the players' interests are diametrìcally opposed (such a game is
called μstrictly competitive"): player 1 wants to take the same actiün as the other
playe乙 whereas player 2 wants to takεthe opposite action.
Head
Head. 1, … 1
Tail -1, 1
Figu 陀 19.2
苟延
-1 , 1
1, -1
Matching Pennies (芜湖mple 19.1)
Chapter 2. Nash Equilibrium: Theory
20
This game may, for example, model the choices of appearances for new products by an established producer and a new firm in a market of fixed size. Suppose
that each firm can choose one of two different appearances for the product. The
established producer prefers the newcomer's product to look different from its
own (so that its customers will not be tempted to buy the r飞机矿comer's product) ,
whereas the newcomer prefers that the products look alike. Or the game could
model a relationship betweεn two people in which one person wants to be like the
oti飞机" whereas the oth仔 wants to be differe时，
。 EXERCISE 20 .1 (Games without conflict) Give some examples of hvo-player strate-
gic games in which each player has two actions and the players have the same pr砰
仅ences， so that there is no con f1i ct between their interests. (Present your games as
tables like the one in Figure 19.2.)
2.5
Example: the 5tag Hunt
A sentence in Discourse on the origin and foundatio l1 s of inequality among men (1755)
by the philosopher 如an-Jacques Rousseau discusses a group of hunters who wish
to catch a stag. (See Rousseau 1988, 36.) Th ey will succeed if they a11 remain
sufficientìy at忧ntive， but each is tempted to d部仔t her post and catch a hare. One
interpretation of the sentence is that the interaction between the hunters may be
modeled as the following strategic game.
EXAMPLE 20.2 (Stag Hunt) Each of a group of hunters has two options: she may
remain attentive to the pursuit of a stag, or she may catch a hare. If a11 hunters
pursue the stag, they catch it and share it equally; if any hunter绸 devotes hεr energy
to catching a hare, the stagεscapes， and the hare belongs to the defecting hunter
alone. Each hunter pref，εrs a share of the stag to a hare.
τhe stra tegic garr飞e that corresponds to this specification ìs:
Ployers The hunters.
ActÎo盯 Each player's set of actìons ìs {Stag , Hare}.
Preferences For each playe乙出e action profile 如 which a11 players choose Stag
(resulting ìn her obtaini吨 a share of thεstag) is ranked highest, followed
by any profile in which she chooses Hare (resulting in her obtaining a har的，
followed by any profile in which she chooses Stag and one or mo陀 of the
other players chooses Hare (resulting in her leaving empty-handed).
Li ke other games with many players , this game cannot easily be presented in a
table lìke that in Figurε19.2. For the case in which there are two hunter瓦 the game
is showr飞 in the left panel of Figure 21. 1.
The variant of 甘孜以To-player Stag Hunt shown in the right panel of Figure 21.1
has been suggested as an alternative to the Prísoner's Dílemma as a model of an arms
race, or, more generally, of the security dilemma faced by a paír of countries.τhe
game differs from the Prísoner ￥ Dílemma ín that a country pr!εfers the outcome in
If
If
2.6 Nash equilibrium
21
5tag
5tag
Hare
2, 2
1, 0
Hare
0, 1
只φη m
1, 1
Arm
只吃fraín
Ann
3, 3
2, 0
。， 2
1, 1
Figure21.1 Leftpanel:TheStagHlIl1 t(主xample 20.2) for the cas号 of two hunters. Right paηel: A variant
()f the tw• player Stag HU l1 t that models the 飞ecurity dilemma 气
which both countries refrain from arming themselves to the one in which it alone
arms itself: the cost of arming outweighs the benefit if the other country does not
arm itself.
2;6
Nash equilibr.i um
Wh at actions w i1l be chosen by the players in a strategic game? We wish to assume ,
as in the theory of a ra tional decisio汗maker (5ection 1.2) , that each player choosεs
the bεst available action. 1n a game, the best action for any given player depends ,
in genera l, on the other players' actions朱 50 wher飞 choosíng an action a player must
have in mind the actíons the other players wi1l choose. That is, she must form a
beli吃f about the other players' actions.
On what basis can such a belief be formed? Th e assumption underlying the
analysis in this chapte玄 and the next two chapters is that each play旧 's belief is
deríved from her past experience playing the game, and that this expεrience is sufficiently extensive that she knows how her opponents will behave. ~o one tells hεr
the actions her opponents will choose, but her previous involvement in the game
leads her to be sure of thεse actions. (The questio只 of how a player's experience can
lead her to the correct beHefs about the other players' actions is addressed brìefly
ìn 5ection 4.9.)
Although we assume that each player has experiencεplaying the game, we
assume that she views each play of the game in isolation. 5he does not become
fami 1iar with the behavior of specì在c opponents and consequently does not condition her action on the opponent she faces; nor does she expect her currcnt action to
affect the other players' future behavio r.
It is helpful to think of the following idealized circumstances. For each player in
the game there is a population of many decision-makers who may, on any occasion,
take that player's role. In each play of the game, players are selected randomly, one
from each population. Thus each player engages ín the game repeatedly, against
ever-varying opponents. H盯 experience leads her to beliefs about the actions of
"typical" opponents, not any specìfic set of opponents.
As an example, think of the repeated interaction ofbuyers and sellers. To a first
approximation, many of the pairings may be modeled as random; in many cases a
buyer transacts only once with any given seller, or interacts anonymously (when
the seller is a large store , for example).
In summary, the solution theory we study has two components. First, each
pl巧er chooses her action according to the model of rational choíce , given her be-
Chapter 2. Nash Equilibrium:τheory
22
lief about the other players' actions. Second, every player's belief about the other
players' actions is correc t. These two components are embodied in the follo认ring
definition.
A Nash equilibrílll17 is an action profile 矿 with the property that no
play挝 i can do better by choosing an action different from a7 , gi ve口
削 every other playεr j adheres to aJ
In the idealized setting in which the players in any 飞fen play of the game are
dra飞IVn randomly from a collection of poplllatìo邸， a Nash equilibrium corresponds
to a steady state. 1f, whenever thεgame is played, the action profile is the samεNash
equilibrium a* , then no player has a reaSon to choose any action different from her
component (\f 旷; t险re is no pressure on the action profile to change. Expressed
differen t1 y, a Nash 叫uilibrium embodies a stable 飞。cial norm": if everyone else
adheres to it, no individual wishes to deviate from it.
The second component of the theory of Nash 叫uilibrium→that the players'
beliefs about each other's actior飞s are corrεct-implies， in particular, that two play翩
ers' beliefs about a third play时 's action are the same. For this reason, the condition Ìs sometimes referred to as the requirement that the players' "expectations are
coordinated气
The situations to which we wish to apply the theory of Nash equilibrium do
not in general correspond 巳xactly to the idealized setting described above. For
example, in some cases the players do not have much experience with the game;
in others they do not view each play of the game in isolation. Whether the no
tion of Nash 叫uilibrium is appropriate 如 any given situation is a matter of judgmen仁 In some cases , a poor fit with the idealìzed sεttìng may be mitigated by
other considerations. For example, inexperienced players may be able to draw
conclllsions about their opponents' likely actions from theirεxperience in other
sÌtu挝ions ， or from other sources. (One aspect of such reasoning is discussed in the
box 0口 page 32). Ultimately, the test of the appropriateness of the notion of Nash
eqllilibrium Ìs whether it givεs us insights into the problem at hand.
Wìth thεaid of an additional piece of notation, we can state the definition of
a Nash 心quilibrium precísely. Let a be an action profile , in which the action of
each player i is aí. Let a; be any action of player i (either equal 始。 Ít or differe时
from it). Tl飞en 时 ， a -Î) denotes the action profile in which every player j except
i chooses her action aj as specifiεd by a, whereas playεr i chooses a;. (The-Ì
subscript on a sta础 for"机cepti".)τhat is, (a;月一 í) is the actio叩时ile in which
a11 the players other than i adhere to while i "deviates" to
a
a:.
2.6 槌ash equilibrium
23
]OHN F. NASH , ]R.
A few of the ìdeas John F. Nash，并， developed while he
was a graduate student at Prìnceton from 1948 to 1950
transfonl1ed game theory. 到ash was bom ìn 1928 in
Bluefield , West Virginia , where he grew 议P 日e was an
undεrgraduate mathematics major at Camegie Institutε
of Technology from 1945 to 1948. In 1948 he obtained
both a B.s. and an M.S. , ar飞d began graduate work in
the Department of Mathematics at Princeton University.
(One of his letters of re∞照mendation， from a professor at Camegìe Institute of τeChnology，以ras a single sen您只ce: "This man is a genius" (Kuhn et a1. 1995 , 282).) A
paper containing the main result of hìs thesis was submitted to the Proceedillgs of the Nationaï Academy of Scíe l1 ces ìn November 1949, fo盯m
teen months after h己 started his graduate work. (" A fìne goal to set ... graduate
students'二 to quote Harolò Kuhn! (See Kuhn et a1. 1995 , 282.)) He completed his
Ph.D. the following year, graduating on his twenty搁 second bírthda y. His thesis
(Nash 1950剖， 28 pages in length, íntroduces the equi1ibrium notion now known
as "Nash equilibrium" and dεlineates a class of strategic gamεs that have 时ash
叫uilibria (Proposition 119.1 in this bool斗. The notion of 时剖h equilibrium vastly
expanded the scope of game theory, which had previously focused on two-player
飞trictly competitive" games (in which the players' Ìnterests are directl丁y opposed).
Wh ile a graduate student at Prìnceton , Nash also wrote the seminal paper ìn bargaining theor予 Nash (1950c) (the ideas of which originated in an elective class in
intemational economics he took as a1飞 undergraduate) He 飞电rent on to take an academic position in the Department of Mathematics at 汉汀~ where he produεed "a
remarkable series of papers" (Milnor 1995, 15); he has been described as one of
the most original mathematical rninds of [the twentieth] century" (Kuhn 1996) He
shared the 1994 Nobel Prize in Economic Sciencεs with the game theorists John C.
Harsanyi and Reinhard Selten.
,
11
,
5日INITION 23.1 (Nash equilibrium of strategic game with ordinal prCj缸'ences) Th e
action profile 旷如 a strategic game with ordinal preferences is a Nash equ i1ibrium
i( for every player i and every action ai of player Î, a* Ìs at least as good according
to player i' s preferences as thεaction profile (a /t a:J in wruch player i chooses ai
whHe every other player j chooses aj. Equivalently, for every player i ,
Ui(旷)主 Ui(ait a~J for every action aí of player i,
(23.2)
where ui Îs a pay。在 function that reprεsents player i's preferences.
Trus definition implies neither that a strategic game necessarily ha日 a Nash
Examples in the next section show that
叫uilibriurn， nor that it has at most one ,
24
Chapter 2. Nash Equilibrium: Theory
some games have a single Nash equilibrium, some possess no Nash equilibrium,
and others have many Nash equilibria
The definition of a Nash equilibrium is designed to model a steady state among
experienced players. An alternative approach to understanding players' actions in
strategic games assumes that the players know each others' preferences , and considers what each player can deduce about the other players' actions from their
rationality and their knowledge of each other's rationalit y. This approach is sludied in Chapter 12. For many games , it leads to a conclusion different from that of
Nash equilibrium. For games in which the conclusion is the same , the approach
offers us an alternative interpretation of a Nash equilibrium , as the outcome of rational calculations by players who do not necessarily have any experience playing
the game
STUDYING NASH EQUILIBRIUM EXPERIMENTALLY
The theory of strategic games lends itself to experimental study: arranging for subjects to play games and observing their choices is relatively straightforward. A few
years after game theory was launched by von Neumann and Morgenstern's (1944)
book, reports of laboratory experiments began to appear. Subsequently a huge
number of experiments have been conducted , illuminating many issues relevant
to the theor y. 1 discuss selected experimental evidence throughout the book.
The theory of N ash eq山librium， as we have seen, has two components: the
players act in accordance with the theory of rational choice, given their be !i efs
about the other players' actions , and these be !i efs are correc t. If every subject
understands the game she is playing and faces incentives that correspond to the
preferences of the player whose role she is taking , then a divergence between the
observed outcome and a Nash equilibrium can be blamed on a failure of one or
both of these two components. Experimental evidence has the potential of indicating the types of games for which the theory works well and , for those in which
the theory does not work well , of pointing to the faulty compo口ent and giving us
hints about the characteristics of a better theor y. In designing an experiment that
cleanly tests the theory, however, we need to confront several issues.
The model of rational choice takes preferences as given. Th us to test the theory
of Nash equilibrium experimentally, we need to ensure that each subject's prefer
ences are those of the player whose role she 1S taking in the game we are examining. The standard way of inducing the appropriate preferences is to pay each
subject an amount of money direc t1 y related to the payoff given by a payoff function that represents the preferences of the player whose role the subject is taking.
Such remuneration works if each subject likes money and cares only about the
amount of money she receives , ignoring the amounts received by her opponents.
The assumption that people like receiving rnoney is reasonable in many cultures ,
but the assumption that people care only about their own monetary rewardsare "selfish"-may, in some contexts at least , not be reasonable. Unless we check
2.6 沁ash equilibrium
25
whether our subjects are selfish in the context of ourεxperiment， we wíll jointly test
two hypotheses: that humans are selfi由-a hypothesis not part of 伊me theory…
and that the notion of Nash equilibrium models their behavior. In some cases we
may ìndeed wish to test these hypotheses jointly. But to test the theory of Nash
号quilibrium alone we need to ensu陀 that we inducεthe preferences we wish to
study.
Assuming that better decisions require more effort , w巳 nεed also to ensure
that each subject fiηds it worthwhile to put in the extra effort required to obtain a
higher payoff. If we rely on monetary payments to pro飞ride incentìves, the amount
of 红lOneya 如同ect can obtain must be sufficiently sensitive to the qωlity of her decisions to compensate her for the effort she expends (paying a flat fee , for example,
is inapproprìate). 1n somεcases， monetary payments may not b巳 necessary: under
some drcumstanc时， subjects drawn from a híghly competitive culture líke that of
the United States may be sufficìently motivated by thεpossibility of obtaining a
high score, e':en if that score does not translate ínto a monetary payoff.
The notion of Nash e气功librium models action profiles compatible with steady
states. Thus to study the theory experìmentally we need to collect observations of
subjects' behavior 'when they have expεrience playing the 伊 m巳 But they should
not have obtained that experience while knowingly facing the same oppone口ts
repeatedly, for the theory assumes that the players consider each play of the game
in isolation , not as part of an ongoìng relationship. One option is to have each
subject play the game against many different opponents, gaining experience about
how the other subjects on average play the game, but not about the choices of any
other gìven player. Another option is to describe the game in terms that relate to
a situation in which the su悻cts already have experience. A dìf主culty with this
second approach is th挑出e description we give may connote more than simply
the payoff numbers of our game. If we descrìbe tl飞e PI台oner's Dilemma ìn terms
of cooperatio口 on a joint PI吗ect， for example, a subject may be biased toward
choosing the actíon she has fOlll飞d appropriate when involved in joint projects,
even ìf the structures of those ìnteractìons were sìgni自cantly different from that of
the Prisoner's Dílemma As she plays the experimental game repeatedly she may
come to app陀cìateh
‘
26
Chapt窑r2. Nash 在quilibrium: 洒在ory
one with the generality of Nash equilibrium. Statistical tests can som位imes aid in
dεciding whether the data are consistent with the theory, though ultimately we re-
main the judge of whether our observations persuade us that the theory e时lancεs
our understanding of human behavior in the game.
2.7
在xamples of Na军h equilibrium
2.7. 1 Prisoner主 Dilemma
By examining the four possìble pairs of actions in the Prisoner's Dilemma (reproduced in Figure 26. 巧， we see that (Fink ,F ink) is the uniq股民 ash equilibrium.
让
31
,,,,
阳门均L
，俨在
'双 20
ω 日
'un
tk
923
4
Fìgure 26.1 The Prisoner's Dilem l1l a.
The action pair (Fù仗， Fink) 始 a Nash 吨uilíbrium because 的 gi飞'en that player 2
chooses Fink , player 1 is better off choosing Fink than Quiet (looking at the right
column of the table we see that FÎnk yields player 1 a payoff of 1 whereas Quiet
yields her a payoff of 台)， and (ii) given that player 1 chooses Fink, player 2 is better
off choosing Fink than Quiet (looking at the bottom row of the table we see that
Fink yields player 2 a payoff of 1 whereas Quiet yields her a payoff of 0).
No other action profile is a N部h equilibríum:
• (Quiet , Quie t) does not satisfy (23.2) because when player 2 chooses Quiet ,
play衍 1's payoff to Fink exceeds her payoff to Quiet (look 挝 the first componer飞ts of the entries in the left column of the table). (Further, wheηplayer 1
chooses Quiet , player 2's payoff to Fink exceeds her payoff to Quiet: player 2,
as well as play时 1 ， wants to deviate. To show that a pair of actions is not a
Nash equilibrium, however, it is not necessary to study player 2's decision
onceweha飞re established that player 1 wants to deviate: it is enough to show
that one player wishes to deviate to show that a pa iJτof actior飞s is r飞ot a Nash
叫uílibrium.)
• (Fink, Quiet) d。但 not satisfy (23 勾 beωuse when player 1 chooses Fi nk,
player 2's payoff to Fink exceeds her payoff tò Quiet (look at the second
components of the entries ín the bottom row of the table).
• (Quiet , Fink) does not satisfy (23.2) beεa时e when pla}础 2 chooses Fin主，
player 1'5 payoff to Fink εxceeds her payoff to Quiet (l ook at the first com懦
ponents of the entries in the rig乞tcolumn of 在εtable).
2.7 Examples of Nash equilibrium
27
In surnrnary, in the only Nash equilibriurn of the Prísoner's Dilemma both players choosε Fink. In particula 毛 the incentive to free ride elirninates the possibility
that the rnutually desirable outcorne (Quiet , Quiet) occurs. In the other situations
discussed in Section 2.2 that may be modeled as the Prisoner 亏 Dilemma， the outcornes predicted by the notion of Nash e弓uilibriur我 are thus as follows: both people goof off when working on a joint project; both duopolists charge a low price;
both countries build bombs; both farmers graze their sheep a lo t. (The overgrazing
of a cornmon thus predicted is sometirnes called the "tragedy of the commons".
The intuition that sorne of these disrnal outcornes rnay be avoided if the sarne pair
of people play thεgarne repeatedly is explored in Chapter 14.)
In the Pr也one了 's Dilemma , the Nash equilibrium action of each player (Fi nk) is
the best action for each player not only if the ot怡r player chooses her equilibriurn action (Fink) , but also if she chooses her other action (Quiet). The actìon pair
(Fínk , Fink) is a Nash 叫uilibriurn because if a player believes that her opponent
will choose Fink , then it is optirnal for her to choose Fink. But in fact it is optirnal for a player to choose Fink regardless of the action she expects her opponent to
choose. In rnost of thεgam硝 we study, a player's Nash equilibriurn action does
not satisfy this conditìon: the action is optimal ìf the other players choose their
Nash 叫uilibriurn actions, but some other action is optirnal if the oth时 play仓rs
choose 到onequilibriurn actions.
? Ex 巳议CISE 27.1 (Variant of Prísoner's Dilemma with altruistic preferences) Each of
two players has two possible actions , Quiet and Fi nk; each action pair results in
the players' receiving arnounts of money equal to the nurnbers corresponding to
that action pair in Figure 坏1. (For exarnple , if player 1 chooses Quiet and player 2
chooses Fink, then play心r 1 receives nothing, whereas player 2 陀cei飞'es $3.) The
players arεnot "selfish"; rather, the preferences of each player i are represented by
the payoff fu配tion mi 伊)今夜 7时吟 where mi 怡) is the arnount of rnoney receivεd
by player i when the action profile is a, j is the other playeζand 武 is a givεnnon­
negativeηurnber. Player l' s payoff to the action pair ( Qu 吼 Quìet) ， for example, is
2 十 2队
a. Forrnulate a strategic garne that models this situation in the case 次1. Is this
garne the Priso l1el飞 Dilemma?
b. Find the range of values of 夜 for 飞4挝ch the 陀sulting game is the PrÎsone山
Dilel111刀a. For values of αfoτwhich the game is not the Prisoner!s Dilemma ,
find the Nash equilibria.
'3., EXE 民ClSE 27.2 (Selfish and altruistic social behavior)
Two people enter a bus. Two
a句acent crarnped seats are free. Each perso川nust decide whether to sit or stand
Sitting alone is rnore comfortable than sitting next 抽出e other person, which is
rnore cornfortable than standing.
。.
Suppose that each person cares only about her own comfort. Model the si始­
ation as a strategic garne. Is 也is garne the Prisoner 孟 Dilemma7 Find its Nash
equilibrium (equilibria 7).
28
Chapter 2. Nash Equilibrium:ηleory
b. Suppose that each p时son is altruistic , ranking thεoutcomes according to the
otl1 er perso孔's comfort, but, out of politeness, prefers to stand than to sit ìf the
other person stands. Model the situation as a strategìc game. Is tl让s game the
Pl' isoner's Oilemma? Find its Nash equilibrium (叫uilibria?)
c.
Compar.εthe people's comfort in the equilibria of thεtwo games.
EXPERIMENTAL EV !D ENCE ON THE Prisoner's Dílemma
The 1勺'isomr's Dilemma has attracted a great deal of attention by economis毡， psychologists , sociologists, and biologísts. A huge number of experir.1 ents have been
conductεd with the aím of discovering how people behave when playing the game.
Almost a11 these experiments involve each subíεct's playing the game repeatedly
agaínst an unchanging opponent, a situation that calls for an analysis signìfìcan t1 y
different from the one 如 this chapter (see Chapter 14).
The eviden忧。到 the outcome of isolated plays of the game is inconclusîve.
No experiment of which 1 am aware carefully induces the appropriate preferences
and is specifically designed to elidt a steady state action profile (see the box on
page 24). Th飞lS in each case the choice of Quiet by a player could indicate that
she is not "selfish" or that she is not experienced in playing the game, ratheτthan
providing evidence against the notion of Nash equilibrium.
1n two experìments wíth very low payoffs , each subject played the game a
small number of tìmes 在gainst different opponents; between 50 and 94% of subjects
chose Fi1仗， d叩 ending on the relati飞'e sizes of the payoffs and some details of the
design (Rapoport, Guye乙 and Gordon 1976, 135…137, 211-213 , and 223…226). 1n
a more recent experiment, 78% of su均 ects chose Fink in the last 10 of 20 rounds
of play against different opponents (Cooper, DeJong, Forsythe, and Ross 1996).
In face-to-face games in which communication is a lIowed , the incidence of the
choice of Fillk tends to be lower: from 29 to 70% depending on the nature of the
communìcation allowed (Deutsch 1958, and Frank, Gilovich, and Regan 1993, 163167). (1ηa11 thesεexpεrimeηts， the subjects were college studer飞ts în the United
States 0 1' Canada.)
One source of the variatior飞出 the results seems to be that some desígns indu优 preferences that dîffer from those of the Prisoner's Dilemma; no clear answer
emerges to th已正luestion of whether the notion of Nash equilibrium is consistent
wìth humans' choìc臼 in the Prisoner's Dilemma. 1f, nevertheless, one interprets the
轨ridence as showing that some subjects in the Prisoηer's Dile l11 ma systematically
choose Quiet rather than Fin走， one must fault the rational choice component or
Nash equílibrium, not the coordinated expectations componen t. Why? Because,
as noted in the text, Fink is optimal 110 matter what a player thinks her opponent
wîll choose , 50 that any mode
2.7 Example军 of Nash equilibrium
2.λ2
29
80S
τo find the Nash equilíbria of BoS (Figure 19.1) ，以1e can examine each pair of actions
m turn:
• (Bβch， Bach): lf player 1 switches to Stravinsky , then her payoff decreases from
2 to 0; if play时 2 switches to Stravinsky , then her payoff decreases from 1 to
O. Thus a deviation by either player decreases her payoff. Th us (Bach , Bach)
is a Nash 吨uilibrium.
• (Bach , Stravinsky): If player 1 switches to Stravinsky , then her payoff increases
from 0 to 1. Thus (Bach , StravÎllsky) is not a Nash equilibrium. (Player 2 can
increase her payoff by deviating , too, but to show that the pair is 口ot a Nash
cquilibl扫 m，. it suffices to show that one player can increasεher payoff by
devìating.)
• (Stravi l/ sky, Bac的: lf pl冯Ter 1 s\vìtches to BaclI, then her payoff i阳reases from
oto 2. Thus (StravÏI放y， Bach) is not a Nash equilibrium.
• (Stravinsky , StrG'DÎnsky): If player 1 switches to Boc l1, then her payoff de~
creases from 1 to 0; if play臼 2 switches to Bach , then her payoff decreases
from 2 to O. Thus a deviation by eith仅 playεr decreases h在 payoff. 丁hus
(Strav的sky， Stravinsky) is a Nash equilibrium
We conclude that BoS has two Nash equilibria: (Bocll , Bach) and (Stravinsky ,
Stravinsky). That ìs, both of these outcomes are compatible with a steady state;
both outcomes are stable social norms. If, ìn every encounter, both players choose
Bach , then no player has an inc自ltive to deviate; if, in every encounter二 both players choose Stravinsky, then no player has an incentive to deviate. lf we use the
garnεto rnodel the choices of men when matched with women, for example, then
the notion of t、.J ash equilibrium shows that two social norms are stable: both pla沪
ers choose the actíon associated with the outcome preferred by wornen, and both
players (hoose the actíon associated with the outζome preferred by men.
2.7.3 Matching Pennies
By checking each of the four paírs of actíons in Matching Pe l1 nies (Figure 19.2) we
see that the ga 附 has no Nash equi1ibrium. For the pairs of actions (Head , Head)
and (Tail , Tt码， player 2 is better off deviating; for the pairs of actions (Head, Tail)
and (Tail， 月仰的， player 1 is beUer off devíating. Th us for this game the notion of
Nash equilibrium isolates no steady state. In Chapter 4 we return to this gamε;
an extension of the notion of a Nash eguilibrium gives us an understanding of the
likely outcome.
2.7.4 The Stag Hunt
Inspection of the left panel of Figure 2 1. 1 shows that the two-player Stag Hunt ha苗
two Nash equ出检ia: (Stag , Stag) and (Hare , Hare). lf one player remains attentive
30
Cha势，ter 2.
Nash Equilibrium:τheory
to the pursuit of the stag, then the other player prefers to remain attentive; if one
player chases a hare , the other 0时 prefers to chase 浅 hare (she cannot catch a stag
alone). (丁he equilibria of the varìar飞t of the game ìn the right panel of Fìgurε2 1. 1
are analogous: (R政ain， R拉伯) and (Ar陀 Arm).)
Unlike the Nash equilibria of BoS , one of these equilibria is bettεr for both play耐
的 than the other: each player prefers (Stag , Stag) to (Hare , Hare). Th is fact has no
bearíng on the equilíbríum status of (Hm飞 Hare) ， since thεconditìon for an eq时，
librium is that a single player cannot gaín by deviating, given the other player' s behavior. Put differently, an equilibrium is immune to any unilateral deviation; coordinated d机riations by groups of players are not contεmplated. However, the existence of two equilíbria raises the possibility that one equilibrium might mo陀 likely
be the outcome of the game than the other. 1 陀turn to thís ìssue ìn Section 2.7.6.
1 argue that the many阳player Stag Hunt (Example 20.2) also has two Nash equìlibria: the action profile (Stag , … , Stag) ìn wh.ich every player joins ín the pursuit
of the stag , and the profile (Hare ,. ., Hm咛 in whích every player catches a hare.
• (Stag ,.., , Sfag) is a 时ash 叫uilibrium because each player prefers this profile
to that in which she alone chooses Hare. (A player is better off remaining
attentive to thεpursuit of the stag than running after a hare if a11 the other
players remain attentÌve.)
• (Hare ,.. ., Hare) isa 到ash equilibrium because each player prefers this profile
to that in whìc趴 she alone pursues the stag. (A play时 is better off catching a
hare than pursuìng the stag if no one else pursues the stag.)
• No other profile is a Nash equilìbrì飞出1， bεcause in any other profile at least
one player chooses Stag 击 nd at least one player chooses Ha 泞， 50 that any
player choosing Stag is better 。在 switching to Hl7I'，乙 (A player is better off
catching a hare than pursìng the stag if at least one other person chases a
hare , since the stag can be caught only ìf everyone pursues it.)
? EXERCISE 30 .1 (飞祉iants of the Stag HUllt) Consíder variants of the n-hunter St愣
Hunt in whìch only 111 hunters, with 2 m < H , need to pursue the stag in order
to catch 泣. (Continue to assume that there is a single stag.) Assume that a captur它d stag ís shared only by th巳 hunters who catch it. Under each of the followìng
a55umptions on the hunters' preferences, find the Nash equilibría of the strategíc
game that models thεsítuation.
a. A5 before , each hunter prefers the fractíon 1/11 of the stag to a hare.
b. Each hunter prefers the fraction 1/ k of the stag to a hare , but prefers a hare to
any smaller fraction of the stag, where k ìs an ìnteger with rn 三 k 'S 以
The following more diffícult exercise enríches the hunters' choices in the Stag
Hun t. This extended game has been proposed as a model that ca严ures Keynes'
basic insight about the possibility of multiple economîc e气uilibria， some of whîch
areundεsìrable (Bryant 1983, 1994).
2.7 Examples of r喝ash equilibrium
31
@EXERCISE 31.1 (Extensìon of the Stag Hunt) Extend the;卜hunter Stag Hunt by giving each hunter K (a positive integer) urúts of effort, which she can allocatεbe­
tween pursuing the stag and catching hares. Denote the effort hunter i devotes
to pursuing the st吨 by ej , a nonnegative integer 叫ual to at most K. The chance
that the stag is caught depends on the smallest of all the hunters' efforts, denoted
minj e;- (" A chain is as strong as its weakest lin k.") Hunter i's payoff to the 肘'
tion profile (el ,. . ., ell ) is 2 minJ ej - ej. (She is better off the more likely the stag is
caught, and worse off the more effort she devotes to pursuing the stag, which
means she catches fewer hares.) Is the action pro恕e 忱.. ./叶， in which eve巧r
hunter devotes the same effort to pursuing the stag, a Nash equi1ibrìum for any
value of e? (What is a pl盯rer's payoff to this pro f!l e? What is her payoff if she
deviates to a lower or higher effort level?) Is any action pro自le in which not all the
players' effort levels are the same a Nash 叫 uilibrium? (Consider a player whose
εffort exceeds the mirúmum effort level of all player5. What happens to her payoff
if she rεduces her effort level to the mirúmun创
2.7.5 Hawk -Dove
Thεgame in the next exercise captures a basic feature of animal conflict.
? EXERCISE 31.2 (Hawk-Dove) Two animals are fighting over some prey.‘ Each can
be passive or aggressive. Each prefers to be aggressive if its opponent is passiv飞
and passive if its opponent is aggressìve; given its own stance , it prefers the outcome in which its opponent is passive to that in which its opponent is aggressive.
Formulate this situation as a strategic game and find its Nash equ让ibria.
2. 又6
A coordination game
Consider two people who wish to go out togethe言~ butwho，飞出like the dissidents
in BoS , agree on the more desirable concert…-say they both prefer Bach. A strategic game that models this situation is shown 白Figure 31.1; ìt is an example of a
coordination gamιBy examining the four action pairs, we see that the game has
two Nash equilìbria: (Bach , Bach) and (Stravinsky , Stravinsky). In particr出价heac唰
tion pair (Stravinsky , Stravinsky) in wlùch both people choose their less-preferred
concert is a Nash equilìbrìu肌
Is the equ挂ibrium in which both people choose Stravinsky plausible? People
who argue that the tεchnology of Apple computers originally dominated that of
IBM comput在5， and 在at the Beta 扣rmat for vidεo recording is better than VHS,
would say "yes". ln both cases users had a strong interest ir飞 adop也19 the same
Bach.
Stravinsky :
Bach
2, 2
0, 0
Stravinsky
。， 0
1, 1
Figure 31.1 A coordínatio l1 game.
Chapter 2. Nash Equilibrium: Th eory
32
FOCAL POINT5
In games wíth many Nash equilibria, the theory isolates more than one pattern of
behavior compatible with a steady state. In some games , some of these equilibria
seem more likely to attract the players' attentions than others. To use the termìnology of Schelling (1960) , some 吨uilibrìa are foca l. In the coordination game ìn
Figure 31. 1, where the players agree on the more desirable 民ash equilíbrium and
obtain the same payoff to every nonequilibrium action pai乙 the preferable 吨Ul­
lìbrium seems more Ii阳ly to be focal (though two examples are gìven iηthe text
of steady states involving the inferior equilibrium). In the variant of this game in
which the two equilibria are 叫ually good (i. e. (2 , 2) is r，巳placed by 口，布， notl飞回
ing in the structure of 出εgame gives any ciue to which steady state might occur.
In such a game , the names or nature of the actions, or other information, may
predispose the players to one equilibrium rather than the other.
Consider, for example, voters in a民 election. Pre-election polls may give them
information about each other's intended actions , pointing them to one of m在ny
Nash equílibria. Or consider a situation ìn which two players independently di伽
vide $100 into two piles, each receiving $10 if they choose the same divisions and
notl飞ing otherwise.τhεstrategic gδme that models this situation has many Nash
equílibria , in each of which both players choose the same di飞怡ion. But the equílibrium in which both players choose the ($50，悍的 division seems likely to command
the players' attentions, possibly for esthetic reasons (it is an appealing division) ,
and possibly because it is a steady state in an unrelated game in which the chosen
division detennines the players' payoffs.
The theory of N ash 叫uilibrium is neutral about the equilìbrium that wíll occur
in a game with many equilibria. If features of the situation not modeled by the
I飞otion of a strategic game make some equilibria focal , then those equilibria may
be more líkely to emerge as steady states, and the rate at which a steady state is
reached may bεhigher than it otherwise would have been.
standard , and one standard was better than the other; in the steady state that
emerged in each case, the inferior technology was adopted by a large majority
of users.
i挝fh毛w
仰
op
严
eo
叩
ple played 出
thi岱sg
伊
am
刀1e in a laboratory it seems 芷
1 ik
挝
el坊
yt出
ha
挝t the outcome
飞认
川
4
r
steady state: if 白
e i让thεr action pair is reached , there is no reason for either player to
devia te from i1.
2.7.7
Provision of 0 pubfic good
τhe model in the next exercise captures an aspect of the provision of a 飞:mblic
good" , like a park or a swimming pool , whose use by one person does not diminish
2.7 军xamples
of Nash equilibrium
33
哩i
唱BA
AHV 咱I
iιτihU
咆ZA
IAHV
咱
fAAHV
1itinuv
IFJ
i''F
TB
,,
Figure 33.1 人 game with a unique Nash equilìbrìum, whiεh ìs not a strict equilibrium
its value to another person (at least, not ur飞til it is overcrowded). (Other aspects of
public good provísion are studíed in Section 2.8 .4.)
? EXER Cl SE 33.1 (Contributing to a public good) Each of n people chooses whether
to contribute a fixed amount toward the pro飞rision of a public good. 丁he good is
provided if and only if at least k people contribute, where 2 三主主 n; if it is not
provided , contributions are not refunded. Each person ranks outcome岳 from best
to worst as fo11ows: 的 any outcom号 in which the good is provìded and she does
not contrìbute, (ií) any outcome in whìch the good is provided and she contributes,
(iii) any outcome ìn wruch the good is not provided and shεdoεs not contribute ,
(i v) any outcome in which the good is not provided and she contributes. Formulate this situation as a strategic game and find its Nash 叫uilibria. (1s there a
Nash equilibrium ín which more than k people contribute? One in which k people
contribute? 臼le iηwhich fewer than k peoplεcontribute? (Be caref哇 1) )
2.7. 8 5trict and nonstrict equilibria
1n a11 the Nash e司uilibria of the games we have studìed so far, a deviation by a
player leads to an outcome WO l'se for that pl巧er than the equilibrium outcome.
The definition of J飞-.Jash equilibrium (23.1), however, requires OI飞ly that thεoutcome
of a deviation be 110 betfer for the deviant than the equilibrium outcome. And,
indeed , some ga凯εs have equilibria in which a play就 is indifferent between her
equilibrium action and some other action, given the other players' actíons.
COI飞sider the game in Figure 33. 1. This game has a unique Nash equílibrìum,
namely (了， L). (Forεvery other paìr of actìons , one of the players is better off
changing her action.) When play衍 2 chooses L, as she does in this e伊ilíbriu!口，
player 1 is equally happy choosing T or B (her payoff is 1 in each case); if she
deviates to B, then she is no worse off than she is in the equilìbrium. We say that
the Nash equilibrium 仔，均 is not a sfrict e号uilibriu l11.
For a general game, an equilibrìum is strict jf each player's equilibrium actìon
is better than all her other actíOI飞s， gìven the other players' actíons. Prεcisely， aη
action profile a* ìs a strict Nash equilibrium if for every player i we have U i(a*) >
Ui(aí , a':.) for every action ai 升。i of player i. 在 ontrast the strict in叫ualit更 in this
de主nitìon with the weak inequality in (23.动. )
2.7.9 Additìonal examples
The following exerCÌses are more difficult than most of the pr.εvìous ones. In the
first two , the number of actions of each pl可er is arbitrary, so you cannot mechan-
Chapter 2. Nash 在quìlibrium: Th eory
34
ícally examìne each actìon profîle índividually, as we did for garnes in whích each
player 1飞船 two actiOI飞s. Instead , you can consider groups of action profiles that
have features in common, and show that all actíon profiles in any gíven group are
or are not equilibria. Deciding how best to group the profiles into types calls for
some intuitíon about the character of a likεly equílìbrium; the exercises coτltain
suggestions on how to proceed.
( EXERCISE 34.1 (Guessing two-thlrds of the average) Each of three people an咛
nounces an integer from 1 to K. If the three integers are different, the person whose
i附g盯 ís closest to ~ of the ave咱们泛位阳e integ在s wins $1. If t认
附
w
100
仪r门I狡I 削
integers are 白t}乞1e 臼
s ame， $1 is 叩 lit equally between the people whose integer is closest to ~ of the averageìnteger. Is there any integer k 切蚓
仿
d们
h1川
巾t阳
t
(仪k ， ι
以k，立支均)， in which every person announces the same integer k, 总 a Nash equilibrium? (If k ~三 2， what happens if a person announces a smaller number?) Is any
other action pro Íile a 时ash equilibrium? (What is the payoff of a person whose
number is the highest of the three? Can she increase this payoff by announcing a
different number?)
Game theory is used 飞句lidely in politícal scier飞优， especially in the study of electìons. The game i只 the following exercise explores citìzens' costly decisions to
vote.
苦 Ex五反C!SE 34 .2 (Voter participatiOI飞)
Two candidates, A and B, compete in an elec搁
k) support candidate B.
Each citizen decides whether to vote, at a cost, for thεcandidate she supports, or
to abstain. A citizen who abstains recei飞res the payoff of 2 if the candidate she
supports 飞'V ins， 1 if this candidate ties for first place, and 0 if this candidate loses
A dti之en who votes receives the payoffs 2 … c, 1 c, and -c in these th挫e cases ,
where 0 < c < 1.
tion.Ofthε11 citizens , k support candidate A and 111 (= 11
a. For k
m
去.
111 , find the set of Nash equilibrìa.
(Is the action profìle in whích
ev世yo时 votes a Nash 叫 uilibrium? Is there any Nash equilibrium in whích
the candidates tie and not everyone votes? Is there any Nash 吨uilibrium in
which one ()f the candidates wins by one 飞'ote? Is there any Nash equilibrium
in whích one of the candidates wins by two or more 飞rotes?)
c.
"气That is the set of Nash equilibria for k < m?
1, is the game the same (except for the names of the actìons) as
any considered so far in this chapter?
For k
If, when sittìng in a traffic jam, you have ever thought about the time you might
sa飞罗e if another road were buílt, the r飞毛xt exercise may lead you to think again.
② EXERCI臼 34.3 (Choosing a rou时 Four people must drive from A to B at the same
time. Ea(丁h of them must choose a route ‘ Two routes are available , one vìa X and
one via Y. (Refer to the left par飞el of Figure 35. 1.)τhe roads from A to X, and from
Y to B are both short and narrow; in each case , one car takes 6 minutes, and each
2.8 Best response f位 nctíons
3S
守A7
守'hq伽
/句/』
机句3
Fígure 35.1 Getting from A to B: the road networks in Exerdse 34.3. The numbers beside each road are
the travel times per ca l' when 1, 2, 3, or -l cars take thaì road.
additìonal car increases the travel time per car by 3 mìnutes. (If two cars drive from
A to X, for example , each car takes 9 minutes.) 在ìe roads from A to Y, and from X
to B are long and wìde; on 人 to Y one car takes 20 mìnutes , and each additìonal car
increases the travel iime per car by 1 minute; on X to B one car takes 20 minutes,
and each additional car increases the travel time per car by 0.9 minutes. Formulate
this situati∞ as a strategic game and fìnd the Nash equilibria. (If all four people
take one of the routes, can any of them do better by taking the other route? Wh at
ìf three take one route and one takes the other route , or if two take each route?)
?、Jow suppose that a relatively short wíde road ìs buìlt from X to Y, gìving each
person four options for tra飞'el from A to B: A-X…B，人一Y-B ， A-X…Y-B , and A-YX-B. Assume that a person who takes A-X…Y-B travels the A…X portion at the
same time as someone who takes A… X-B , and the Y-B portìon at the same tìme as
someone who takes A-Y… B. (η1Ìnk of thεre being constant flows of traffic.) On the
road between X and Y one car takes 7 minutes and each additional car increases
the tr那'el time per c的. by 1 minute. Find the Nash equilìbria in this new situation.
Comparεeach person's travel tìme with her travel time ín the equilibrium before
the road from X to Y was built.
2.8
Best response functions
2.8.1
Defìnition
We can find the Nash equilibrìa of a game in which each player has only a few
actions by exam如ing each action profile in turn to see if ìt satistìes the conditions
for equilibrium. In more complìcated games, it ìs often bett缸 to work with the
players' "best response functions 气
Consider a player, say player i. For any gìven actions of the players other than i,
player i's actions yield her v盯lous payoffs. We are interested in the best ac远ons­
those that yield her the highest payoff. In Bos , for example, Bach is the best action
for player 1 if player 2 chooses Bach; Stravills~守 ìs the 快st action for player 1 if
pl町er 2 chooses stravinsky. 1n partìcular, in Bos , player 1 has a single best action
Chapter2. 挝矗$狂草quilibrìum:
36
Theory
for each action of player 2. By contrast in the game in Figurε33.1 ， both T and B are
best actions for player 1 if player 2 choosεs L: they both yield the payoff of 1, and
player 1 has no action that yields a higher payoff (in fact , she has no other action).
We denotεthe set of pJayer í's best actions when the list of the other players' actions is a_/ by B/(a_i)' Thus in BoS we have B 1 (Bach) 口 {Bach} and
B1 (StravÎnsky) = {StravÎ l1 sky}; in the gamεin Figure 33.1 we have B1 (L) = {了， B}
Precisel予 we define the function Bi by
BI 伊
= {ai inμí(aitLi) 2: 叫 a; ， a-i) for all a; in Ai} :
any action in Bi(Lí) is at least as good for player i as every other action of player i
when the other players' actions are given by Li' We call Bi the best respon时
function of player i.
The function B; is set-valued: it associates a set of actions with any list of the
other players' actions. Every member of the set Bi(a-í) is a best 1哩部ponse of
player i to a… i: if each of the other players adheres to a… i, then player i can do no
better than choose a member of BííLJ In some games , 1ike BoS , the set Bj(a
consists of a singJe action for evεry list Lj of actions of the other players: no matter
what the other players do , player í has a sìngle optimal action. In other games , like
the one in Figure 33.1 ，阳。一 j) cor飞tains more than one action for some lists i1 -i of
actions of the other playεrs.
2.8.2 Using best responsθ funζtÎoη5 to define Nosh equifíbrium
A Nash equilibrium is an action profile with the property tha t no player can do better by changing her action, given the other players' actions. Using the ten页inology
just developεd ， we can altemati飞rely define a Nash equilibrium to be an actior飞 pro
file for which evεry player's action is a best response to the other players' actions
That is, we have thεfollowing result.
• PROPOSITION 36.1 The action pl听le 旷 is a Nash 叫 uilibrium of a strategic ga111e with
ordí l1 af preferel1c臼扩α nd 0111y 扩。eryplaye自 actiol1 is a best respollse to the other players'
actío 11 s:
(36.2)
。 j is in Bi(a*-i) for 已刊ry player i
lfεach player í h飞as a si只gJ 巳 best respοn飞se to each list 泛 i of the 时0)吮ther players
actìor口lS鸟F 飞w
々咣
e can \'"川气叩付气
V盯
y咛r门
呐1 怆
te 位
t h飞e con
飞di挝
tìons 阳
i 贝 (怡
36ι.2
斗) as equations.
In this case , for each
player i and each 1ist a… i ofthεother players' actions, denote the sìngle member of
BI
by b/(Li) (that is , B i (。→ i)
{b/(a_i)}). 丁hen (36.2) is equivale时 to
aj 口 bi(a*-;) for every player Î,
(3已.3)
a col1ection of n equations in the n unknowns aj , where 11 is the number of players
in the game. For example, in a game with two players, say 1 and 2, thesεequations
are
αi
= b1(ai)
aî = b2 (aÎ)
2.8 Best respons告 functions
37
T
M
B
L
1 ， 2不
C
R
2飞 γ
,1
o , 1*
1*, 0
o ,0
o ,1
o ，。
γrγ
Figure 37.1 Using best response functions to find Nash equilibría in a two-player game ín whichεach
player has thrcc actíons、
That is , in a tw吗-player game in which each player has a single best rεsponse to every action of the other play吼叫 ， a;) is aNash equìlibrium if and only ifplayer l's
action ai is her best resp。在se to player 2's action ai , and player 2's action ai is her
best 陀sponse to playerγs action aî
2.8.3 Using best r，ε spo门se functions to 污nd Nash equilibria
The definition of a Nash equ i1ibrium in terms of best response functions suggests
a method for finding Nash equìlibria:
• find the best response function of each play臼
• find the action profiles that satisfy (36勾 (which reduces to (36 .3) if each
player has a single bestτesponse to each list of the other players' actions).
τo illustrate this method , consider the game in Figure 37. 1. First fìnd the best
response of player 1 to each action of player 2. If player 2 chooses L, then player l's
best response is M (2 is the highεst payoff for player 1 in this column); indicate the
best response by attaching a star to player l's payo旺 to (M , L). lf player 2 chooses
C, then player l's best response is T , indicated by the star attached to playerγs
payoff to (T. C). And if player 2 chooses R, then both T and B are best 陀sponses
for player 1; both are ìndicated by stars. Second , 在nd the best respo口se of player 2
to each action of player 1 (for each row, find highest payoff of player 2); thesε
best respons部 are indicated by attaching stars to player 2's payoffs. Finally, find
the boxes in which both pl盯rers' payoffs arεstarred. Each such box is a Nash
equilibrium: the star on player l's payoff means that player l's action is a best
response to player 2's action, and the star on player 2's payoff means that player 2's
action is a best response to player
action. Thus we conc1 ude th批 the game has
two Nash equilibria: (M , L) and 恼，只) .
φEXERCISE 37.1 (Finding Nash equilibria using best response functions)
a. Find the players' best response functions ìn the Prisoner 给 Dilemma (Figure
15.1) , BoS (Figure 19 斗 ， Matching Pennies (Figure 19.2), and the two-player
Stag Hunt (left par飞el of Figure 21. 1) (and verify the Nash 吨uilibria of each
game).
b. Find the Nash e司uilibria of the game in Figure 38.1 by finding the players'
best response functions.
Chapter2. N矗sh Equilibrium:τheory
38
了
M
β
Fì伊始 38.1
L
2, 2
3, 1
1, 0
j之
C
1, 3
。， 1
。， 0
0, 0
0, 0
。，。
The game ín Ex苦rcise 37 .l b
,
τhe players' best response functions for the game in Figure 37.1 are presented
i口 a dífferent format 如 Figure 38.2. In thís figure, player l's actíons are on the hor-
izontal axís and player 2's are on the vertical axis , (Thus 血
thε ∞i汕
um
饺
mηη1订ns
to choíces of playe盯r 1, ar丑ld the rowsc∞orrespor只ld 怡
to choices of 抖
p laye
衍r 2, whereas the
reverse is true in Figure 37. 1. 1 choose this orientatíon for Figure 38.2 for consis翩
tency with the convention for 在gurε5 of thís type.) Player l's best responses are
índícated by cirdes , and player 2'5 by dots. Thus the circ1 e at (T , C) reflects the
fact that T ís player l's best respon5e to player 2's choíce of C, and the circles at
R) and (B , R) reflect the fact that T and B are both best resp∞ses of player 1 to
player 2'5 choice of R. Any action pair marked by both a circ1 e and a dot ís a Nash
equilibrium: the cirde means that player l's action i5 a best response to player 2's
actiol1, and the dot indícates that player 2's action is a best respo口se to pl玛rer l's
actìon.
ø EXERCISE 38.1 (Constructing bεst response functions) Draw the analogue of Figure 38.2 for the game in Exercise 37.1b.
( EXERCISE 38.2 (Dividing money) Two people have $10 to di飞址e between therr飞"
selves. They use the following procedure. Each person namεs a number of dollars
(a nonnegative integer), at most equal to 10. If the sum of thεamounts that the
people name is at most 10, then each person receives the amount of money she
named (and the remainder is destroyεd) ‘If the sur我 of the amounts that the peoω
ple name exceeds 10 and the amounts named are differe时， then the perso口 who
R
0
A2< C
0
L
@
-e
M 〉叫
A
了
B
Figure 38.2 The players' best response functíons for the game in Figure 37 1 Player 1's hest responses
,
在re índicated by círcles, aηd pJayer 2、 by dots. The action paírs for whìch there is both a circle and a
dot are the Nash equilíbría
,
2.8 Best response functions
39
nam叫 the smaller amount receìves that amount and the other person receives the
remaining mone予If the sum of the amounts that the people name exceeds 10 and
the amounts named are the same, then each person receives $5. Determine the
best response of each player to each of the other play时 's actions, plot them in a
diagram like 俨igure 38 .2, and thus find the Nash equilibrìa of the game.
A diagram like Figure 38.2 is a convenient representation of the playε刊， best
response functíons also in a game in whích each player's set of actions is an interval
of numbers , as the next example 过lustrates.
EXAMPLE 39.1 (A synergistic relationship) Two individuals are involved in a synergistic relationship. If both individuals devote more effort to the relationship , they
are both betteτoff. .For any given effort of individual j , the return to individual i's
effort first incr巳ases， then deζreas能 Speci自cally， an effort le\叶 is a nonnegative
number, and iI飞dividual j's preferences (for i = 1, 2) are represented by the payoff
function 叫 c 寸的叫， where Gj is j's effort 1εvel， Gj 1S the other individual's effort
levet and c > 0 is a constant.
τhe following strategic game models this situation.
Players τhe two individuals.
Actions Each player's set of actions is thεset of effort levels (non我役军ati飞'enuI曰"
bers)
Pnεferences Pl町er í's preferences are represented by the payoff function Gi(C 个
叫一 Gi) ， for i = 1, 2.
In particula艾， each player has infinitely many actions , so that we cannot present the
game in a table like those used previously (Figure 38 .1, for example)
To fir飞d the Nash equilíbrìa of the game, we can construct and anaìyze the play校5' best response functions. Gíven aj , índìvidual j's payoff is a quadratic function
C 十町， and reaches a maximum ín
of Gí that is zero when Gj 口 o and when Gj
between. The symmetry of quadratic functions (see Section 17.3) implies that the
best respo只se of each individual i to Gì is
bj(GÚ =
!(c + aj)'
(If you know calculus, you can reach the same con cJ usion by settir飞gthe deriγative
of player í's payoff with respect to Gj 叫ual to zero.)
τhe best response functions are shown in Figure 40. 1. Player l's actions are
plotted on the horizontal axis and player 2's actions are plotted on the vertical axis.
Player l's best response function assodatεs an action for player 1 with every actior飞
for player 2. Th飞1s to interpret the function b1 in the diagram , take a point a2 0到
the vertical axis, and go across to the line labeled b1 (the steeper of the two lin.e吟，
then read down to the horizontal axis. The point on the horizontal axis that you
best
reach is bl (a纣， the best action for player 1 when player 2 chooses a2. Player
response function, on the other hand, assodates an action for player 2 with every
action of playe主 1. Th us to interpret this function, take a point al on the horizontal
Chapter 2. Nash Equilibrium: Theory
40
1
句
C
ic
。
C
al 一→
The players' best response functíons for the game in Example 39.1τhegameh在s a umque
Nash equilibrium ，吨， a;i 二仗 ， C)
Figur告 40.1
axis, and go up to b2 , then across to the vertical axís. The point on the 飞lertical axís
that you reach is 句(叫， the best action for player 2 when player 1 chooses a1
At a point (aj , a2) where the best response functíons intersect in the figure , we
have al = b1(a2) , because (al , a2) is on the graph of b1, player l's best response
b2 (ad , because (aJ ， tψis on the graph of b2 , player 2's bεst
function , and az
response function. Thus any such point (aj ，向) is 在 Nash equilibrìum. ln this
game the best response functions intersect at a single point so there is one N吕sh
equilibrium In genera l, they may ìntersect more than once; every point at whích
they intersect 1S a Nash equilibrium.
To find the point of intεrsection of the bεst response functions precisely, we can
solve the two equations in (36 .3):
‘
a1 口 !(c + az)
32rj(C 十 al)
i
S山stìtuting the seco时叫ua tion in the fírst, we get Gl
(c 十七c 十 SINt jc 斗m
!Gj , so that G] 工 C. S阳títuting 拙 value of aJ into the second 叫uation， we get
a2 c. We conclude that the game has a unique Nash 吨ui1ibrium (肉，电) = 卡， c ).
(To reach this conc1 usion , it suffices to solve the two equations; we do not have
to draw Figure 40 .1. However, the diagram shows us at once that the game has a
unique equi1ibriur叽 i~ which both players' actions exceed ~c， facts that ser刊 to
check the results of our algebra.)
In the game in this example, each player has a unique best response to every ac帽
tion of the other playe乙 so that the best response functions are lines. If a player has
many best responses to some of the other players' actions, then her best response
function is "thick" at some points; 能veral examples 如 the next chapter have this
2.8 Best response functions
41
a2
东
A? 一
- a2
求?
nu*7h
nM峙
@年
*?加
*
fl:.2 、
冲
@争鸣BEA
牢
中T
品咱'A
本 τi
nu
ANH
4AT
mMH
A1
Figure 4 1. 1 人民 example of the best response functions of a two-player g在 me in which eac扒 player's
set of actions is an interval of numbers. 丁'he set ofNa忍sh 趴e q叫归
μL川
u
üωi
让山
lib
忻n凶
a (l、迁f the 伊g主喜栩
lIT
mη1哇 consists of tt怡飞泼硅 F
红
严
ai泣rs of
&仄ct加
iωG
∞只陌s(归问
巧j ， aη
d
叩;引)μand (付问
巧抖*气， 唁
g
a 2 叮)， ar叫 all tl挝
1程
ep
严
a 山 of 仄
a ctiμ阳
4
2
， a ')and(归问
叫; *归问气
a
a袍'，
簿气.飞 a
property (see, for example, Figure 66.1). 古飞e game in Example 39.1 is special also
because it has a unique Nash equilibrium-the best response functions cross once.
As we have seen, some games have more than one equilibrium, and others have
none. Figur功是1. 1 shows a paìr of best response functions that illustrates some of
the possibilìties ， ηle shaded area of player l's best response function indic挝es
that for a2 between 再2 and 空2' player 1 has a range of best responses. Forεxample，
a11 actions of player 1 greater than aì* and at most ai 料盯e best responses to the actÌon a2** of player 2 , For a ga血e with these best 陀sponse functions , the set of r\Jash
equílíbria consists of the pairs of actions (ai , aî) and (aî 飞 aT 飞 and a11 the pairs
of actions on player 2's best response function between (aj* , aî*) and (ai 忡，可可
@ EXERCISE 4 1.1 (Strict and nonstrict Nash 叫uilibria) Wh ich of the r、妇sh equilibria
of the game whose best response functions are given in Figure 41 .1 are strict (see
the definition on page 33)?
Another featu挖出at differentiates the best response functions in Fìg飞lre 4 1. 1
from those in Figure 40.1 is that the best response function b1 of player 1 is not
continuous. Wh en player 2's action is 言之， pl可er l's best response is aì* (indicated
by the small disk at (a ì*, a2 月， but when player 2's action is slìghtly g陀ater than
坠， player l's best response is si伊ificantly less than ai气 (The small circle indicates
a point exc1uded from the bεst response function.) Again, several examples in
the next chapter have this feature. From Figure 4 1.1 we s时 that if a player's best
response func悦。n is discontinuous, then depending on where the disc∞tinuity
occurs, the best response functioí1s may not intersect… the game may, like Matching
Pennies , have no Nash equilibrium.
Chapter2. N在 sh Equilibrium: Theory
42
? EXERCISE 42 .1 (Finding Nash 叫uilibria 飞lsing best respon.<;e fun位ions) Find the
?飞、Jash equilibria of the twoωplayer strategic game in which each play缸 's set of
actions is the set of nonnegati飞!e numbers and the players' payoff functions are
llj(Gj , G2) Gl( 何一句) and 盯圳， {的 = 112(1 …♂ j - (/2
? EXERCISE 42.2 (A joint projeεt) Two p巳 ople are engaged in a joi时 project. If each
person i puts iηthe effort Xj , a nonnegative number equal to at most 1, which
costs her c( 刊 I the outcome of the project is 认rorth f(工 L X 2). τhe worth of the
project ìs split equally between the two people , regardless of their effort levels.
Fonτmlate this situation as a strategic game. Find the Nash equilibria of the game
when (a) f(X1t X2)
3工1X2 and C(Xi) = 工?fof l 口 L 2, and (的 f(X1 ， X2) = 4X1X2
and C(Xi) = Xi for i = 1, 2. In each casε， is there a pair of effort levels that yields
higher payoffs for both players thar飞 do the 技ash equilibrium effort levels?
2.8 .4
1/Iustration: contributing to a public good
Exercise 33.1 modeís decisions on whether to contribute to the provision of a public good". We now study a model in which two people decide not ∞ly whethεr to
contribute , but also 110W m!l ch to contribut巳
Denote person i's wealth by 'W i, and the amount she contributes to the public
good by Cj (0 <二 CIf二 zc飞); sl飞εspends her remaining wealth Wi - ci on "private
goods" (l ike clothes and food , whose consur丑ptio口 by one person prεcludes their
consumption by anyone else). The amount of the public good is equal to the sun飞
of the contributions. Each person cares botl飞 about the amount of the public good
and her consumption of private goods.
Suppose that perSOlì i's prefere只ces are represented hy the payoff function
Vi( c1 十门)十 Wj 一白， where Vi is a口 increasing fUlìction. Because Wj is a constant ,
person í's prεferences are alternatively reprεsented by the payoff function
!f
Ilj 仙 ， (2)
Vi(C1 + 句) - [j.
(42.3)
This situatíon is modeled by the following strategic game
PJayers The t认'0 people
Actions Pl 吧'er 1\s set of actions is the set of her possible contributions (nonnegative numbers less than or equal to Wj) , for i = 1, 2.
Pr.εferences
Player i's prefer创ìCes are repτesented by the payoff function !I í
= 1, 2.
gi飞'en in (42.3) , for i
To find the Nash equilibria of this strategic game， ωnsider the players' best
response functions. Pla yer 1'5 best rεsponse to the contribution [2 of player 2 is
thεvalue of C1 that ma)也时zes Vl ([1 十 (2) … [1. With肌1t 守
sP
严
权
e
cif守
ym
咔
g the form of
the function v) we cannot explicitly calculate this optimal v叶 ue. However, we can
determine how it varies with [2.
2.8 8est -response functions
43
主
主
If ì:工‘!， Iq
户L
:(
)
AU
\
G
b} (k) b} (0)
W1
C1 一→
Figure 43.1 The relatìon between player 1'5 best responses b1 (0) and b] (k) to c2 口 o and C2 = k in the
game of contributing to a public good.
Fírst consider player l's best response to C2 工 O. Suppose that the form OÍ
thεfunction v} is such that the function 叭(句，的 increases Up to its maximu日1，
then decreases (as in Figure 43 .1 ).τhe我 player l's best response to C2
0, which
1 denote b} (哟， i5 unique. This best respon5e is the value of C} that maximizes
u}(c} ， 的口 V} (c}) - c} subject to 0 三 c} < W1' Assume th挝 o <句顶) < W}:
player l's optimal contributi岭
on 沁
t o the public good wh
怡
1况
en pl协
咛
a
y啊 2r孜1飞冶
akes no cont恬
f民i儒
bu
时tiω01η1i沁s positive and less than her 创
en让巾
t悦Ìri
陀
e wealth.
Now consider player 1'5 best response to C2 = k > O. 丁his best response is
the value of c1 that maxirnizes L川 C1 ， 叫
V1 (C1 十 k)
C1. Now, we have 问( C} 十
支， 0) := V}(C1 十 k) - c} - k by the definition of u} , so that
z叭 cl ， k) 口叫 C1 十主，的十 k
That is , the gr咛h of u1 (C} , k) as a functi∞ of c} is the translation to thεleft k unìts
and up k units of the graph of 叫 (C1 ， 0) as a function of C1 (陀fer to Figure 43.1).
Thus ìf k 三 bl(的， 白
then
to k, ther口1 player 1γ，、s best respor口lse decreases by k. If k > b1 (0) , then, given the
form of ul ( 叫，时， we have b1 (k)
O.
Wεconclude that if player 2 increases her contribution by k, then player l's best
respon5e is to redu但七er contribution by k (or to ze邸， if k is larger than player 1'5
original contribution)!
在le same analysi5 applies to player 2: for every unit more that play时 1 con州
tributes, player 2 contributes a unit less, so long a在 her contributìon is nonnegative.
The functìon V2 may be different from the function 叫， so that player 1'5 best contribution b1 (0) when c2
0 may be different from player 2'5 be5t contribution b2 (0)
when C1 = O. But both best response functìon5 have the same character: the slope
of each function is -1 where the value of the function is positive. They are shown
in Figurε44 .1 for a case in which b1 (0) >妇(的.
o
‘
Wed
岱
εed
缸
uc
岱
ε t市
沁&挝f低近 b句1丘削(检
h
0) > b2 (0
的)， 拉
th础
岱
e
nt出
h飞.eg
伊
am
引1e
忖础
hasa 飞u江
凶
m
挝
n1泣i坞
罕时
q
u1淀
eNas由
h 吨u
川硝
i注li如
briu皿
1旦
m
丑
m1
(作岛剖削
1刊(0
创)， 的0): 抖
p la
咛
ye
衍r2 co
∞
时
Eη
1t饺古i澎
b时e
岱sr∞
1冶
ot出h
挝
泌i王鸣
1宅
g. Si邸larl予 if b1 (的 < b2 (时， then the unìque
Chapter 2. Nash 主.quilibrium: Theory
44
b1 (0)
\\M
。
Figure44.1τhe
b110)
best response functions for the game of contributing to a public g∞d in a case ìn whìch
b2 (O). The best response function of player 1 i革 the black lin日; that of play♀r 2 is the gray line
过没sh equ i1i briurn is (0 , b2 (O)): player 1 contrib时es ∞thi略
That is , the person
who contributes rnore when the other person contributes n悦hìng is the only one
to rnake a cor由ibution i口 a Nash equilibriurn. Only if b1 (0) :;:: b 2 (哟， which is not
likely if the functíons Vl and V2 differ, is there an equilibriurn in which both people
contribute. ln this case the downward-sloping parts of the best response functions
coincide, so that any pair of cor巾ibl时 ons (C1 , C2) with C] 十 C2 记 b 1 (0) and Ci 主 O
for i = 1, 2 is a r、Jash equilibriurn.
ln surnrnary, the r飞otíon of Nash equilibriurn prεdicts that, except in unusual
cirεurnstances， only one person contribub号s to the provision of the public good
when each person's payoff functio口 takes the forrn Vi( 肉牛句}十叫… Ci ， each fun仑
tion Vi( Ci) … Ci increases to a rnaximu叽 then decreases, and each person optirnally
contributes less than her entire wεalth when the other person does not contribute.
The person who contributes is the on号 who wishes to contribute more when the
other p轩son does not contribute. ln particula 乙 the identity of the person who
contributes does not depend on the distribution of wealth; any distribution in
wt飞ich each person optimally contributes less than her entire wealth when the other
p仔son does not contribute leads to the same outcome.
τhe next exercise asks you to consider a case in whích the amount of the public
good affects each person'sεnjoyment of the private good. (丁he public good rnight
bεclean air, which improves each person's enjoyrnent of her free time.)
? EXE汉CISE 44.1 (Contributing to a public good) Consider thε1τlOdel in this section
when 叫句
is the Sl口n of three parts: the amount c1 十巧 of the public good
prO\址ed ， the amount 叫一 Ci pers∞ í spends on pr认rate goods, and a terrn (ω
Cj)(Cl 十 C2) that reflects an interaction between the amouñt of the public good and
her pri飞rate consumption-the greater the amount of the public good , the more she
values her private consumptio只. In summ盯y， s泣ppose that person i's payoff is
C1 十 C2 十 ωJ … Ci 十 (Wi- C i)(C1 十 C2 )， or
Wi 十 cf 十 (Wi - Ci)(Cl 十 C2) ，
2.9 Dominated actions
45
,
where j is the othεr perso孔 Assume that w = 'W2 艺工 w , and that each player i's
contribution Cí may be any nurr飞b巳r (positive or negative, possibly larger than W).
Find the Nash equilibrium of the game that models this situation. (You can calculate the best responses explicitl y. Imposing the sensible restriction that Cí lie
between 0 and w complicates the analysis but does not change the answer.) 5how
that in the Nash equilibrium both players are worse off than they are when both
contribute half of their wealth to the public good. lf you can , extend the analysis to
the case of n people. As the number of people increases, how does the total amount
contributed in a J飞.Jash equilibríum change? Compare the players' equilibrium payoffs with theír payoffs whèn each contributes half her wealth to the public good ,
as n increases withot!t bound. (The game is studied further in Exercisε388. 1.)
2.9
Dominated actions
2. 旦 1
Strict domination
You drivεup to a red traffic ligh t. The left lane is free; in th芝 right lane there ìs a
car that may turn right when thεlightchangεs to green , ìn which case it \'vill have
to wait for a pedestrian to cross the side strεe t. Assuming you wish to progress
as quickly as possible, the action of pulling up in the left lane 飞trictly dominates"
that of pullìr飞g up in the right lane. If the car in the right lane turns right, then you
are much bett臼 off in the left lane , where your progress will not b巳 impeded; and
even if the car in the right 1在只e does not turn right, you are still better off in the left
lane , rather than behind the other car.
In any game, a player's action "strictly dominates" another action if it is superior, no matter what the other players do.
DEFIN lTl ON 45.1 (Strict domi l1 ation) In a strategic game with ordinal pref，εrences，
player i's action a;' strictly dominates her action a; if
Uí(a了，仁 í) >叫ι a~í) for every list a-í of the other players' actions,
where uí is a payoff function that represents player ì's preferences. We sa)' that the
actìon a; is strictly dominated
1n the Prisoner's Dilemma , for example, the action Fink strictly dominates the
action Quiet: regardless of her opponent's a c: tion , a player prefers the outcome
whenshecl飞oosεs Fink to the outcome when she chooses Quiet. 1n BoS , on the otÌ1er
hand , neither action strictly dominates the other: Bach is better than Stravinsky
if the other player chooses Bach , but is worse than Stravínsky if the other player
chooses Stravinsky.
A strictly dominated action is not a best response to any actions of the other
players: whatever the other players do, some other action is bette汇 5ince a playεr's
Nash 叫uilibrium action is a best response to the other players' r、Jash equilibrium
actions,
a strictly dominated action is not ωed in any Nash equilibr均m.
46
Chapter2. t归sh 在quilibrium: Theory
013
IU1 品?』句 3
只
L121i
TMB
RO142
TMB
whích player 1'5 action Tis strictly dominated by M. (Only player }'s payoffs
are given.) In the left-hand gan泼， B ís better than M if player 2 chooses R; in the right七在ndgame， M
itself is strictly domínated ，问! B.
事igure46.1τ节。 gamεsin
When looking for the Nash equilibria of a game , we can thus eliminate fror到 con­
sideration all strictly dominated actions. For example ，阳 the Prisoner's Dilemma we
can eli盯lÍnate Quiet for each player, leaving (Fi此 Fink) as the only âcti0n pair that
can possibly be a Nash equilibrium. (As we know, this action pair is indeed a Nash
equilibrium.)
The fact that the action a;' strictly dominates the action 叫 of course does not
imply that a;' strictly dominates .'111 actions. lndeed ，叫I may itself be 由ictly dominated. ln the left斗land game in Figure 46.1 , for example, M strictly dominates T,
but B is better than M if player 2 chooses R. (1 give only the payoffs of player 1
in the figure , because those of player 2 ar.ε 窍。t rele飞rant.) Since T is strictly dominated , the game has no 时ash equilibrium in which play佼 1 uses it; but the gam廷
may also not have any equilibrium in which player 1 uses M. ln the right-hand
game, M strictly dominates T, but is itself strictly dominated by B. ln this case ,
in any Nash equilibrium player l's action is B (her only action that is not strictly
dominated).
A strictly dominated action is incompatible not only with a steady state, but
also with rational behavior by a player who confronts a game for the first time.
τhis fact is the first step in a theory different from Nash equilibrium , explored in
Chapter 12.
2.9.2
Weak domination
As you approach the red light in the situation at the start of the previous section
(2.9.1) , there ìs a car in cach lane. The ca主如 the rìght lane may, or rnay not, be
turnìng right; if it is, it rnay be delayed by a pedestrian crossing the side stre枕
The car in the left lane cannot turn righ t. In this case your pulling up in the left
lane 气veakly dorninates" , though does not strictly dominate , your pulling up in
the right lane. If the car in the right lane does not turn righ t, then both lanes are
equally good; if it does, then the left lar飞e is better.
ln any game, a player's .a ction "weakly dornìnates" another actiOI飞 if the fìrst
action is 必 least as good as the second action, no matter what the other players do,
and is better than the second action for some actions of the other players.
D盯INJTION 46.1 (Weak domination)
player i's action
a;'
1n a strategic gamεwith ordinal preferences,
weakly dominates her action if
a;
ui(a;'， a_d 主 Ui(a;， a-i) for every list a-i of the other players' actions
2.9 Dominated actions
47
L
T
M
B
G
2
2
c
Figure 47.1 A game iIl ustrating weak domination. (On 坏" player 1's payoffs are given.) The action tvt
kly domínates T; B weakly dominates M. The actíon B strictly dominates 了，
WCi1
and
叫。;'， a-í) > Ui(叫， Lj) for some list a_寸。f the otr附 players' actìon
where Uj is a payo日 function that represents player i's preferences. We say th挝 the
actìon a~ ìs weakly.dominated
For example, in the ga陀、e in Figure 47.1 (in which, oì飞ce again , only player 1's
payoffs are gíven) , M w相 kly dominates T, and B weakly dominat部 M; B strictly
dominates T.
In a strìct ì、-Jash equilibrium (Section 2.7.8) no play时 's 吨 uilibriurn ac坟。只 is
weakly dorninated: for every pl咛代 the payoff to each non叫uilibrium action is
less than her e气uìlibrium payoff, so that no none守主ilibriurn actior飞 weakly dominates her e气uilibrium action.
Can an action be w岱kly dominated in a nonstrict Nash e气过ilibrium? Defi翩
ηitely. Consider the games in Figure 47 .2. In both games B wea kly (butηot strictly)
dominates C for both players. 吕时 in both games (c, C) is a Nash 均 ui1ibrium:
given that player 2 chooses C, player 1 cannot do better than choosε C， and given
that play轩 1 chooses player 2 cannot do better than choose 巳 Both garnes also
have a 出ash equilibrium, (B , B) , in wlùch neìther player's action is weakly dom唰
inated. In the left蛐hand game this e电uilibrium is bett位如r both players than the
equilibrium (巳 C) in which both players' 阳
a ct创ions ar挖
e weakly dor血
ni扭
na
挝te
创
d， whereas
in the righ飞t-七a栩
1引飞d game 让
it 始
i swo
时Y部se 妇
forbo
仗thpi妇
玛盯
a
ye
仔rs than (c,
.
( EXERCISE 47.1 (Strìct 吨ui1ibria and dominated actìons) For the game in Fìgure 48 .1, determìne, for each player, whether any actìon ìs strictly dominated or
weakly dominated. Find the Nash equilìbrìa of the game; determine whether any
叫uilibrium is stric t.
( EXERCISE 47.2 (Nash equi豆brium and weakly domìnated actions) Give an example of a two-player strategic game ir飞 which each player has finitely many actions
and ìn the only T、如sh equilibrium both players' actìons are weakly dominated.
B
C
B 1, 1 0, 0
C
0, 0
0, 0
B
C
B
1, 1
0, 2
C
2, 0
2, 2
Figure 47.2古wo s泣ategic gβmes with a Nash equilibrium (C, C) in whiçh both players' actions are
weakly dominated.
Ch昌 pt，世r 2.
48
L
T 0, 0
丸1 . 1, 1
B 1, 1
Nash Equilibrium:τheory
C
R
1, 0 1, 1
1, 1 ì 3, 0
2, 1: 2, 2
Figure 48.1 The g吕meinEx白rcise 47.1.
2.9.3
II/ustratioη:γoting
Two candidates, A and B, vie for office. Each of an odd number of citizens may
vote for either candidate. (Abstention is not possìble.) The candidate who obtains
the most votes wins. (Because the number of citizens is odd , a tie is impossible.) A
majority of citizens prefer A to win.
The following strategic gamεmodels the citizens' voting decìsions in thìs
situatíon.
Players
τhe citizens.
Actions Each player's set of actions consists of voting for A and voting for B.
Preferences AJl players are indifferent among all actìon profiles in which a majority of players vote for A; all players are also indifferent among all action
pr。在les ìn which a majority of players vote for β. Some players (a majority)
prefεr an action profile of the first type to one of the second type, and thε
others have the reverse preference.
1 cJ aim th剖 a citizer飞's vot放飞g for her less preferred candidate is weakly dominated by her votiηg for her favorite candidate. Suppos仑 that citìzen i prefers candidate A; fix the votes of all citìzens other than i. If citizen i switches from 飞loting for
B to voting for A , then , depεnding on the other citizens' votes , either the outcomε
does not change, or A wins rath位 than B; such a switch cannot cause the wìnner
to change from A to B.τhat ìs , citizen i's switching from voting for B to voting for
A either has no effect on the outcome , or makes her better off; it camlot make her
worse off.
τhe game has Nash equilìbria in which some, or all , citizens' actions are weakly
dominated. For example , the action profile in whìch all citizens vote for B is a Nash
equilibrium (no citizen's switching her vote has any effect on the outcome).
? EXERClSE 48.1 (Voting) Find all the Nash equilibria of the game. (Fi rst consider
action pr。在les in w hich the winner obtains one more 飞!ote than the loser and at least
one citizen who votes for the winner prefers the loser to the winner, then profiles in
which the winner obtair飞s one more vote than the loser and all citizens who vote for
the wirmer prefer the winner to the loser, and finally profìles in which the winner
obtains three or more votes more thar飞 thεloser.) Is there any equi1ìbrium in which
no player usεs a weakly dominated action?
Consider a variant of the game in which the numb衍 of candidates is grεater
than two. A variant of the argument above shows that a citizen's voting for 1飞机
2.9 Dominated actions
4吉
least preferred candidate is weakly dominated by her votíng for her favorite candi翩
date. The next exercise asks you to show that no other action is weakly dominated喃
? EXERCISE 49.1 (Voting between three candidates) Suppose thεre are three candidates, A , B, and C, and no citizen is indifferent between any two of them. A tie for
first place is possible in this case; assume that a citizen who prefers a win by x to a
win by Y ranks a tiεbetween x and y between an outright win for x and an outright
win for y. Show that a citizen's only weakly dominated action is a vote for her least
preferred candidate. Find a Nash 吨uilibrium in which some citi之en does not vote
for her favorite candidate, but the action she takes is not weakly dominated.
7 EXERCISE 49.2 (Approval voting) In the system of "approval voting'二 a citizen
may vote for as m血y candidates as she wishes. lf there are two candidates, say
A and B, for example, a citizen may vote for r飞eithεr candidate, for A for B, or for
both A and B. As before , the 囚犯didate who obtains the most votes wins. Show that
any action that includes a vote for a citizen's least preferred candidate is weakly
dominated , as is any action that does not include a votεfor her most p肥ferred
candidate. More dífficult: show that if there are k candidates, then for a citizen
who prefers candidate 1 to candidate 2 to . .. to candidate k, the action that consists
of votes for candidates 1 and k … 1 is not weakly dominated.
2.9.4
tIIustrotion: co/lectiviε decisior卜making
τhe members of a group of people are aHected by a policy, modeled as a number.
Each person í has a favorite polic予 denoted xi; she prefers the policy y to the
policy z if and only if y is closer to xi than is z. The number nof people is odd
The following mechanism is used to choose a policy: each person names a policy,
ar飞公 the policy chosen is the median of those named. (Th at is, the policies named
are put in order, and the one in the middle is chosen. If, for example, th号re are
five people, and they name the policies … 2， 0 ， 0β ， 5， and 10, then the policy 0.6 is
c且osen.)
What outcome does this mechanism inducε? Does anyone have an incentive
to name her favorite policy, or are people induced to distort their preferences? 有e
can answer these questions by studying the following strategic game.
Players The n people.
Actions Each person's set of actions is the set of policies (numbers).
Preferences Each person i prefers the action profile a to th乒 action pr。我le a' if
and only if the median policy named in a is c10s在 to xi than is the median
policy named in a'.
1 claim that for each play的， the action of narni吨 her favorite policy xj weakly
dorninatεs all her other actions. The reason is that relative to the situation in whlch
she names xi , she can change the me也an only by narning a policy卢Jrther from her
Chapter 2. Nash Equilibrium: Theory
50
favorite policy than 仙台 current median; no change in the policy she names moves
the median closer to her favorite polic沪
Precisely, 1 show that for each action Xi 乒
of player i , (a) for all actions of
the otl赔 playe眈 player i ìs at least as 附11 off naming 咛 as she is namíng Xi ,
and (的 for SO I1H二 actions of the other players she is better off nami吨巧 than she is
nami吨 Xi. 1法 e xz>xj
。.
For any list of actions of the players other than player i, denote the value of
the ~ (n … 1) th highest actio汕 y 旦 and the value of the (11 十 1 )th highest
action by 百 (so that half of the 陀 maining players' actions are at most 经 and
half of them are at least ã).
i
•
If ã 三习。 r 旦主 Xj ， then the median policy is the same wheth时 player í
names xj or Xj.
If互 > x7 and g < XÍt then when player i names
the median po 1icy ís
• 吕 t most the greater of
与 and 旦 and when player i names xi the median
policy is at least the lesser of 工 i and ã. Thus play衍 i is worse off naming
Xi than she is naming
b. Suppose that half of the remaining players name policìes less than xj and
half of them name policies greater than Xi. Then the outcome Îs
nam岱巧， and Xi if she names Xj. Thus she is better off nam妇 g
namm.g Xi'
if player i
than she is
A symmetric argument applies when Xi <
If we think of the mechar飞 ism as asking the players to name their favorite
polici 时， then the result is that telling the truth weakly dominates a11 oth衍 actions.
人双 implicatíon of the fact that player i's naming her favorite policy 巧 weakly
dominates all her other actions is that the action profile in which every player
names her favorite policy is a Nash eguilibrium. That is , truth-tellìng is a Nash
eguilibrium, in the interpretation of the previous paragraph.
Q) EXERCIS 我 50.1 (Other N 吕 5h 叫 uilibria of the game modeling collective decisior子
making) Find two Nash eguilibria in which the outcome is the median favorite
policy, and one in which it is not
? EXERCISE 50 .2 (Another mechanism for collecti飞!e decision 一 making) Consider the
variant of the mechanism for col1ective decision-making i 到 whid飞 the policy cho
sen is the mea l1, rath衍 than the median, of the policies named by the playεrs.
Does a player's action of naming her favorite policy 认reakly dominate all her other
actions?
2.10
Equilibrium in a single population: syr宫lmetric games and
symmetric equilibria
A Nash eguilibrium of a strategic game corresponds to a steady state of an interaction between the members of several populations , one for each player in the
2.10 主quilibrium
in a single population: symmetric games and symm岳重ric equilibria
A
W,W
B
x, y
B
y, X
2， 立
A
Quiet
Fink
Quiet
Fink
2, 2
3, 0
0, 3
1, 1
Stag
Hare
51
Stag
Hare
2, 2
1, 0
0, 1
1, 1
Figure 51.1 The general form 0 1' a t帘。-player symmetrîc g在 me (Ieft), ilnd two cxamples, the PrísO/)(吁
DilC11l1l11l (mìddle) and the tw心-player Stag Hll l1 t (right).
game; each play of the game involves one member of 位ch population. Sometimes
we want to model an ìnteraction in which the members of a single homogeneous
population are involved anonymously and symmetrically. Consider, for example,
pedestrians approaching each oth肝 on a sidewalk or car drivers arriving simultaneously at an intersection from different dìrections. In each case, the members
of an encounter (pairs of pedestrians who meet 柏油 other， groups of car drivers
who simultaneously approach intersections) are drawn from a single population
andhaγe the same ro险，
1 restrict attention here to cases in which each interaction involves two partic叫
ipants. DefiI飞e a two-play佼 game to be 飞ymmetric" if each player has the same
set of actions and each player's evaluation of an outcome depends only on h臼
action and that of her opponent , not on whether she is player 1 or player 2. That
iS t player] feels the same way about the outcome (alta2) , in which her action is
al and her opponen t' s action is {j 2, as player 2 feels about the outcome (旬，叫，如
which her action is al and her oppo口ent's action is a2. In particula几 the play衍s'
preferences may be represented by payoff functions in which both players' payoffs
are the same whenever the players choose the same action: ul (a ， a) 土问 (a， a) for
every action a.
DEFIN ITl ON 5 1. 1 (Symmetric two-player strategic ga仔仔 with ordinal pr咕rences) A
two-player strategic game with ordinal preferences is symmetric if thεplayers'
sets of actions are the samεand the players' preferences are reprεsented by payoff
func悦。ns Ul and U2 for which Ul (肉 ， a2) = U2( 吨 ， al) foreveryactionpair (al , a2)'
A two-player game in which each player has two actions is symmetric ìf the
players' preferences are rεpresented by payoff functions that take the form shown
in the left panel of Fi gure 51 where w , x , y, and z are arbitrary numbers. Several of the two-pìayer games we havεconsidered are symmetric , including the
Priso l1 cr's Dilemma and the two-player Stag Hunt (given again in the middle and
right panels of Fìgure 51. and the game in Exercise 38 立 BoS (Figure 19 .1) and
Maich价g PCHnìes (Figurε19.2) are not symmetric.
n
@EXE汉CISE 51. 2 (Symmetric strategic games) Which of the games in Exercises 31.?
and 42 .1, Example 39 .1, Section 2.8 .4, and Figure 47.2 are symmetric?
认Then the players in a symmetric 队lo-player game are drawn from a single
population, nothing distinguishes one of the players in any given encounter from
the other. 专Ve may call them Ifplayer 1" ar飞d Ifplayer 2" , but these labels are only
for our convenience. There is only one role in the game, so that a steady state
Chapter 2. Nash Equilibrium:τheory
52
ιε卢
Right
Lcft. 1, 1
R首ht
0, 0
0, 0: 1, 1
Figure 52.1 Apprm主chingp告destrìans
is characterized by a singlc action used by every participant whεnever she plays
the game. An actìon a* corresponds to such a steady state if no player can do
better by usìng any other action, given thai a11 the other players use a*. A到 action
a* has 也必 property ìf ar飞d only ìf 作飞 a*) is a Nash 叫uilibrium of the game. 1n
other words; the solution that corresponds to a steady state of pairwise interactions
between the members of a single population is 11 symmetric 法ash equilibrium": a
Nash equilibriurn in which both players take the same action. The idea of this
notion of equilibrium does not depend on the game's havir飞g only two players, so
1 give a definition for a game with any nurnber of players.
DEFINIτION 52.1 (Syl1l1ηctric Nash equilibrium)
An action profile a* in a strategic
game with ordinal preferences ín whích each player has the same set of actions is
as
叼
阴
yF吁I
m
町
哨咄
川n
町附
I
every player i.
As ar飞 example， consider a model of approaching pedestríans. Each participant
in any given encounter has two possible actions-to step to the right, and to step
to the left-and is better off when participants both st叩 in the same direction
than when they step in different directions (in wl飞ich case a collision occurs). The
resulting symmetric strategic game is given in Fi gurε52.1.τhe game has two
symmetric r也sh 叫uilibria， namely (L伊， Lcft) and (Rigllt , Rigllt). That 院 there
are two steady stat咽， in one of which every pedestrian steps to the left as she
approaεhes another pedεstrian， and in another of which both participants step to
theright. (τhe latter steady state seems to prevail in thεUnited States and Canada.)
A symmetric game may have no symmetric 提ash equilibrium. Consider, for
example, the game ín Fi gure 52.2. This game has two Nash equilibria, (X, Y) and
(Y , 蜀， neither of which is symmetric. You may wonder iC in such a situatíon,
there is a steady statεin which each player does not always take the same action
in every interaction.τhís question is addressed in Sectior飞 4.7.
( EXERCISE 52.2 (Equilibrium for pa让阴阳 interactions in a single population) Find
a 1l the Nash equilibria of the game ín Figure 53. 1.有hich of the equilibria, if any,
correspond to a steady state if the game models pairwise interactions between the
members of a single population?
X
FÎgure52.2 人 symmetric
Y
X
0, 0
t1
Y
1, 1
0, 0
game with no symmetric Nash equilibrium
Notes
53
A
B
C
A 1, 1 2, 1 4, 1
B
C
Figur牵 53.1
1, 2
1, 4
5, 5
6, 3
3, 6
0, 0
The game in Exercise 52.2
Notes
Th e notion of a strategic game originated in the work of Borel (1 921) and von Neumann (1928) 必 τhe notion of 到ash equilibrium (and its interpretation) is due to
Nash (1950的. (τhe idea that underlies it goes back at least to Cournot 1838, Ch. 7.)
The Prisoner's Dilemma appears to have first been considered by Melvìn Oresher
and Merrill Fl ood , who used it in an experiment at the RAND Corporation in Ja升
uary 1950 (Fl ood 1958/忡， 11一17); it is an example in Nash's Ph.O. thesis (r也sh
1950树 f submitted in M町 1950. 丁he storγassociated with it ìs due to Tucker (1950)
(see Straffin 1980). O'Neill (1994 , 1010-1013) argues that there is no evidence that
game theory (and iηparticular the Prisoner's Dilemma) influenced U.s. nuclear
strategists i只 the 1950s. The precise analysis of the idea that common property
will be overused was ínitiated by Gordon (1954). Hardin (1968) coined the phrase
μtragedy of the commons气
BoS , like the Prisoner:五 Dilemma ， is an example in Nash's Ph.D. thesis; Luce
and Raiffa (1957, 90-91) name it and associate a story with it. Matching Pe1llúes
was first consid位ed by von Neumann (1928). Rousseau's sentence about hunti鸣
stags is interpretεd as a descriptíon of a game by Ullmaru卜Margalìt(1977， 121)and
Jervis (1977/7弱， followi口g discussion by Waltz (1959 , 167… 169) and Lewis (1969 ,
7, 47)
τhe information about JoOO Nash in the box on page 23 comes from Leor飞ard
(1994) , KuOO et al. (1995) , Kuhn (1996) , Myerson (1996) , r、Jasar (1998) , ar飞d Nash
(1995). Hαwk…Dove is known also as "Chicken" (two drìvers approach each other
on a narrow road; the one who pulls over first is 气hicken"). It was first suggested
(i贝 a more complicated form) as a model of animal conflict by Maynard Smitl飞
and Price (1973). The discussion of focal points in the box on page 32 draws on
Schelling (1960， 5ι58)
Games modeling voluntary contrìbutions to a public good were first consìdered by 01son (1965 , Section 1. 0). The game in Exercise 33 .1 ìs studied ìn detail by
Palfrey and Roser飞thal (1984). The result in Section 2.8 .4 ìs due to Warr (1983) and
Bergstrom, Bl ume, and Varian (1986).
Game theo巧1 was fìrst used to study votir飞gbεhavior by Farquharson (1969)
(whose book was completed in 1958; see also Niemi 1983). The system of "approval voting" in Exercise 49.2 was first studied forma l1 y by Brams and Fishburn (1978, 1983).
Exercise 18.1 is based on Leonard (1990). Exercise 27.2 is based 0
S4
Chapter 2. Nash Equilibrium: Theory
talio, and Beil (1990). 丁he game in Exercise 34.1 is taken from Moulin (1 986b ,
72). The gar口号 in Exercise 34.2 was first studied by Palfrey and Rosenthal (1983).
Exercise 34 .3 is based on Braess (1968); see also Murchland (1 970). The game i旦
Exercise 38.2 is 如ken from Brams, Ki1 gour, and Davis (1993).
Nash Equilibrium: lII ustrations
3.1
3.2
3.3
3.4
3.5
3.6
ζournot's model of oligopoly
55
Bertrand's model of oligopoly 63
日 ectoral competition 70
The War of Attrition 77
Auctions 80
Accident law 91
Prerequisite: Chapter 2
N THIS CHAPTER I discuss in detail a few key models that use the notion of Nash
equilibrium to study economic , political, and biological phe口omena. The discussion shows how the notion of Nash equilibrium improves our understanding of
a wide variety of phenomena. It also i1l ustrates some of the many forms strategic
games and their Nash equilibria can take. The models in Sections 3.1 and 3.2 are
related to each other, whereas those in each of the other sections are independent
of each other.
3.1
(ournot's model of oligopoly
3.1.1
Introduction
How does the outcome of competition among the firms in an industry depend on
the characteristics of the demand for the firms' output, the nature of the firms'
cost functions , and the number of firms? Wi1l the benefits of technological improvements be passed on to consumers? Wi1l a reduction in the number of firms
generate a less desirable outcome? To answer these questions we need a model
of the interaction between firms competing for the business of consumers. In this
section and the next I analyze two such models. Economists refer to them as models of "oligopoly" (competition between a small number of sellers), though they
involve no restriction on the number of firms; the label reflects the strategic interaction they capture. Both models were studied first in the 19th century, before the
notion of Nash equilibrium was formalized for a general strategic game. The first
is due to the economist Coumot (1838).
3.1 .2
General model
A single good is produced by n 自rms. The cost to 自rm i of producing qi units of the
good is Ci(qi) , where Ci is an increasing function (more output is more costly to
Chapter 3. Nash Equilibrium: lII ustrations
5岳
produce). All the OUtput is sold at a single price, determined by the demand for the
good and the firms' tot在 1 outpu t. Specificall如 if the firms' total output is Q, then
the market pri优 is P( Q); P is called the "inverse demand function". Assume that
P is a decreasing function when it is positi飞'e: if the firms' total output increases ,
then the price decreases (unless it is already zero). If the output of each firm i is
% then the price is P(q] 今…十 qιso that fìrm i's revenue is qíP( 们才-" .十 q斗
Thus fírm i's pro丑t， equal to its re飞renue minus ils cost, is
注1( 守 1 ，.. ., ql1) = Qi P (ql 十
十号 11) - Ci(qi)
(56.1)
Cournot suggested that the industry be modeled as the followìng strategìc
game , which l refer to as Coumot's oligopoly game
Players The fìrms
Actions Each firm's set of actions ìs the set of its possible outputs (nonnegative
numbers).
Preferences Each firr叭's prεferences ar巳 represented by íts pro主 t， given in (56.1).
3.1 .3
Example: duopoly with constant unit cost and linear inverse demand function
For specific forms of the functìons Ci 在nd P we can compute a Nash equilibrium
of Courn时 's game. Suppose there are two firms (the industry is a "duopolY")f
each fírm's cost function is the same, gi飞ren by C i ( 村工 cqi for all qì ("unit cosγ' is
constant, equal to 叶， and the inverse demand functiOl飞 is linear where it is positi飞re，
given by
r ([ - Q if ♀三 α
(56.2)
P(Q)=l~;
刊 >i飞
where 在> 0 and c > 0 are constants. This inverse demand function is shm飞rn in
Figure 57二1. (Note that the price P(Q) cannot be equal toα … Q for all values of Q ,
for then it would be negative for Q >队) Assumεthat c <疚， so that there is some
value of total output ♀ for which the market price P( Q) is greater than the firrr飞s'
common 过 nÌt cost c. (If c wεre to exceed α， there would be no output for the firms
at which they could make any profit, because the market price never exceeds í\.'.)
To find the Nash equilibria in this example , we can use the procedure based on
the firms' best response functions (Section 2.8.3). First we need to 自nd the firms'
?在yoffs (profits). lf the firms' outputs are ql and q2 , then the market price P 句 1+
q2) is í\'…们一们 if ql 十号2 ::; x and zero if ql 十但〉队 Thus firm 1's profit is
叫 (qlr q2) 且 ql (P(ql 十号2) … c)
=
{廿 C …叫q们
1 …
一 C吁
们证fq叭1 +q2 > α.
q
Tofiηd firm l's best r泛sponse to any given output 悦。f firm 2, we need to study
firm l's profit as a function of its outp的们 for given values of 号2. If q2 工 0， then
3.1 Courno t's model of oligopoly
57
r\
G
Figure57.1τhe
i.l:
Q •
inverse demand function in the example of Courn仗's game studied in Section 3.13.
fírm 1's profit is 汀1 (ql , O) = q1 (α 而 c qd for q1
乱:， a quadratic function that
is zero when ql = 0 and when 们 =α c. τhis function is the b1ack curve in
Figurε58.1. Given the symmetry of quadratic functions (Section 17.3)，往le output
ql of fírm 1 that maxirr归销出 profit Ìs 们 zj 权… c). (If you know calc山眈
you can reach the same conclusion by setting the derivatíve of firm 1's profit wÌth
respect to qj equal to zero and so1ving for q1') 百1US fírmγs best response to an
output of zero for firm 2 is 11 1 拍)口 j 仅… c)
As the output q2 of firm 2 increases, the profit fírm 1 can obtain at any given
ou 悖的 decreases， because more output of fìrm 2 means a 10wer price.τhe gray
curve in Figure 58.1 is an examp1e of 7[1(q1 , q2) for q2 > 0 and Cj 2 < 夜- c. Again
this function is a quadratic up to the output ql
皮一如 that 1eads to a price of
zero. Specifícall其 the quadratic is 乓忡 1 ， q2) 口们拉… c 一段 - ql) , which is zero
when ql 0 and when 号 1 仪 -c … q2. From th台 symmetry of quadra tic functions
(or some calcu1us) , we conclude that the output th 在t maximizes 叫( ql ， 们) is ql =
扫一 c 一时. (W1:附号2 口 0， this is 吨ua1 to i 扫一叶， the best respon恍如 m
output of zero that we found ìn the previous paragraph.)
队1hen q2 > 浅一 c， the value of lX
c - q2 is negativeτhus for such a value of
q2 , we have 们 (α-c 一 q2 …们) < 0 for a11 posìtive values of q( firm 1's pr。如 is
negative for any positive output, so that its best response is to produεeno output.
We ωnclude that the best response of firm 1 to the output q2 of firm 2 depends
on the value of q2: if q2 至在一 c， thenfirm 1's bεst response is (lX - C… Q2) , whereas
if q2 > α … c， then firm 1's best responsεis O. 。乙 more compactly,
!
匀'ι
门叫，
EEFP
、、l、
，，
1
j'a飞
e飞
j7
{ ~(此一 c - q2) if q2 :::;α-c
。
if q2 > 泛… c.
Because firm 2's cost function is the same as firm 1's; its best rεsponse 如nction
与 is a1so the same: for any number 号， we have b2 传 ) = b 1 (的. Of cour机自rm2's
best response function associates a value of 如m 2's output with every output of
firm 1, whereas firm 1's best response function associates a value of firm 1's out帽
put with every output of firm 2, so we plot them relative to different 以侣. Tney
Chapt母r 3.
58
Nash Equilibrium: lI\ ustration军
•
π1 (q1t q2)
号2=0
//1(:;。\
在一 C
11 一→
\
Figure 58.1 Firm 1's profit as a function of its output, given firm 2'主 outpu t. τhe black curve shows the
case '12 江 0 ， whereas the gr在 y cur\"e shows a case in which q2 > o.
are shown in Figure 59 1 (b 1 is black; b2 is gray). As for a general game (seεSec­
tion 2.8 .3), b1 assocíates each point OI飞 the vertical axis with a point 01飞 the hori懈
zontal axis, and b2 assocíates each point on the horizontal axis with a point on the
vertical axis.
A Nash eq凶 ibriur沪tis a pa让(奸 ， q2) of outputs for which qi 总 a best response
to q2: and q2 is a best response to 号1 .
,
qi = b 1 柯 2)
and
qi
如何n
(see (36 .3))丁he set of such pairs is the set of points at which the best response
functions in Figure 59.1 in佼rsect. From the figure we see that there is exac t1 y one
such point v叶飞ich is given by the solution of the t飞γo equations
号1 -
!(a
…
c - q2)
q2 = i(品:一 C … ql ).
Solving these two equations (by substituti只g the second into于 the first and then
isolatiηg ql , for example) we find that qT 口号j z i 何一 c ).
In summary, when there are two firms, the inverse demand function is given
by P(Q) = 在一 Q for Q 三在， and the cost function of each firm is Ci(qi) = ζqυ
Courno t' s oligopoly game has a ur叫ue Nash equilibrium 句:r 记)口(~扫一 c ),
~ ( LX c) ). The total 01均ut in this equilibrium is ~ (α ←叶， so tha t the price at wruch
01吨ut is sold ìs P( ~归一 c))
~(在今 2c). As ct increases (meaning that consumers
are willing to pay more for thεgood)， the equilibrium price and the output of
出ch firm increase. As c ~the uIÙt cost of production) increases , the output of each
firm falls and the price rises; each unit increase in c leads to a two-thirds of 在 unit
increase in the price.
@ EXERCISE 58.1 (Courno t's duopoly game with linear inverse demand and different
unit costs) Find the Nash 吨uilibrium of Courno t's game when there are two firms ,
the inverse demand function is given by (56.2) , thεcost function of each fîrm i is
3.1 Cournot's model of oligopoly
59
?如
东《4
苟­
r
月时，
n时，
/4 、、
在一 C
)
3
、是g
飞egfv
气4
2
n坷，，
h
雹 u
ù--(
Jgs飞、、
。
I
此
C
q1 •
Fig始 re 59.1 The bcst response functions in Courno t' s duopo!y game when the inverse demand fuηc­
tion is given by (56.2) and the cost function of each firm 巧 cq. The unique Nash equílibrium is
(ql ,1/ 2)
(H在
c), ! (且一训
C i (号 d 口 ciqit where c} >句， and C1 <α(τhere 盯e two cases, depending on
the size of C1 relative to C2.) Which firm produces more output in an 叫uilibrium?
What is the effect of technìcal change that low岔路 firm 2's unit cost C2 (while not
affecting 仕m 1's unit cost C1) on the firms' equilibrium outputs, the total output,
and the price?
? EXERCISE 59 .1 (Courno t' s duopoly game with linear inverse demand and a quadra tic cost function) Find the Nωh 叫ui1ibrium of Courr飞ot' s game when there are
tvvo 在rms ， the inverse demand function is given by (56.2) , and the cost function of
each firm i is Ci(qí) = qT
1n the next exercise each firm's cost functíon has a component that ís ìndependent of outpu t. You wìll find in this casεthatCourn时's game may have more than
one Nash equilibrium
EXERCISE 59.2 (Cournot's duopoly game with linear inverse demand and a fixed
cost) Find the Nash ε年lÍlibria of Cournot's game whe在 there are two fi孜孜s， the
inverse demand function is give只 by (56.2) , and the cost function of eaεh firm i is
given by
Ci(qi)
{~忡。
f + cqi if qi > 0,
where c > 0, f > 0, a时 C< 弘 (Note that the fìxed cost f affects only the fir自由 de­
cision of whether to opεrate; it does not a丘ect the output a firm wishes to producε
if it wishes to operate.)
Chapter 3. Nash 在quilibrium: lII ustrations
60
50 far we have assumed that each firm's objective is to maximize its profi t.
τhe next exercise asks you to consider a case in which one f让m's objective is to
maximize its mad使 t share.
? EXE 议Cl SE 60.1 (Variant of Courn肘's duopo1y game with market懈share心laximizing
firms) Find the 时ash 吨uilíbrium (equilìbria?) of a 飞rariar飞t of the examp1e of Courno t' s duopo1y game that differs from the one in this section (linear inverse demand,
constant unit cost) on1y i口 that one of the two firms chooses its output to maximize
its market share subject to not making a 105s, 1前挝r than to maximize ìts profi t.
What happens if each firm maximizes its market share?
3.1 .4
Properties of Nash equifibrium
τ'wo economically interesting properties of a Nash equilibrium of Courn仗's garhe
C01飞cern the re1ation bet飞veen the firms' equilibrium pro自ts and the profits th叫
cou1d obtain if they acted collusive1y, and the character of an equilibrium when
the number of firms is 1arge.
Co斤lparísorz ofNash equilibriu111 with collusive outcomes In Courn时's gamewith 队'0
firms , i5 there any pair of outputs at which both firm5' profits exceed their 1evels in
a Nash equilibrium?τhe next exercise asks you to show that the answer Ìs "yes" in
the example considered in the p陀飞'ious section (3.1 巧 .5peci自cally， both firms can
increase their profits re1ative to their equi1ibrium 1evεIs by reducing their outputs
‘
Q) EXER Cl SE 60.2 (Nash equilibrium of Courno t' s duopo1y game and collusive out
comes) Find the tota1 output (ca11 it Q 可 that n飞往 ximizes the firms' total profit in
Courn时's game when theτe are two firms and the in飞rerse demτ
η1a
引
汩
ar飞d 阳
fur时
1汇ction飞 and
邸t fun.刀
1只飞ω川akε E出
he
时
ef扣
f orm冶
sa
础附部阳ume
时
d5e仗
$
out甲
pu
时t 趴
il 1 拉
t h飞eN\
\Jas
虱
he
叫
q口山ilibr忖i飞um， ands
仙
ho寺沁
w
￥r 应
tha
挝t each fír
杠rnγse
叫
qui出
i豆ibrium
泯1 pro 缸 i岭s 1eε
岱S
岱
阳li忡ofìti挝只
叫
M
出
t1h飞e'ν
IFωlus
邸肌
纣
S
siv
M
认
弘叫
飞vy吧
e
elF气飞
F气o忧O
削
1饺
m
引1陀
e
e i川hic
削
he
刊
时 fi让
e
怕rm
h
阳口
扛
m
卫叩飞
i岛s 出
t hi沁s collus
刽i飞ve outcome not a Nash 叫uilibrium?
The same is tr时 more generally. For r飞on.linear inverse demand functions and
cost functior毡， thεsh叩es of the firms' best response functions diffe乙 in genera1 ,
from those in the examp1e studied in the pre飞rious sectiol1. But for mar飞y invers台
demand fu日ctions and cost functions , the game has a 队Jash equilibrium and , for
any 叫uìlibrium， there are pairs of ωtputs in 飞^lhich each firm's output is 1ess than
its equilibrium level and each firm's profit exceeds its equilibrium le飞!el.
To see why, suppose that 时 ， qì) is a Nash equìlibrium and consider the set of
pairs (轧 q2) of outputs at which firm l's profit is at least its equilibrium profit
The assumption that P is decreasing (higher total output leads to a lower price)
implies that if (们 ， q2) is in this set and q~ < 啦， then (们 ， q;) is also 如 the se t. (We
have ql 十 qj<ql 十号2 ， arid hence P(如牛毛) > P(q1 十时， so that firm l's profit at
垢，而 exceeds its profit at (ql , q2)') Th us in Figure 61 .1 the set of pairs of outputs
at wruch firm l's pro曲 is a t least ìts 吨uilibrium profit lies on or below the line
q2 q~; an examp1e of such a set is shaded lìght gray. 5imila巾 the set of pairs of
3.1 Cournot's model of oligopoly
61
↑但
Firm 2's profit
告xceeds its
吨uílibrium
leγel
Firm l's p,rofit exceeds
its equilibrìum level
。
ql
ql 叫
Figure 岳 1.1 TI飞e
川
is a Nash equiìíbrium; along each gray curve one of the firm's profits ís
constant, equal to its profit at the equilibrium. The area shaded dark gray is the set of pairs of outputs
atwhiζh both firms' profìts exceed their equilibri泣m leveJs
outputs at which firrn 2's pro缸 is at least its 吨位libri 认rnpro位 líes on or to the left
of the line ql 钊， and an exarnple is shaded light gra 予
We see that if the parts of the boundaries of these sets indicated by the gray
1ines in Figure 61.1 are srnooth, then the two sets rnust intersect; in the figure the
intersectìon is shaded dark gra y. At every pair of outputs in this area, each firrn's
output is less than its equi1ibriurn leve1 (qi < qi for i 1, 2) and each firrn's profit
is highεr than its 吨uilibriurn profi t. That is , both firrns are better off by restricting
their outputs.
Depeηdence 写fl\fash equili衍'ium on number of卢 rms
How does the equilibriurn outcorne in Cournot' s garne depεnd on the nurnb校 of firrns? If each firrn's cost function has the sarne constar飞t unit cost c, the best outcorne for consumers cornpatible
with no firm's rnaking a 10ss has a pric巳 of c and a tota1 output of tt 一 c. The next exercise asks you to show that if, for this cost function, t检 inverse dernand function
is linear (as in Section 3. 1.3) , then the price in the Nash equilibriurn of Courn仗 's
game decreases as the nurnber of firrns increases, approaching c. That
frorn
the 飞riewpoint of consurners , the outcorne is better the larger the nurnber of firms ,
and when the nurnber of firrns is 飞咄 y large, the outcorne is close to the best one
compatible with nonnegative profits for the 如ms.
:1" Ex芜汉CISE 61.1 (Courno t' s game wi也 rnany firrns) Consider Courno t' s garne in
the case õf an arbitrary nurnber n of 如ms; 把tain the assurnptions that the in伽
verse dernand functìon takes the form (56.2) and the cost function of each firm i
is Ci(qi) 口 Cqi for all ql' wit乞 c<α. Find the best 距sponse function of each 如那
and set up the conditions for (吭， . . . ， q~) to be a Nash equilibríurn 书时 (36.3片， assuming that there is a Nash equilibrium in which all firrns' outputs are positive.
Solve 出ese equations to find the Nash equilibriurn. (For n = 2 your answer should
Chapter 3. Nash 主quilibrium: lIIust隐tions
62
be (1(α - C) ， 1(泛
c) ), the equilìbrium found ìn Section 3. 1.3. Fì rst show that in
注n equilibrium all firms produce the same output, then solve for that output.
If
you cannot show that all firms produce the same output, simply assume that they
do.) Find the price at which output is sold in a Nash 吨uilibrium and show that
this price decreases as n increases, approachìng c as the number of firms increases
without bound.
丁he main idea behind this result does not depend on the assumptìons about the
inverse demand function and the 位ms' cost functions. Supposε， more generally,
that the Înverse demand function is any decreasÎl飞g function , that each firm's cost
function is the same, denotεd by C， ar飞d that there is a single output, say q, at which
the averagεcost of production C( q)/q is mi只ima l. ln this case , any given total
output is producεd most efficiently by each firm's producing 小 and the lowest
price compatible with the firms' not making losses is the minimal 飞ralue of the
average cost.丁he next exercise asks you to show that in a Nash equilibrium of
Courno t's gamεin which the firms' total outp时 is large relative to 号， this is the
pricεat which the output is sold
苦 EXERCIS五 62.1 (Nash equilibrium of Cournot' s game with small firms)
Suppose
that there are infinìtely many firms , all of which have the same cost functìon C.
Assume that C( O) = 0, and for q > 0 the fu挝tìon C (q) / q has a unique mìnir时zer
q; denote the minimum of C 问) / q by p. Assume that the inverse demand function
P is decreasing. Show that in any Nash equìlibrium the firms' total output Q 不
satisfies
P( 矿十 q) 三 p :::;; P( ♀勺，
(丁h在t is, the price is at le在st the minimal value p of the a飞罗erage cost, but is close
enough to this minìmum that ìncreasing the total output of the fìrms by q would reduce the price to at most p.) 丁o establish these inequalities, show th时 if P( 矿 )<p
or P( 矿十 q) > p, then Q大 is not the total output of the firms ìn a Nash eq山librium，
because in each case at least onεfirm can deviate and increase its profi t.
3.1.5
A generalization of Coμmot's game: using commo门 property
1n Courno t' s game, the payoff functio烈。f each firm i is QíP(Ql 十. .十 Qn) … Ci(qi)'
In particular, each firm's payoff depends only 0口 its output and the sum of all
the firm's outputs, not on the distribution of the total output among the 在rr邸，
and decreases when this sum increases (given that P Îs decreasing). That is, the
payo丘 of each firm i may be written as !i(qi ， ql 十... +qιwhere the function!i
is decreasing in its second argument (given thεvalue of its first argument, qi)'
This general payoff function captures many situations in which players compete in using a piece of common property whose value to any one player d Îlninìshes as total use ìncreases. 百le property might, for example, be a village green
that is less valuable to any given farmer the higher the total number of sheep that
gra之e on it.
3.2 Bertrand's model of oligopoly
63
The first property of a Nash equilibrium in Cournot's model discussed in Section 3.1 .3 applies to this general model: common property is "overused" in a Nash
equilibrium in the sense that every player's payoff increases when every player reduces her use of the property from its equilibrium leve l. For example , all farmers'
payoffs increase if each farmer reduces her use of the village green from its equilibrium level: in an equilibrium the green is "overgrazed". The argument is the
same as the one illustrated in Figure 61. 1 in the case of two players , because this
argument depends only on the fact that each player's payoff function is smooth
and is decreasing in the other player's action. (In Cournot's model , the 气ommon
property" that is overused is the demand for the good.)
( EXERCISE 63.1 (Interaction among resource users) A group of n firms uses a common resource (a river or a forest , for example) to produce outpu t. As the total
amount of the reSO Uf('e used by all firms increases , any given firm can produce less
outpu t. Denote by Xi the amount of the resource used by firm i (= 1,..., 11). Assume speCl自cally that firm i's output is xj(l 一 (Xl + . . . + .l ll)) if Xl + . . . + X I1 三 1 ，
and zero othenvise. Each firm i chooses .l j to maximize its outpu t. Formulate this
situation as a strategic game. Find the Nash equilibria of the game. Find an ac
tion profile (町， . . ., X Il ) at which each firm's output is higher than it is at any Nash
equilibrium.
3.2
Bertrand's model of oligopoly
3.2.1
General mode!
In Cournot's game, each firm chooses an output; the price is deterIIÙned by the
demand for the good in relation to the total output produced. In an alternative
model of oligopoly, associated with a review of Cournot's book by Bertrand (1883) ,
each firm chooses a price , and produces enough output to meet the demand it
faces , given the prices chosen by all the firms. The model is designed to shed light
on the same questions that Cournot's game addresses; as we shall see, some of the
answers it gives are differen t.
The economic setting for the model is similar to that for Cournot's game. A
single good is produced by 11 firms; each firm can produce q units of the good at
a cost of Ci(qJ It is convenient to specify demand by giving a "demand flllLction"
0 , rather than an inverse demand function as we did for Cournot' s game. The
interpretation of 0 is that if the good is available at the price p, then the total
amount demanded is 0 (p).
Assume that if the firms set different prices , then all consumers purchase the
good from the firm with the lowest price , which produces enough output to meet
this demand. If more than one firm sets the lowest price, all the firms doing so
share the demand at that price equally. A firm whose price is not the lowest price
receives no demand and produces no output. (Note that a firm does not choose its
output strategically; it simply produces enough to satisfy all the demand it faces ,
,
Chapter 3. Na军h 革quilibrium: lII ustrations
64
在iven the prices , even if its price is below its unit cost , in which case it makes a
loss‘百lÌs assumption can be modified at the price of complìcating the model.)
1n summary, Bertrand's oligopoly game is the following strategic game
Players The firms.
Actions Each firm's set of actions is the set of possible prices (nonnegative
numbers).
Prεferences
日rm j's preferencεs are repτesented by its profit which is equal to
PiD(pi)/m … C1( D (p 1) / m) if firm i is one of m firms setting the lowest price
(m
1 ìf firm i's price Pí is lower than every other price) , and equal to zero
if some firm's price is lower than Pí.
3 .2.2
Example: duopoly with constant unit cost and linear demand flυnction
Suppose, as in Section 3. 1.3, that there are t认10 firms , each of whose cost function
has constant unit cost c (that is , Ci(qi) cqí for Î = 1, 2). Assume that the demand
fu配 tion is D(p) = α - p for p 三 αand D( 抖出 o for p > 底， and that c <在
B伫cause the cost of producing each unit is the same, equal to c, firm i makes the
profit of Pi - c on every unît it sells τhus íts profit is
( (Pí- C )( α … p iJ
πí(Pl ， P2) 口 {j(12i-c)(α
l0
Pi)
if Pí < Pj
if Pi = Pj
if Pl > Pj'
where j is the other firm (j 口 2ifi=l ， andj 1ifi 2).
As befor毛， we can find the Nash equilibrìa of the game by finding the firms'
best response functíons. If firm j charges Pj' what is the best price for firm i to
charge? We can reason informally as follows. If firm i charges Pi , it shares the
market with firm j; ìf it charges slightly less, it sells to the entire marke t. Thus if Pj
exceeds c, so that firm Î mak岱 a positive profit selling the good at a price slìghtly
below Pj' firm i ís defínite\y better off ser飞IÌng all the market at such a price than
serving half of the market 在 t the price Pj制If Pj is very high, however二位rmimaybe
able to do evεn better: by reducing its price significantly be\ow Pj it may increase
its profit, because the extra demand en萨nderεd by the lower price may more than
compensatεfor the lower revenue per unit sold. Finall予 if Pj is less than c, then
firm i's profit is negative ìf it charges a price less than or equal to Pj' whereas this
profìt is zero ìf it charges a hìgl恕r price.τhusiηthis case fír双1 i would like to charge
a到y price greater than Pj' to make sure that it gets no customers. (Remember that
if customers arrive at its door it is obliged to serve them, regardless of whether it
makes a pr。在t by so doing.)
We can make these arguments precise by studying firm i's payoff as a function
of its price Pi for various values of the price Pj of firm j. Denote by p1n the value
of P (price) tha t maximizes (p c) (IX - p). Tnis price would be charged by a firm
with a monopoly of the market (beca出e(p-c)( α- p) is the profit of such a firm)
3.2 Bertrand's model of oligopoly
65
↑
i
汀
l[
i
πi
O Pof白
.
。
(
p"
。
ι
Pi 1)111
ìì:
pm
在
。--­
Pí
在
PI •
Pj < C
C
Pj> pm
< Pj :S: P川
Figur号岳5.1 T头 ree cross sections (ín black) of fìrm Is payoff fuηction in Bertrand's duopolv game
Where the payoff functío只 jump'>， its value ís gìv芒n
七εsrτlall disk; the small circles ind 汇在 te poìnts
that are excluded 占5γ在 lues of the functions.
Three cross sections of firm i's payoff function , for different values of Pj' are shown
in black in Figure 65. 1. (The gray dashed line is the function (Pi - c)(在一 pJ)
• If Pj < c (firm j' s price is below the unit cost) , the口 firm i's profit ìs negative
if Pi <二 Pj and zεro if Pi > Pj (see the left panel of Fìgure 65.1). Thus any
price greater than Pj is a best respoηse to Pj. That is, the set of firm i's
responses is Bi(pj) = {Pi: Pi > pj}.
• If Pi 口 c， then the analysis is similar to that of the previous case except that
Pj , as well as any price greater than 町， yields a profit of zero, and hence is a
best response to pr Bi(pj)
{Pi: Pi 主 Pj}
• If c < Pj 三 p111 ， then firm i's profit increases as Pí ìncreases to 酌， then drops
abruptly at Pi (see the middle panel of Fìgure 65.1). Thus there ìs no best
response: 主rm i wants to choose a price less thar飞抖， but is better off the
closer that price is to Pj- For any prìce less than Pj there is a higher price that
ìs also less than Pj , so there is no best price. (1 have assumed that a firm εan
choose any number as its price; in particula乙 it is not restricted to charge an
integral number of cents.) Thus Bí(pj) is empty (has no members).
• If Pi > pnl , then pm is the unique best response of firm i (see the right panel
of Figure 65. 曰: Bí(pj) 口 {p m }.
1nsummar予 firm
best response fur飞ction 点 given by
nι
户
ι-'f
，
4A
制1·1·1·
正 i
附
FJt
〈一一 'J<
psnv'cp'
。
114
‘1
正 ArφA
、 ga，、 B 飞gEF
p' 刀'
…一
《trJLFU 比
、a
r
，
EE
EB
eegzzt
E
J飞、
、Et
刊Va
、h!''
d'
』'飞、
nnu
>>…
ppjp
pplf
-r.rm
ll
<-nm
nr
where ø dεn飞ot拮e部s the set with 只r飞omi
丑1em
丑1b
怡
ers (the 写mηlp
严ty se
仕t"
which 白
thi总sb
快
es
纣t res印?严
ons
飞沁
龄
S
ε 妇E挝
lct柱io
∞
nd
沓if险
t轮e
假挝
主郑sq飞圳
i泌
al运it城在挝均
t挝i扣V
咔
毛r宅ely 企
from
主丑丑1卢 a fir口rr泯l'S best respor邱
1怒
se
functi∞ inCoum时， s game: for some actions of its opponer役， a firm has no best
response, and for some actions it has multiple best responses.
Chapter 3. Na古h 在quilibrium: lIIustrations
66
P2
P2
pm:
pm I
o
B2(Pl)
B1 (P2)
C
C
C
pnI
Pl 叫
O
C
pnl Pl •
Figure 66.1 The firms' best resp∞ se functions in B告 rtrand's dηopoly g吕 me. Firm 1'5 best response
function i5 in the left panel; fìrm 2'5 is in th告 rig沁问我el.
τhe fact that fírrr飞 i has 110 best response when c < Pj < pm is an artifact of
modeling price as a continuous variable (a firm can choose its price to be any nonnegative number). ff instead we assume that each firm's price must be a multiple of
some indivisible unit E (e.g. price must be an integra1 number of cent坊， then firm i' s
pn1 is Pj 一 ε. 1 model price as a corν
optimal response to a price Pj with c < Pj
tinuous variab1e because doing so simp1ifies some of the analysis; in
67.2
you are asked to study the case of discrete
When Pi
c, firm i's set of best responses is the set of a11 prices great时 than
pi- In partiε飞11ar， pricεs between Pj and c are best rεsponses‘ You may object that
setting a price
than c is not very sensib1e. Such a price exposes firm i to the
risk of posting a 10ss (if firm j chooses a higher price) and has no advantage over
the price of c, 陀 gard1ess of firm j' s price. That is, such a price is weakly dominated
(Definition 46.1) by the price c. Nevertheless, such a price is a best response!τhat
is, it is optimal for firm ì to choose such a price, give l1 firm j' s price: there is no price
that yields firm i a high凹 profi七 given firm j' s price. 丁i飞εpoint is that when asking
if a p1ayer' s actíon Îs a best responsεto her opponenγs action,- we do not consider
the "risk" that the opponent will take some other action.
Figure 66.1 shows the firms' best response functions (firm l's on the left, firm 2's
on the right). The shaded gray area in the lert panel indicates that for a price P2 less
than c, any price greater than P2 is a best response for firm 1. The absence of a b1ack
line a10ng the s10ping 1eft boundary of this area indicates that only prices Pl greater
than (not equal to) P2 are included. The black line a10ng the top of the area indicates
that for P2 = c any price greater than or equal to c is a best response. As before, the
dot indicates a point that is included , whereas the smal1 circle indicates a poir飞t that
is exduded. Firm 2's best response functionhas a similar interpretation.
A Nash equilibri飞1m is a pair (芮 ， P;J of prices such that pr is a best response to
P2' and P2 is a best response to PÎ -that 也 pi is in B1(月) and pì is in B2(p j')
(36.2)). If we superimpose the two best response functions, any such pair is in the
intersection of their graphs. If you do so , you will see that the graphs have a sing1e
3.2 Bertrand's model of oligopoly
67
point of intersection, namely (町，芮
(c， c). That is, the game has a si鸣le r地-Jash
equilibrìum , in which each firm charges the price c.
The method of finding the Nash equilibria of a game by constructing the players' best response functions is systemati c. As long as these functions may b巳 com­
puted , the method straightforwardly leads to the set of Nash e司uilibria. Ho耳听ver，
in some games we can make a direct argument that avoids the need to construct
the entire best response functions. Using a cornbination of intuition and trial and
error, we 住址 the action profiles that seem to be equilibria; then we sl飞ow precisely
that any such profile is an equilibrium and every other profile is not an equilibnum.τo show that a pair of actions is not a Nash equilibrium we need only find a
better response for one of the players… not necessarily the best resp∞se.
In Bertrand's g é\ me we can argue as follows. 的 First we show that (Pl , P2) =
(c , c) is a 刊出h 叫uilibrium. If one firm cha丐es the price c, then the other firm
can do no better th汶口 charge the prìce c also, because if it raises its price ìt sells no
output, ar飞d if it lowers its price it posts a loss. (ii) r过εxt we show that no other pair
(Pl , P2) is a Nash equilibrium, as follows.
• If Pi < c for either i
1 or í 口 2， then the profit of thεfirm whose p对ζe is
lowest (or the profit of both fir出s， if the prices are the sarne) is nega拍电 and
this firm can increase its pro缸 (to zero) by raising its price to c.
,
c and P > c, then firm i is better off increasing its price slightly,
rnaking its profit positive rather than zero.
• If Pi
• If Pi > c and Pj > c, suppose that Pi ;:: Pj. 在飞机飞 firm í car飞 Íncrease its profit
l1l
by lowering Pi to slightly below P if D(pj) > 0 (i.e. if P，
时 and to p if
D( 白) = 0 (i.e. if Pj 主 a)
,
In conclusiOI飞， both arguments show that when the unitωst of production is a
constant c, the same for both firms , and demar飞d is linear, Bertrand' s game has a
unique t也 sh 叫 uilibrium， in which each firm's price is equal to c葡
( EXERcrSE 67.1 (Bertrand's duopoly garne with COI飞stant unit cost) Consider the
εxtent to which the analysis depends upon the demand function 0 taking the specific form 0 ( 纣 :α … ρSuppose that D is a巧 function for which O(p) 三 o for
a11 P and there exists p > c such that 0( 1') > 0 for a11 l'主乒 Is (c , c) still a Nash
equilibrium? Is it still the only Nash e守tilíbrium?
:1. EXERCISE 67.2 很ertrand's duopoly game with discrete prices) Consider the variant of the example of Bertrand's duopoly garne in this section in which each firrn is
restrict
3.2.3 Discussíon
For a duopoly in wmch both firms have the same constant unit cost and the demand function is linearr the Nash 叫uilibria of Coumo t' s and Bertrand's gam出
68
Chapter3. N事sh Equìlibrium: lIIustrations
generate different economic outcomes. The 叫uilibrium price in Bertrand's game
is egual to the common unit cost c, whereas the price associated with the equi1 ibrium of Cournot's game is j (α 十 2叶， whiche附eds Cbeca附 c < lt. 1叩articula巳
thεequi1ibrium price in Bertrand's game is thεlowest price compatible with the
firms' not posting losses , whereas the price at the equ i1ìbrium of Courno t's game
is higher. In Cournot' s gam芒， the price decreases toward c as the number of firms
íncreases (Exercise 61.刀， whereas in B盯trand's game it is c even if there are only
two firms. 1n the next exerci使 vou are asked to show that as the number of fírms
increas部 in Bertrand's game , the price remains c.
① Ex 照CISE 68.1 (Bertrand's oligopoly game) Consider Bertrand's oligopoly game
when thεcost and demand functions satisfy the conditions in Section 3.2.2 and
there are ηfirms ， with n 二~ 3. Show 血
tha
挝t the set of N
证a
邵
sh equí出
i挂ib
坟ri归
a 沁
i s the set of
严rofiles (p
P
内1 ，...， 扣
P n) of 严
p nce
部s for 飞w
矿N'l
企挝}
仅l严
ua
挝1 怡
t o c. (Show that any such profile is a Nasl飞 equilibrium， and that every other
profile is 只ot a Nash equilibrium.)
What accounts for the differencεbetween the Nash equ i1ibria of Courno t's and
Bertrand's games?τhe key point is that different strategic variables (output 加
Cournot' s game , price in 如rtral址 's gam功 imply different types of strategic reasoning by the firms. ln Courn时's game a firm 乞hanges its beha飞rior if it can increase
its profit by changing its output , on the assumptìon that the other firms' outputs
will remain the same and the price w il\ adjust to c1 ear thοmark性 In Bertrand's
game a firm changes its bεha飞rior if it can increase its profit by chan苍白g its price ,
on the assumption that the other firms' prices w il1 remain the same and their 0时，
puts will adjust to c1 ear th巳 market. Which assumption makes more sense depends
on the contex t. For example , the wholesale market for agricultural produce may 在t
Cournot's game better, whereas the ret在il market for food may 白t Bertrand' s game
betì号:
Under some variants of the assump柱。ns in the prevíous section (3.2.号， Bertrand's game has no Nash 吨uilibrium. 1n one case the firms'ωst functions have
constant unit costs, and these costs are different; in another case the cost functions
have a fixed cornpor飞en t. ln both these cases , as wel1 as in some other cases, an
e气uilíbri口m is rεstored if we modify the way in which consumers are divided between the firms \vhen the prices ar'εthe same, as the follovving exercis仍 show.
(We can think of the di飞rision of consumers between firms charging thεsame price
as being det衍mined as part of the e气uilibrium. Note that we retain the assumption that if the firms charge different prices, then the one charging the lower price
陀ceives a11 the demand.)
(V Ex 狂 RCIS 巨 68.2 (Bertrand's duopoly game with different unit costs) Consider 苔仔
trand's duopoly game under a variant of the assumptions of Section 3.2.2 in which
the firms' unit costs are di迁ff坠er陀e创
阳I眈
1
t仕he
巳 pric优
e 位
tha
挝t ma拟川
刘imi泣
X
ze
臼s (p 一 C1ρ)(归
a 一p
抖)， and assume 白
tha
挝t 白
c 2 < Pl ar时 t出
ha
挝t the
function (p … C1 )怡'一 p) is increasi吨 ir
3.2 B曹公限稳 d's model of oligopoly
69
a. Suppose that the rule for splitting Up consumers when the prices are equal
assigns a1l consumers to firm 1 when both firms char伊 the pricε C2. Show
that (阳，向) = (C2 , C2) is a Nash equì苔brium and that 丑。 other pair of pric出
is a r‘\l ash equilibrium.
b. Show that no Nash equilibrium exists if the rule for splitting up consumers
when the prices are equal assigns some consumers to firm 2 when both firms
charge c2.
7í EXERCISE 69.1 (Bertrand's duopoly game with fixed costs) Consider Bertrand's
game under a variant of the assumptions of S时 tion 3.2.2 in which the cost function
f + cqi for qí > 0, and Cj(O) = 0, where f is
of each firm i is given by Ci(qj)
positive and less tl:tan the maximum of 印… c) 何一 p) with respect to p. Denote
by 15 the price p that satisfies (p …c) (a:… p) f and is less than the maxÎmizer of
(p - c) 扫一抖 (see Figure 69.1). Show that if firm 1 gets a11 the dema时 when both
firms charge the same price, then 币， 15) is a Nash equilibrium. Show also that no
other pair of prices ìs a 1飞-Iash e弓uìlibrìum. (Fi rst consider cases in which tl有如ms
charge the same price, then cases in which they charge different prices.)
(p - c)(α - p)
。
Fi警挝 re 69.1
C
p
αp …→
The determination of the price p in Exercíse 69 .1.
COURNOT, BERTRAND，人民10 NASH: SOME HISτORICAL NOTES
Associating the names of Cournot and Bertrand with the strat心萨那部 in Sections 3.1 and 3.2 invites two conclusions. First, that Cournot, writing in the first
half of the 19th century, developed the concept of Nash equilibrium in the contεXÌ
of a model of oligopoly. Second, that Bertrand, dissatisfied with Courno t's game,
proposed an alternative model in which price rather than output is the strategic
variable. On both points the history is much less straightforward.
Cournot presented his "equilibrium" as the outcorne of a dynamic adjustment
process in which, in the case of two firms, the firms alternately choose best responses to each other's outputs. During such an adjustment process , each 自rm，
when choosing an output, acts on the assumption that the other finn's output will
remain the same, an assumptìon shown to be incorrect when the other firm subse-
Ch撞pt最r 3.
70
Nash Equilibrium: lIIustrations
quently adjusts its outpUt. The fact that the adjustment process rests on the firms'
acting 0口 assumptions consta时ly show口 to be false was the subject of criticism in a
leading presentation of Cournot' s model (Fe丑ner 1949) available at the time Nash
was developing his idea.
Certainly 妇在 sh did not literally generalize Courno t' s idea: the e飞ridence suggests that he was completely unaware of Courn时's work when developing the
notion of Nash equi1ibrium (Leonard 1994, 502… 503). ln fact, only graduall予
as Nash's work was absorbed into mainstream economic theory, was Cournot's
solution ínterpreted as a Nash equilibrium (Leor飞ard 1994, 507斗。坷，
The association of the price唰setting model with Bertrand (a mathematician)
rests on a paragraph he wrote in 1883 in a review of Cournot' s book. (The book,
P泣bli沁shed iη1838， had previously be
优
εn飞 largely ign飞ore
仅d.) The 1
Ber泣tr拮
ar负1d 岭
í s unde
町r 位
thε ìm
吉η1pr陀
essioI口飞 t出
ha
挝t 站
i η Co飞urno
时t's m
丑:lodel the firms cωom
孜1pe
仕te in
pr
刘ic
优
es
孔， undercuttìng each other to attract more business! He argues that the陀 is
"no solution" because there is no limit to the fall in prices , a rεsult he says that
Cournot' s formulation conc出ls (Bertrand 1883, 503). ln brief, Bertrand's understanding of Cournot's work is t1awed; he sees that price competition leads each
firm to undercut the other, but hís conclusion about the outcome is incorrect.
Th rough the lens of modern game theory we see that the models associated
with Cournot and Bertr汶口d are strategic games that differ only in the strategic
variable, thεsolution in both cases being a Nash eq山 librium. Ur飞til Nash's work,
the picture was much murkier.
3.3
在lectoral competition
What factors determine the numb时 of political parties and the policies they propose? How is the outcome of an election affected by the electoral system and the
voters' preferences among policies? A model that is the foundatiol飞 for many theories of political phenomena addresses these questions. In themodel , each of several candidates chooses a policy; each citizen has preferences over policies and
votes for one of the candidatεs.
A simpl台 version of this model is a strategic game in which tl飞e players are tÌ1 e
candidates and a policy is a numbe几 referred to as a "position". (The compression
of a11 policy differences i口 to one dimension is a major abstraction , though political positions are ofter飞 categorized on a left-right axis.) After the candidatεshave
chosen positions, each or a set of citizens votes (nonstrategica l1y) for the candidate
whose position she likes best. The candidate who obtains th较 most votes wins.
Each candidate cares only about winning; no candidate has an ideological attach喘
ment to any position. Specificall予 each candidate prefers to win than to tie for first
place (in which case pεrhaps the winner is determinεd randoml抖， and prefers to
tie for first place than to lose; if she tiεs for first placε， she prefers to do so wÎth as
few othεr candidates as possible.
3.3 Electoral competition
71
- k
x*
x* + k
x •
Fig￥ re 7 1.1 τhe payoff of a voter whose fa飞'orite position is x 袍， as a function of the winning position , x.
There is a continuum of voters , each with a favorite position. The distribution
of these favorite positions over the set of all possible positions is arbitrary. In partic飞11ar， this distribution may not be uniform: a large fraction of the voters may
hav伫 favorite positions close to one point while few voters have favorite positions
close to some other poin t. A positioηthat turns out to have special significance is
the median favorite position: the position m with the property that exactly half of
the vo始rs' favorite positíons are at most n1, and half of thεvoters' favorite positions ar它 at least m. (1 assurr飞εthat the distribution of favorite positions ìs such th时
there is only one position with this property.)
Each voter's distaste for any posi挝on is given by the distance between that
position and her favorite position. 1n particula几 for any value of k, a voter whose
favorìte positio丑 is x* is indifferent betw时n the positions x* … k and x* 十 k. (Refer
to Figure 71. 1.)
Under this assumption , each candidate attracts the \'otεs of a11 citizens whosε
favorite positions are closer to her position than to thεposition of any other candidate. An example is shown in Figure 71.2. In this example there a陀 three candidates, with positions X} , X2 , and x3 Candidate 1 attracts the votes of every citizen
whose favorite position is in the interval , labeled "Votes for 1" , up to the midpoint
~ (Xl 十 X2) of the line segme时 from Xl 始工2; candidate 2 attrac总 the votes of ev航
向T citizεn whose favorite positio们s in the interval from ~ (工1 十 X2)to~(月十 X3);
and candidate 3 attracts the remaining votes. 1 assume th挝 citizens whose favorite
position is ~(Xl 十 X2) dividεtheir votes equally between candidates 1 and 2, and
those whose favorite position is ~ (巧十巧) divide their votes 叫旧llyb时weencan­
didates 2 and 3. If two or morεcandidates take the same position, then they share
equally the votes that the position attracts.
In summary, 1 consideτthεfollowing strategic game , which , in honor of its
originato乙 1 call Hotelling's model of electoral competition.
‘
Players
在le candidates.
Actions Each candidate's sεt of actions is the set of positions (numbers).
工1:{xlf X2)
工2j(X27X3)X3
←一 Votes for 1 叫←- Votes for 2 一斗扣一 Votes for 3 一→
Figure 71.2 The allocation of votes bet时een three candidates, with posi位。而 句，巧， and x3.
Chapter 3. Nash Equilibrium: lII ustrations
72
Preferen但5
Each candidate's preferences arεrepresented by a payoff function
that assigns n to every action pro白le in which she wins outright, k to every
action profile in which she ties for first place with n - k other candidates
(for 1 :二 k 三 n-l) ， andOtoεvery action profiìe in which she loses , where
positions attract votes in the way described in the previous paragraph.
Suppose there are two candidates. We can find a 到ash equilibrium of the game
by studying the players' best response functions. Fix the position X2 of candidate 2
and consider the best position for candidate 1. First suppose that 工2 < m. If
candìdate 1 takes a posìtion to the left of X2 , then candidate 2 attracts the votes of
allc阳enswho肥 favorite positions a陀 to the 句htoq(Xl 十对， a 时t that indudes
the 50% of ci tÌzens whose favoríte posítions are to the rìght of m, and more. Thus
candidate 2 wins, and candidate 110ses. If car飞dída te 1 takes a position to the right
of 巧， then she wins as long as the dividing line between her supporters and those
of candidate 2 ìs less than m (see Figure 72.1). lf she ís so far to the right that this
divìding line lies to the rìght of m, then she loses. She prefers to win than to lose,
and is indifferer飞t between a11 the outcomes in which she wins , so hεr set of best
responses to x2 is the set of positions that causes the midpoi时 !(X} 十工2) of the
line segment from X2 to x} to be less than m. (If this midpoint is equaI to 111 , then
the candidates tie.) The condition (x} 个工2) < m is equivalent to X} < 2m x
soca
肌
Iηldidat沁台 1'、s set ofb
加
es纣t 珩
res印
po
∞口川S附
仍st怜
e
O 工巳2 is the set of all
挂1 po
仍sì凶
t挝ìon飞s be
创twε
优
en飞 X2 ar叭飞d
2m
ηI 一 X2 位
( excζcludi拉
r只19 the poi出
nt怡S 工均2 and 2m
η 町…叩 工ú
A symmetric argument applies to the case ìn which x2 > 衍. ln this case candidate 1's s纪 t of best respor飞ses to x2 is the set of a11 positions between 2m ←与
and X2 曹
Finally consider the case ín which 句= 111. 11飞 this εase candídate 1's unique
best response is to choose the same position, m! If shεchoosεs any other posìtion,
then she loses , whereas if she choo优S 11l, then she tíes for first place.
ln summary, candidate 1's best response function is defined by
!
<口>
••
，$豆
‘，
毛t
qfh
叫/』肉，&气，待
-/\
EI--
。，ι匀，岛
" .，可飞，‘，
、-4，
噜
eEAB
立
X
、飞IJ
/\…
叮ι
'1·1·l
2x
<
/\ XX
xnx
XXX
71
177
111
zn
lf
l
x
ra飞飞 fdtf
一-
JI--飞F2tt 飞
a飞、
1
x 1/
叫fh
口nu
,d
•••
d
Car飞didate 2 faces exactly the same ìncentives as candidate 1, and hence has the
same best response function. The candidates' best response functions are shown
in Figure 73. 1.
If you superimpose the two best response functions , you wíll see that the gan飞e
has a uni年le Nash equilibrium, in which both candídates choose the position n1,
m
X
τZBA
•- Votes for 2 一叶4
x}
ι
饨，
巧，阳
( x-
十 il
1·
工2
Votes for 1
figure 72.1 An action pro自 le (.1'],.1'2) for which candìdate 1 wì口S.
3.3 Electoral competition
•
73
。
O
工2
X2
B 1 (工2)
m
•
m
BZ(Xl)
m
Xl …•
figure 73.1 The candidates' 忱st response functions i到 Hotelling's model of 21ectoral competitio烈的 th
two candidates. Candidate }'s best response function is in the left panel; candidate 2's is in the right
pane J. (The edges of the shaded areas are εxcluded.)
the 飞悦ers'm付出n favorite position. (去emember that the edges of the shaded area ,
which correspond to pairs of positions that result in ties, are excluded from the best
response functions.) The outcome is that the election is a tie.
As in the case of Bertrand's duopoly game in the previous section, we can make
a direct argument that (m , m) is the unique Nash equilibrium of the gam己 without
constructing the best response functio肌 Fi烛， (In, In) is an equilibrium: it re如
sults in a tie , and if either candidate chooses a position different from n1, then she
loses. Second, no other pair of positions is a Nash 叫uilibrium， by the following
argumen t.
• lf one candidate loses, then she can do better by moving to n1, where she
eithεr wins outright (if her opponen t' s position is different from m) or ties
for first place (if her opponen t' s position is 111).
• If the candidates tie (because their positions are either the same or symmetric
about 111) , theηeither candidate can do better by moving to n1, where she wins
outright.
Our conclusion is that the competition between the candidates to secure a ma如
jority of the votes drives them to select the same positiOl1, equal to the median of
the citizens' favorìte positions. 日otelling (1929, 54t the originator of the mode l,
writes that this outcome is 飞trikingly exempli在ed." He continues, "The competition for votes between the Republican and Democratic parties [in the Urdted States]
does not lead to a c1 ear drawing of issues, an adoption of 们年o strongly contrasted
positions bεtw时n which the voter may choose. Instead , each party strives to make
its platform as much lì始 the other's as possible."
(jj) EXERCISE 73.1 (Electoral competition with asymmetric voters' preferences) Consider a variant of Hotelling's model in which voters' prefεrences are asymmetric.
Speci直cally， suppose that each 飞roter cares twice as much about policy differences
74
。1apter 3.
Nash Equìlibrium: lIIustrations
to thεi己ft of her favorite positìon than about polìcy diffεrences to the rìght of her
favorite position. Hmv does this aft，εct the Nash equilibrium?
1n the model considered so far, no candidate has the option of staying Ollt of the
race. SllppOSe that vve give each candidate thìs optio陀 assllme that it is better than
losÍl飞g and worse than tying for first plac巳 Thm the Nash eqllilibrillm remains as
before: both players enter the race and choose the position m. The direct argument
differs from the one bεfore only ìn that in addition we need to check that there is
no equilibr沁m ìn which one or both of the candidates stay out of the race. If one
candidate stays Ollt , then, given the other candìdate's position, she can enter and
心 ither win outright or tie for first place. If both candidates stay out , then either
candidate can enter ar飞d 认'in outright.
τhe next exercise asks you to consíder the Nash equilìbria of this variant of the
model when there are th陀e candidates.
? EXERCISE 74.1 (Electoral competition with threεcandidates) Consider a variant
。f Hotelling' s model in 飞Nhich there are three candìdates ar飞 d each car吐 idate has
the option of staying out of the race , 'which she regards as betteτthan losìng and
worse tl冶n tyíng for fírst place. Show that if less than one-third of the citìzens'
favorite positions are 吨ual to the median favorite positio巳 (111) ，泣len the game has
no Nash 吨uilibrium. Argue as follows. Fìrst , show that the gamεhas no Nash
equìlibrium in \^.'hich a single candidatεenters the race. Second, show that in any
队Jash equilibrium in which more than one candidate enters , a11 candidates that
entεr tie for first place. Third , show that there is no Nash equilibrium in whìch
two candidates entεr the race. Fourth , show that there is no Nash 叫uilibrìum in
which all three candidates enter the race and choose the same position. Finally,
show that there ìs no Nash equilibrium in which all three candidates enter the race
and do not all choose the sal我e positìon.
?t EXERCISE 74.2 (U
.s. presidential election) Consider a variant of Hotelling's model
that captures features of a U .5. presidential election. Voters are di飞'ided between
two states. State 1 has mo陀 elect时al college votes than does state 2. 丁he winner is thεcandidate \'\'ho obtaÌns the most electoral coUegε\'otes. Denote by 111 i
the median favorite position among the citizens of state Î, for i 二 1 ， 2; assume
that 1112 < 11l!. Each of two candidates chooses a single position. Each citizen
votes (nonstrategically) for the candidate whose position is closest to her favorite
posítíon. The candidate who wins a majority of the votes in a state obtains all
the electoral college votes of that state; if for some state the candidates obtain the
same number of 飞'otes， they each obtain half of the electoral college votes of that
state. Fí l} d the Nash 叫uilibrium (equilibria?) of the strategic game that models
this sìtuation.
50 far we have assumed that the candidates care only abollt winning; they are
not at all concerned with the wìnn肘 's positio孔 The next exerds仑 asks yOll to
cor飞sider the case in whìch each candidate cares only about the winner's position ,
and not at a11 about winning. (You may be surprised by the equilibrium.)
3.3 日曹ctoral
competition
7S
育 EXERClSE 75 .1 (Electoral competitíon between candìdatεs who care only about the
wìnnìng positíon) Consider the variant of Hotelling's model in which the candidates (like the citízens) care about the winn衍 's posìtio日， and 1飞ot at all about
wìnning per se. There are two candidates. Ead飞 candídate has a favorite positio日; her dis1ike for other positions increases with their distancεfrom her favorite
positìon. Assume that the favorìte positìon of one candìdate is less than m and
the favorite positìon of the other candidate ís greater than n1. Assume also that
ìf the candidates tìe when they take thεpositions X1 and 工2 ， then the outcome is
the compromise polìcy ~(X1 十工2)' Fìnd 泣le set of Nash equílibria of the strategic
game that models this situa tÌ on. 在irst consider pairs (巧，均) of positior飞sforwhich
either X1 < m and X2 < 盹 or Xl > m and X2 > 况 Next considεr pairs (X1 , X2)
for which eìtl飞时 X} < m < X2 , or X2 < m < X} , then those for which x} = m ar飞d
X2 并 m ， or X} 泸 m and X2 = m. Finally consider the pair (衍，叫 j
The set of candidates in Hotelling's model is gìven. The next exercise asks
you to analyze a model ín which the set of candidates is generated as 伊rt of an
equìlìbrium.
若 EXERCISE 75.2 (Cí tizen-candidates) Consider a game ìn which the players are the
ci tizens. Ar飞y citizen may, at some cost c > 0, bεcome a candidate. Assume that
the only position a citi泛泛n can espouse is her favorite position, so that a citizen's
only decision is whether to stand as a candidate After all cìtizens have (约multa­
neously) decided whether to become candidates, each citìzen votes for her favorite
candidate , as in Hotelling's mode l. Cítizens care about the positio日 of the winning
candidate; a citizen whose favorite position is x loses Ix x.1 if the winnìng candi伊
date's posìtion ìs x*. (For any number z, Izl denotes the absoh~出 val树。f z: Izl = z
if z > 0 and Izl = -z if z < 0.) Winning confers the benefit b. Th us a citizen who
becomes a candidate and ties with k - 1 other candìdates for first place obtains
the payoff bjk c; a citizen wìth favorite posítion x who becomes a candidate and
1.S not one of the candìdates tied for first place obtains the payoff -1 x - x~ I … c,
where x* is the winn啊 's posìtion; and a citizen with favorite position x who does
not become a candidate obtains the payoff … Ix … x* í, where x* is the 飞与inner's po喃
sition. Assume that for every position 工 there is a cìtizen for whom 工 is the favorite
position. Show that if b :::; 2c, then
‘
Hotelling's model assumes a basic agreement among the voters about the ordering of the posìtions. For example, if one voter prefers x to y to z and another
voter prefers y to z to x, no voter prefers z to x to 沪白1e next exe主cise asks you to
studya model that does not so restrict the voters' preferences.
@ EXERCISE 75.3 (Electoral competition for more general preferences) Suppose that
there is a f切ite number of positions and a finite , odd, number of voters. For any
Chapter 3. Nash 在quilibrium: lIIustrations
76
positions x and y, each voter 的ther prefers x to y or prefers y to x. (That ÌS , no
voter regards any two positions as equally desirable.) 认le say that a position x*
is a CO l1dOJ衍t wÎnner if for every position y different from x* , a majority of voters
prefer x* to y.
a. 5how that for any configuration of preferences there is at most one Condorcet
飞，y mner.
b. Give an example ìn which no Condorcet winner exists. (5uppose th剖 there
are three positions (x , y , and z) and three voters. Assume that vot仅 1 prefers
x to Y to z. Construct preferences for thεother two voters with thεproperties
that one voter prefers x to y and th芒。ther prefers y to x, 。在伫 prefers x to z
and the other prεfers z to x, and one prefers y to z and the other prefers z to
yτhe prefiεrences you construct mu处， of course, satisfy the condition that a
voter who prefers a to b and b to c aÌso prefers a to c, where G, b, and c are any
positions.)
c. Consider the strategic game in which two candidates simultaneously choose
positions , as in Hotelling's mode l. If the candidates choose different positÌon队 each voter endorses the candidate whose position she prefers, and the
candidate who 陀ceives the most votes 认，1 Íns. If the candidat巳 s choose the
same position, they ti已 5ho叭. that this game has 及 unique Nash e气uilibriu血
if the voters' preferences are such that there is a Condorcet winner, and has no
时ash equilìbrium if the voters' preferences are such that there is no Condorcet
wlnne r.
A variant of Hotelling's model of elεctoral competition can be used to analyze
the choices of product characteristícs by compεting firms in situations in which
price is not a significant variable. (Think of radio stations that offer diffεrentstyles
of music, for example.) The set of positions is the range of possible characteristics
for the product, and the citizens are consum台rs rath盯 than voters. Consumers'
tastes differ; eachζonsumer buys (at a fixed price, possibly zero) one unit of the
product she likes bes t. The modeJ differs substa对ially from HotelJing's model of
electoral competition in th挝 each firm's objecti飞re is to maximize its markεt share,
rather than to obtain a market share larger than that of any other firm. 111 the
next exercise you are asked to show that the Nash equilib1'ia of this game in the
case of two 0 1' three firms are the same as those in HotellÌI吧's model of electoral
competition.
φEXERCISE 76.1 (Competition in product ch盯acterìstics) 1n the variant of Hotelω
ling's model thatεaptures competíng 自 rms' choìces of product ch衍acteristics，
show that when there are two firms the unique Nash equilibrium is (17l, m) (b oth
firms offer the consUI丑ers' median favorite product) , and when there are three
firms therεis no Nash equilibrium. (5tart by arguing that when there are two fìrms
whose products differ, eìther firm is better off ma挝ng its product more similar to
that of its riva l.)
3.4 The WarofAttrítíon
77
3 .4 The War of Attrition
The game known as the War Of Attl' ition elaborates on the ideas c叩 tured by the
game Hawk-Dove (Ex衍cise 31. 2). It was originally posed as a model of a con丑ict
between two animals 在ghting over pr句. Each animal chooses the time at which it
intends to gi飞'e up. When an animal gi飞咄叩， its opponent obtair飞s a11 the prey (and
the time a t w hich thεwinner intended to give up is irrele飞rant). If bothanimalsgive
up at the same time, then each has a只 equal chance of obtaining the prey. Fighting
is costly: 指ch animal prefer:; as short a fight as possible
The game models not only such a conflict between animals , but also many other
disputes. The "prey" can be any indi\也ible object , and "fighting" ca只 Uεany costly
activity-for example , simply waiting.
τb define the game precisely, let time be a continuous variable that starts at
台 and runs indefinitel y. Assume that the value p泣 ty i attaches to the object in
dìspute Îs vi > 0 and the ￥.alue it attaches to a 50% chance of obtaining the object
is vJ2. Each unit of time that passes before the dispu饺 is settled (i.e. one of the
parties cOl飞cedes) costs each party one unit of payoff. 百1US if player i concedes
first , at time ti' her payoff is - t{ (she spends ti units of time and does not obtain
the object). If the other player co口ceàes first , at time ti' player i's payoff is Vi tj
(she obtains the object after t j 泣 nits of time). If both players concedεat 出e same
tir盹 player j's payoff is ~刘叫川here t i is theωnmon conc岱创刊ime. The War
ûf Attrition is the following strategic gamε.
‘
Players
百1e two parties to a dispute.
Actions Each player's set of actions is the set of possible concession times (non-
negative numbers).
Prefere门ces
Player i's preferences are represented by the payo日 function
(
ifti<tj
Ui(t1rfdz < jol-fl ifflz fl
lui-ti
if fi>tlF
1年here j is the other player
丁o find the Nash 叫uilibria of this game , we start, as before, by finding the
players' best response funcnons. Intuitively, if player j' S intended concession time
isεarlyeno吨h (t j is small) , then it is optimal for player i to wait for player j to
concede. That is, in this case player i should choose a concession timεlater than
t j ; any such time is as good as any other. By contrast, if playεr j intends to hold
out for a long time 的 is large) , then player i should Co配ede imrnediately. Because
player i values the object at V Ít th心 length of time it is worth her waíting is Vi.
To make these ideas precise , we can study player i's payoff function for variou垂
直xqd 飞ralues of tj' thεconcession 出le of player j. The three cases that the intuitive
argument suggests are qua1itatively differe时 are shown in Figure 78.1: 与< Vi in
Chapter 3. Nash Equilibrium: lII ustrations
78
yui
Ui
$
7d
l
fj
O
c←一一句-
ti
.
•
tj < Vi
tí •
tj > Vi
tj = υi
Figure 78.1 Three cross sectíons of p)ayer i's payoff function in the War of Affrition.
the left panel , tj = Vi in the middle pa 只el， and tj Vi in the right pa口el. Player i's
best responses in each case are her actions fOL which her payo旺 is highes t: the set
of times after ti if ti < Vi
tí = Vi , and 0 if tj > Vi
Vlf, 0 and the 咒 t of times after tj
irJ if 1I
In summary, player i's best responsεfunction is given by
]
J.J.
"']
.........
V
U J.. Ll..A
.....1."- ...;1'-1. L' J.
l.J.J..l l..... <.-1
"-<l.ll......s.
( {ti:ti > tj}
Bí (t j)
= \ {t;: tí = 0 or t 1 > tj}
l {O}
.1.1.
if tj < V;
if t， 口叫
iftj >忽i
For a case in which '01 > V2 , this function is shown in the left panel of Figure 78.2
for i 口 1 and j = 2 (player l's best response function) , and in the right panel ror
2 andj 之二 1 (player 2's best response function).
Superìrnposìng the players' best response functions，飞Ne see that there are two
areas of intersection: the vertical axis at 汶口d above z'l and the horizontal axis at
and to the right of V2. Thus ([1 , t2) is a Nash eguilibriurn of the garne if and 0口ly if
εither
t 1 = 0 and t2 2: Vl
。r
t2 = 0 and t 1 2: V2.
?ih
•
!h
B1 (t 2)
17 1
>
0--仰而--刷刷翩翩喃.
V2
V1
O
B2 (t l )
V?
-•
p)ayers' best response functions in the War of Attriliol1 (for 在 case in which V1 > 内}
1'5 best resronse function is in the left pane); player 2'5 is in the right panel. (The slopìng edges
are excluded.)
Figur告 78.2τhe
PI在yer
3.4 The War of Attr;t;on
79
ln words, ín every equilíbrium either player 1 concedes immediately and player 2
con也des at time V1 or later, or player 2 concedes immediately and player 1 concεdes at time 句 or later.
( EXERCI5E 79.1 (Di rect argument for Nash equilibria of ~毛主1 1' of Attrítion) Give a
direct argument, not using informatìon about the entire best response functions ,
for the set of Nash equilibria of the War of Attrition. (Argue that if t1 = t2 , 0 <
t i < 句， or 0 口 ti < tj < Vi (for i 口 1 and j = 2, or í 2 and j 1), then the pair
(杠，马) is not a Nash equilibrium. Then argue 出at any remaining pair is a Nash
equilibrium.)
τhree features of the 吨 uilibria are notable. Fir纹， in no 合气uílibrium is there
any figh t: one player always concedès immediately.是 Second， either player may
concede first, regardless of the players' valuatio邸. 1n particular, there are always
equilibria in which the player who values the object more highly CQI飞cedes firs t.
Third , the equilibria arε asymmetric (the players' actions are diffeτent) ， even when
右1
时， in which case the game is symmetric… the players' sets of actions are
the same and player 1's payoff to 仇，与) is the same as player 2's payoff to (t2' tl)
(Definition 5 1.月. Given this asymmetr予 to in饺rpret the Nash equilibria as action
profíles compatible with steady states, the populations from which the two players
are drawn must be distinct. One player might be the current owner of the 。如Iject
in dispute and the other a challenge瓦 for example. In this case the equiLibria corr夺
spor飞d to the two conventions that a challenger always gives up immediately and
an owner always does so. (Some evídence is discussed ín the box on page 409.) If
all players-those in the role of player 1 as well 部 those in the role of player 2-are
drawn from a single population, then 0贝ly symmetric equilibria are relevant (see
Secti∞ 2.10). The 以îr of Attritiol1 has no s泣ch equilibria , so thε 口otion of N asr飞
equilibrium makes r飞o prediction about the outcome in such a situation.
子 EXERCISE 79.2 (Variant of War of Attrition) Consider the variant of the War of AttrÎ-
tion in which each player attaches no value to the time spent waíting for the other
player to concede, but the object in dispute loses value as tìme passes (Think of a
rot位ng ar飞imal carcass or a melting ice cream cone.) Assume that the value of the
o均ect t。但ch player i after t units of time is Vj … t (and the value of a 50% cha时e
of obta凶吨 the object is ~ (Vi 一切. Specìfy the 蛇ati怡巳吨gicg
伊a附 t出h
沁
归a挝t打
拟rr阴d
命已l沁st佼位
h飞i怒5 姑
u泊
挝tion (take care wi让th the payoff 扣
a
fun飞ct刽ior趴lS玛). Construct the analogue of Figure 78.1 ,
find the players' best response functions , and hence fînd the Nash equilibria of the
game.
‘
The 甘'ar of Attritíol1 is an example of a 飞ame of tinùng" , in which each player's
action is a number and each player's payoff depends sensitively on whether her
action is greater or less than the 0也er play归 's action. In many such games , each
pl玛咐 's strategic variable is thεt垃le at which to act, hence the namε"ga 江 le of
tinùng". 在le next two exercìses are further examples of such games. (In. the first
the strategic variable is time, whereas in t'he second it is no t.)
Chapter 3. Nash Equilibrium: lII ustratio罚$
80
EXERCISE 80.1 (Timing product 陀le捕时 Two firms are dεveloping competing
products for a market of fixed size.τhe 10nger a firm spends on developme时，
the better its produc t. But the first firm to release its product has an advantage: the
customεrs it obtains will not subsequently switch to its riva l. (Once a person starts
using a product, the cost of switching to an alternative , even one signifìcantly bet饺乙 is too high to make a switch worthwh i1e.) A 自rm that releases its product first ,
at time t, captures the share h (t) of the market , where h is a function that increases
from time 0 to time 了， with h(O) = 0 and h(T) = 1. The remaining market share
is left for the other 在rm.lf 注le firms release their products at the same time，部ch
obtains half of the marke t. Each firm wishes to obtair飞出ε!飞ighest possible mar懈
ket share. Model this situation as a strategic game and íind its 时ash equi1ibrium
(e f"J.uilibria?). (飞岛'hen finding firm i's best response to firm f' s re1ease time tj' there
are three cases: that in which h (t j) < (firm j gets less than half of the ma此时
ìf it is the first to re1ease its product) , that ìn which 1I 叫
i ， and that in whìch
i
h 问) >
i.)
? EXERCISE 80.2 (A fight) Each of two people has one unit of a resource. Each person
chooses how much of the resource to use in fightìng the other individual and how
much to usεproductively. 写出ch person í dev: otes Yi to fighti吨， then thεtotal
output is 月 Yl ， Y2) 三 o and person i obtains the fraction Pi(Yl , Y2) of the output,
where
(1 jf Yi > Yí
Pi(Yl ， Y2) 口~
i if Yi = 约
l0
if Yi < Yj'
ηle functio口 f is continuous (small changes in Yl and Y2 cause smal1 changes in
f 打八斗V2 日， ís dec陀asing ín both Yl and y2 (the more each player devotes to fighting ,
the less output is produc时)， and satisfies f(l , 1) 口 o (if each player devotes all her
resource to fighting , then no output is produced). (If you prefer to deal with a
specific functior飞 f， take f(Yl , Y2) = 2 - yl - Y2') Each personεares on1y about the
amoUl飞t of output she receives , and prefers to receive as much as possible. Specify
this situation as a strategic game and 在nd its Nash equilíbrium (equilibria?). (U能
在 direct argumen t: first consider pairs (Yl ， 只) with Yl 并 Y2 ， then those with Yl =
Y2 < 1, then those with Yl y2 口1.)
3 .5 Auctions
3.5.1
Introduction
1n an "auction气 a good is sold to the party who submits the highest bid. Auctions ,
broadly defined , are used to al1 0cate significant economic resources , from works of
art to short-term government bonds to offshore tracts for oil and gas exploratiOl飞
to the radio spectrum. They take many forms. For example , bids may be called
out seque对ially (as in auctions for works of art) or may be submitted in sea1ed
enve10pes; the price paid m町 be the highest bìd , or some other price; if more thao
3.5 Auctions
81
one unit of a good is being sold , bids may be taken on aU units símultaneously,
or the units may be sold sequentiall予 A game-theoretic analysis helps us to understand the consequences of various auction designs; it suggests , for example ,
the design likely to be the most effective at allocatíng resources , and the one likely
to raise the most revenue. In this section I dìscuss auctions in which every buyer
knows her own valuation and every other buyer' s valuation of the item being sold.
Chapter 9 develops tools that allow us to study, in Section 9 刷 6， auctions in which
buyers are not perfectly informed of each other's valuations.
AυCTIONS FROM BABYLONJA TO EBAY
Auctìoning has a very long histor予 Herodotus， a Greek writer of the fjJth century B. C. who, togεther with Thucydides, created the intellectual field of history,
describes auctions in Babylonia. He wrìtes that the B浅bylonians' "most sensible"
custom was an annual auctíon in each 飞lillage of the women of marriageable age.
The women most attractive to the men were sold first; they commanded positive
prices , whereas men were paid to be matched with the least d础irable women. In
each auction , bids appear to have been called out 附quentiall予 the man who bid
the most winning and paying the price he bid.
Auctions were used also in Athens in the fifth and fourth centuries B. C. to
sell the rights to co l1ect taxes, to dispose of confiscated proper句号 and to lease land
and mines. The evidence on the nature of the auctions is slim, but some interesting
accounts survive. For example , the Athenian politician Andocides (c. 440-391 B. C.)
reports collusion in an auctìon of tax附collection rights (see Langdon 1994, 260).
Auctions were frequent in ancient Rome, and continued to be used in medieval Europe after the end of the Roman empire (tax蛐collection rights were annually auctioned by the towns of the medieval and ea r1 y modern Low Countries, for
example). The ea r1 iest use of the English word "auction" given by the 0写ford El1glish Dìctìo l1 ary dat部 from 1595 and concerns an auction "when will be sold Slaves,
household goods , etc."我ules surviving from the auctions of this era show that in
some cases, at least, bids were called out sequentially, with the bidder remaining
at the end obtaining thεobjeζt at the prìce she bid (Cassady 1967, 30…31). A variant
of this mechanism , ir飞 which a timεlimit is imposed on the bids, is reported by the
English diarist and r飞机γal administrator Samuel Pepys (1633-1703). 在le auctiorν
eer lit a short candle, and bids were valid only if made before thεflame went out.
Pepys reports that a fl盯ry of bidding occurred at the last moment. At an auctìon
on September 3, 1662, a bidder "cunninger than the rest" told him that just as the
flame goes out, "the smoke descends" , signaling the moment at which one should
bid, an observation Pepys found "veηT pre忧y" (Pepys 1970, 185-186).
The auction houses of Sotheby's and Christie's were founded 拍拍emi小 18th
century巳 At the beginning of the 21st centur予 they are being eclipsed , at least
in the value of the goods they sell, by online auction compa时的. For exampl已F
eBay, foun
82
Chapter 3. Nash Equilibrium: lII u辈辈rations
auctions during the second quarter of 2000, roughly double the nurnbers for the
second quarter of the prevíous year; Sotheby's and Christie's together sold around
U.S.$l billion of art and antiques each quarter.
了he rnechanisrn used by eBay shares a feature with the ones Pepys observed:
a11 bids rnust be received before sorne fixed tirne. The way in whïch the price is
deterrnined differs. In an eBay auction, a bidder subrnits a "proxy bid" that is not
revealed; the prevailing price is a srnall incrernent above the second皿highest proxy
bid. As in the 17th-century auctions Pepys observεd， rnany bidders on eBay act at
the last rnornent一-a practíce known as "sniping" in the argot of cyberspace. Other
online a uction houses use different termination rul部. For exarnple , Arnazon waits
ten rninutes after a bid before closing an auction. The fact that last-rninut，εbi过ding
is rnuch less cornrnon in Arnazon auctions than it is 1..'1 eBay auctions has attracted
the attentìon of garne theorists; Roth and Ockenfels (2002) explain it ín terrns of the
difference in the 在uctions' terrnination rules.
ln recent years, rnany countries have auctioned the rights to the radio spec栅
trurn, used for wireless cornrnunìcation. 丁hese auctions havεbeen rnuch studied
by garne theorísts; thεy are discussed in the box on page 300.
3.5.2 Second-price seoJed司bidouζ tions
ln a comrnon forrn of auction, people sequentially subrnit increasing bíds for an
object. (The word "auction" cornes frorn the Latin augere, rneaning "如 incrεase".)
Whεn no one wishes to submit a bid higl飞机. than the current bìd, the person maki且g the current bíd obtaíns the object at thεprice she bíd.
Given that every pεrson ís certain of her valuation of the objeεt before the bidding begins, duríng the bidding no one can 1εam anythìng rele飞1ant to her actions.
Thus we can rnodel thεauction by assurnìng that each person decides , before bid翩
díng begins, the rnost she is willing to bid-her "rnaxirnal bid". Wh en the players
carry out their plans, the winner is the person whose rnaxirnal bid is highes t. How
rnuch does she need to bid? Eventually only she a1飞d the person with the second
highest rnaxirnal bìd will be left cornpeting against each other. To win, she there扣 re needs to bid slìghtly rnore than the second h垃hest rnaxirnal bid. 盯
1ft役佑
h飞eb
切ìdd
函ll1
incrernent 巳
í s srn
丑1冶
all
丑1 ， we can take the príce the winner pays to be equal to the second
highest maxirnal bid.
丁hus we can model such an auctiOI飞 as a strategìc garne in which each player
chooses an arnount of rnoney, ìnterpreted as the maxilllal amount she is willing to
bid, and the player who chooses the highest arnount obtains the object and pays a
priceεqual to the second highest arnoun t.
This garne rnodels also a sítuation ín which the people símultaneously put bids
ìn sealed envelopes, and the person who subrnits the highest bid wìns and pays a
prïce equal to the second highest bid. For this reason thεgarne ìs called a second-price
sealed-bid auction.
3 .5 Auctions
83
To define the game precise)y, denote by Vi tl飞e value player i attaches to the
object; if she obtains the object at the prícε p her payoff is Vi … p. Assume that
the players' 飞!aluations of the object are all different and all positive; number the
players 1 through n in such a way that Vj > V2 > … > V I1 > O. Each p\ayer i
submits a (sealed) bid bi . If player i's bid is hîgher than every other bid , she obt在ins
the object at a price equal to the s心cond-highest bid , say j岛jl and hence receives the
payoff vi …乌. If some other bid is higher than player i's bid , player i doεs not
obtain the object and receìves the payoff of ze ro. If player i is in a tìe for the highest
bid , her payoff depends on the way in which ties are broken. A simple (though
arþitrary) assumptìon is that the winner is the player among those submìtting the
highest bid whose number is sm剖 lest (i.e. whose valuation of the object is highest)
(If the highest bid is subrr旧
r
pμlδ叮
ye
盯r 2.) 飞Unde
时r this assur熙
Zηlption， player i's payoff when she bìds bi and is in a tie
for t怡 highest bid is Vi - bi if her number is lower than that of i1 ny other pl巧er
submitting the bid bir and 0 otherwise.
In summar予， a second-p对优 sealed-bid auction (with perfect information) is
the following strategic game.
Players
丁he 11 bidders , where 11 2:: 2.
Actions The set of actions of each player is the set of possible bids (nonnegati飞fε
n议mbe
臼r咛.
Preferences Denote by bi the bid of player i and by b the highest bid submitted
by a playεr other than i. If eíther (的 hi > b or (b) bi 工艺 b and the number of
every other player who bids b is greater than i, then player i's payoff is Vi - b.
Otherwis心 player i's payoff is O.
丁his game has many Nash equilibria. One equìlibrium is (屿，. . ., b l1 ) 工 (V1' .
V I1 ): each player's bid is equal to her valuation of the object. Bεcause V1 > 巧
> Vll , the outcomε 总 that player 1 obtains the object at the price b2; her payoff ìs
2气… b 2 and every other player's payoff is zero. This profile is a Nash eguilibrium
bv the follow加g argumen仁
• If player 1 changes her bid to some other price at 1εastεqual to 172 , then the
outcome does not change (recall that she pays the second highest bid , not the
loses and
highest bid). If she changes her bid to a price !ess than b;. , then
obtains the payoff of zero.
• If some other play臼 lowers her bid or raises it to some price at most 叫ual to
的 I then she remains a loser; îf she raises her bid above b11 then she wios but,
în paying the price b1, makes a loss (because her valuation is less than b1).
A∞t由
her川
:e
叫
qu
旧i1ib
祉主i凶
um
obtains the 忡
0bj扣
ect and 阳
p ay
归st位
hεpr扎ic
优
e of zero. The profile is an equilibrìum because
if player 1 changes her bid , then the outcome remains the sam役， and if any of 世le
ζhapter 3.
84
Nash Equilibrium: lIIustrations
remaining players-say player í-raises her bid, then either the outcome remains
the same (if player í' s new bid is at most V1) or causes player i to obtain the object at
a price that exceeds her valuation (ifher bid exceeds Vl). (The auctioneer ob飞riously
has an incentive for the price to be bid up , but she is not a player in the game!)
In both of these equilibria , play校 1 obtains the object. But there are also equilibria in which player 1 does not obtain the object. Consider, for example, the action
profile (V2 , V1 , 0尸. ，时， in which player 2 obtains the object at the príce V2 and every player (i挝luding player 2) 陀ceives the payoff of zero. This action profile is a
Nashe弓ui1ibrium by the foìlowing argumeI飞仁
.
• If player 1 raises her bid to 坷。r more, she wins the object but her payoff
remains zero (she pays the príce v1 , bid by player 2). Any other change 回
her bid has no effect on the outcome.
• If player 2 chang岱 her bid to some other price greater than V21 the outcome
does not change. If she changes her bid to V2 or less she loses, and her payoff
re口lams zero.
• If any othεr player raises her bid to at most 屿， the outcome does not change.
If she raises her bid above V1 , then she wins, but í口 paying the price V1 (bid
by player 2) she obtains a negative payoff.
? EXERCIS巨 84.1 (Nash equilibrium of second-príce sealed-bid auction) Fî nd a Nash
叫uilibrium of a second-price sealed-bid auction in which player 11 obtains the
object.
Player
bid in this equilibrium exceeds h杠飞raluation， and thus may seem a
little rash: if player 1 were to increase her bid to any valuεless than 叫， player 2's
payoff would be negatìve (she would obtain the 0问 ect at a prìce greater than
her 飞raluation).τhis property of the action profile does not affect its status as an
equilìbr如m， because in a Nash equilibrium a player does not consider the "risk"
that another player will take an action different from her equilibrìum actìon; each
player simpiy choosεs an action that is optìma l, given the other players' actions.
But the property does suggest that the 吨uilibrium is less plausible as the outcome
of the auction than the equilibrium in which every player bids her valuation.
The same point takes a different form when we i时机pret the strategíc game as a
model of events that unfold over time. Under this interpretation, player 2's action
V1 means that she wîll continue bidding until the price reaches vl. lf player 1 is sure
that player 2 will continue bidding until the price is 坷 I then player 1 ratìonally
stops bidding when the prícεreaches V2 (。毛 indeed， when it reaches any other
level at most 吨ual to Vl) 刷 But there is little reason for player 1 to believe that
playεr 2 will in fact stay in the bidding if the price exceeds V2: player 2's action is
not credible, because if the bidding were to go above l勺， player 2 would rationally
withdraw
‘
3.5 Auctions
lI j
85
专2 …主
/1
告/1
。
。
Vj
/J -,
payoff to the bid Vi
。
告→
(~τ
i's payoff to the bid /J;
í's payoff to the bid 可
b;
7} i
Figure 85.1 Player i's payoffs in a se吃ond-price s已aled-bid a 口 ctíon as a function of the highest of the
other players' bids , denoted 击， for bids of "j (left pa肘。，告: < Vi (middle pa时巧， and 白了 > vi (right
panel)
The weakness of the equilibriur古i ís reflected in the fact that player 2's bíd Vl is
weak!y dominated by the bíd V2. 词。re generall予
m 疗 second-price sealed而id auctio l1 (wilh pcrfect i呐 rmatio肘 ， a r刊叨「内 bid
equal to her valzwtio l1 wcakly domÎnatcs all hcr other bids.
That is, for any bid b i 并 Vi' player i's bid Vj ís at least as good as bμ110 l1I atter what
the other players bid , and is better than /J i for some actions of the othεr players. (See
Definition 46. 1.) A player who bids less than her valuatior飞 stand三口。t to win in
some cases in which she could profit by winning (wheηthe híghest of the othe 1'
bids is betì气(een her bid and her val飞lation) ， and never stands to gai日 relati飞F巳 to
the situation in which she bids her 飞写luation; a player who bids more than her
valuation stands to win i口 someωses in whích she obtains a negative payoff by
doing so (when the highest of the remaining bids is bet认Teen her valuation and
heτbid) ， and never stands to gain relative to the situatior飞 in 飞il'hich she bids her
valuation. 丁he key point is that in a second翩price auctìon, a player who changes
her bid does not 10认'er the price she pays , but only possibly changes her status
from that of a winner into that of a loser, or vice versa.
A precíse argument is illustrated in Figurε85.1 ， which compares player i,s pa 沪
offs to th台 bid Vi (left panel) with her payoffs to a bid 可 < Vi (middle panel) , and
with her payoffs to a 七id bjf 〉巳'i (right panel) , as a function of the highest of the
other playe凹， bids, denoted b. Wεseεthat for all values of the highest of the other
players' bids, p]ayer i's payoffs to Vi are at least as large as her payoffs to any other
bid and , for any bid b i 乒乓 ， there are some values of the highest of the other players' bids for which her payoffs to Vj exceed her payoffs to b尸 τhus player i's bid Vi
wεakly dominatεs all her other bids.
ln summary, a second-price auction has many 民ash equilibria , but the equilibrium (b 1 尸 .， b n ) 尘仆 1 尸‘，如) ，如 which every player's bidβequal to her valuatioηof theobject, is distinguished by the fact that every play帘! S action weakly
dominates all her othe玄 actions.
CD EXERCISE 85.1 (Seco叫-price sealed七id auction with two bidders) Find all the
Nash equilibria of a second-price sealed-bid auction with two biddεrs. (Construct
Chapt告r 3.
毒6
Nash Equilibrium: lII ustrations
the players' best response functions. 人part from a difference in the tie部breaking
rule , the game is the same as the one in Exerci民 79.2.)
? EXERCIS巨 86.1 (Auctioning the right to choose) An actiOl飞 affects each of two peo-
p!e. The right to choose the action is sold in a second-price auction. That is , the
two people simultaneously submit bids, and the one who submits the higl飞机‘ bid
chooses her favorite actiün and pays (to a third party) the amount bid by the other
person , who pays nothing. (AsSU111e that if the bids are the same , pεrson 1 is the
winner.) For i
1, 2, the payoff of p盯S01飞 i when the action is a and person i
pays 111 is ul(a) - 11l. In the game that modεIs this situatìon, find for each player a
bid that we此1y dominates 在11 the player s oth杠 bids (and thus find a Nash equilibrium in which each p1ay巳r's 吨uilibrium action weak1y dominates all her other
actìons).
l
3.5.3
Firs 卜prìζe sealed也 ìd auctions
A first-price auction differs from a second-price auction only in that the winner
paysth ε price she bids not the second highest bid. Pr阳eci沾sely予予:二， a first-佩 price sealedbid auction (机毛认
川vi
w
飞
才i让th
川 pe
衍rf必巳ct 怡
i I口口1飞扣
f白rτma
fo
川ti怡
or吟
1斗) 沁
i s defin时 a
削s fo1芷10
仍毛认
w
飞v嗡$
l
Player巧
5
The 1η1 bidders , \，飞γrhe创杠r‘elη 三三 2.
Actions The set of actions of each player ís the set of possible bids (nonnegative
numbers)
Prεferences
Denüte by b1 the bíd of player i and by /J the highεst bid submitted
bya player other than i. If either 衍) bi > /J or (b) 句巳 b and the nllmber
of every other player 飞飞，ho bids b ís greater than Î, then player i's payoff ís
Z!j … bí.Other
丁his game models an allction in which people submit sealed bids and the high叩
est bid wins. (You conduct such aηauction when you so!icit offers for a car you
飞气lish to selJ, or, as a buyer, get estimates from contractors to fi 戈 your leaky basement , aSSUIηing in both cases that you do not info l" m potential bidders of existing
bids.) 丁he game models also a dynamic auction in which the auction巳er begins by
anno飞的cing a high price , which she gradually 10\飞rers Uηtil son飞仅me indicates her
willingl飞的s to buy the objec t. (Flowεrs are sold in this way in the 时 eth创'lands.) A
bid in the stratεgic game is interpreted as the price at which the bidder will indicate
i飞机. will Ìl飞gnεss to buy the object in the dynamic auction.
One Nash equilibrium of a fírst阳price sealeιbíd a uction is (b] 尸. , bl1 ) = ('('21
V2 ，"i.协… ， V II )， in whîch player l's bid is player 2's valuation V2 and εvery othεr
play挝、 bid is her ow口 valuation. The outcome of this equilibrium is that player 1
obtains thεobject at the prícε 句
.
。 EXERCISE 86.2 (Nash equilibrium of first-price sealed-bid auction) Show that 俏，
. ., bl1 ) = (1:弘吨， I弘. ., V n ) is a Nash equilibrium of a first-price sealed-bid auction
3.5 Auctions
87
A fìrst-prìce sealed-bid auctíon has many other equilibria , but in all equilibria
the wi妇只er is the player who values the object most highly (player 门， by th泛 fol­
lowing argumen t. ln any action profile (17 1,..., bll ) in which some player í 并 1
wìns , we have bl > bl . If bí > '[勺， then i's payoff is negative, so that she can do
better by reducing her bid to 0; ìf hí <二 V2 ， then player 1ε3n increase her payoff
from 0 to vJ … hí by bìdding bi' ìn which case shεwins. Thus no suζh action profile
is a Nash eguilibrìum.
φEXERCISE 87.1 (First-price sealed-bid auctiOl飞) Show that in a Nash 吨uilibriurn of
a first-price sealed七id auctìon the two higl飞est bids are the samε， one of these bìds
is subrnitted by player 1, and the highest bid is at least v2 and at rnost vJ. Sho飞气F
also that any actíon profile satisfying these conditions ìs a Nash equilibriurn.
In any e斗uilibríurn i只 which the winning bid exceeds 吨， at least one player's
bid exceeds her 飞吼 uation. As in a second-prìce s部 led七id auction, such a bid
seerns "risky" because it would yield the bidder a negative payoff if it were to win.
ln the equilibriun飞 there is no risk, because the bid does not win; but, as before, the
fact that the bid has this property reduces the plausibìlity of the equilibriur口，
人s in a second-price sealed-bid auction , the pote口 tialμrisk挝ess" to player i of
a bid bí > Ví is rεt1ected in the fact that it is weakly dominated by the bid zμas
shown by the following argurner飞七
• lf the other players' bids are SL1 ch that player i loses 飞气.r hen she bids 171, then
the OutCOIηe is the sarne whether she bids bí or Z与"
• If the other players' bids are such that player i wins when she bids bú then
her payoff is negative when she bids bí and zero when 计\e bids Ví (时gardless
of whether this bid wins).
However, in a first-price auction, unlìke a second-price auction, a bid bì < vi
of player i is 110t 认reakly dorninated by the bià ví. 111 fact , such a bid is not weakly
dornina时 by any bid. It is ∞t weakly dorninated by a bid 与 < bí because 运 the
otl飞机. players' highεst bid is bet附en /1; anà b" then 17; loses, whereas bí wins a口
yields player í a positi 附 payoff. And it is not weakly dorninated by â bid b; > bi
because if the ot仙he盯r play问
仅芷$正川山，
e
飞
t
highest
bìd ìs less than 句， ther飞 both bì and /1; win and
bí yiεlds a lower prìo;.
Furthe飞机'εn though the bid l' i \veakly dominates higher bids , this bid is itself
weakly dominated, by a 10飞Ner bid! If playεr í bids Ví her payoff is 0 regardless of
the other players' bids , whereas if shεbids less than Vj her payoff is either 0 (if she
loses) or positi飞'e (if she wins).
1n surnrnary,
ma 卢rst-príce sealed俨bíd auctìon 仰Íth peJfect Ì1价 tmatioll)， a playe r's bid of
at least her val沁μωω
atiω仰
O ηi必Sí衍
附
U
eakly 由
dOω7时
y
i必5 IηlO tW足沱
Jet
仇
3气a欢
lcly d01 仅时111σ ted
‘
Ar飞出lplication of t hÌs result ìs that in e出ry r如sh equìlibrium of a first-price
sealed-bid auction at least one player's action is weakly àorninated. Hovγever，
Chapter 3. Nash Equilibrium: lII ustrations
88
this property of the equilibria depends on the assumption that a bid may be any
number. ln the variant of the game i 到 which bids and valu拭ions ar，εrestrìcted
to be multiples of some disC1性e monetary unit ε 怡 .g四 a cent) , an action profile
(V2 - t:， V2 … ε， b3 , .. . , bll ) for any 乌三 Vj - e for j = 3,..., 11 is a Nash equilibrium in which no player's bid is weakly domìnated. Further, every equilìbrium
i民 which no player's bid is weakly dominated takes this form. Whenεis smalt
台ach such equìl加ium is close to an equilibrium (吨，叫J 鸟，. .., bn ) (with bj 三句
for j = 3,.…, 11) of the gamεwith unrestricted bids. On this (somewhat ad hoc)
basìs, 1 select action pr。在i创 (V2 ， V2 , b3, .. ., bn ) with bj 三 Vj for j
3, . . . , 11 as
" distir飞guished" equilibria of a first-prìce sealed七id auction.
One conclusion of this analysis is that while both second-price and first-price
auctions have many Nash equilibrìa, yielding a variety of outcomes, their distin叩
guished equilibria yield the same outcome. (Recall that the distinguished equilibrium of a second-price sealed-bid auction is the action profile in which every
player bids her valuation.) lnεvery distinguished equilibrium of each game，在廷
。bject is sold to player 1 at the pric心 V2. ln particula乙 the auctione它r's revenue is
the same in both cases. 丁hus if we restrict attention to the distir飞guished equilibria ,
the two auction forms are "revenue equivalent". The rules are differe址， but the
players' equilibrium bids adjust to the differenc台 and lead to the same outcome:
‘
the single Nas Ì1 e号 uilibríum in which no player's bid is weakly dom Ìl zated í11
a seco11d-príce auction yields the same outcome as the distinguished e号uilibria
ofa 卢 rst-príce auctioll.
? EXERCISE 88.1 (Third-price auction) Consider a tlúrd-price sealed-bid auction,
which differs from a first- and a second-price auction only in that the winner (the
person who submits the highest bid) pays the thir咽d highest price. (Assume that
theτe are at least three bidders.)
a. Show that for any player i thεbid of Vi weakly dominates any lower bid but
does not weakly dominate any higher bid. (To show the latte巳 for any bid
bi > Vi , find bids for the oth位 players such that player i is better off bîdding
鸟 than bidding Vi.)
b. Show that the action profìle in which each player bids her valuation is not a
?、Jash equilìbrium.
c. Find a Nash equilibrium. (There are ones iηwhich every player submits the
same bid.)
3.5.4 Variants
U11certain valuations The models 如 this section depart from reality with resp仅t
to the assumption thatωch bidder is certain of both her own valuation ar飞d every
other bidder's valuation. In most, if not all, actual auctìons , information is surely
less complete.τhe case 如 whìch the players are uncertain about each other's valuations has been thoroughly explored and is discussed in Section 9.6. 在le result
3.5 Auctions
89
that a player's bidding her valuation weakly dominates a11 her other actions in a
second相price auction survives when players are uncertain about each other's valuations, as does the revenue equivalence of first- and second-price auctìons under
some conditions on the playεrs' prefereηces.
Common valuations In some auctions the main dìfference between thεbidders is
not that they value the object dìfferently but that they have dìfferent information
about its value. For example, the bìdders for an oìl tract who put similar values on
any gìven amount of oìl may have different inform时ìon about how much oil is in
the trac t. Such auctions ìnvolve informatìonal consid旧ations that do not arise in
the model we have studied in this section; they are studied in Section 9.6.3.
All-payauctíons Some situations may be modeled as all-pay auctions in which every bidder, not only the winner, pays. An example is competition betweer飞 lobby
groups for government attention: each group spends resources in attempt to win
favor; the one that spe扒出 the most is successfu l. (Other n飞odels that cast lobbying
as an auction are consider已d on page 91.) You are asked to study both first- and
second-price versions of an all-pay auction 出 the following exεrcise.
ø EXERCISε89.1 (A11-payauctions) Consìder variants of the sealed七id auctions 如
Sectìons 3.5.2 and 3.5.3 ìn which all bidders pay the winning price. Show that for
two bidders the varia时 of a second-price auction differs from the Wat of Attrition
shtdied in Section 3.4 only in the p在 yoffs when the players' bids are equal, a difference that does not affect the set of Nash equi1ibria. Show a1so that the variant of a
first-pricεauction has no 出3sh equilibrium.
Multiunít auctio时 In some auctions, like those for U.s. Treasury bills (short-term
government bonds) , many units of an object are availab1e , and each bidder may
value positive1y more 出an one uni t. ln each of the types of auction described ìn
the followìng list, each bidder submits a bid for each unit of the good. That is, an
action is a list of bids (b 1,. . ., b门， where b1 is the player's bìd for the first unit of
the good, b2 is her bid for the second unit, and so on. Thεplayer who submìts the
highest bid for any given unìt 0己tains that uni t. The auctions differ in the prices
paid by the winners. (The first type of auctíon generalizes a first-price auctíon,
whereas the next two generalize a second-price auctìon.)
。 iscriminatory audion
The price paid for each urut is the 11\岱tIling bid for that
unit.
Unifonn-price auction The price paid for each unit ìs the same, equal to the
highest rejectεd bid among all the bids for a11 unìts.
Vickreyauction A bidder who wins k objects pays the sum of the k highest rejected bids submitted by the other bidders.
The next exercise asks you to study these auctions when two units of an object are
avaìlable.
Chapter 3. Nash Equilibrium: lII ustrations
90
@E
缸
XERCIS臼E 锦
9O
在轮.1 (M
叫咄\I1
仇仙
u山i 飞m
圳
l贝1让叫州
i坟
臼t auctions) Two units of an 伪
0bj扣e
仅C时t are available. There
are 11 bìdders. Bìdder í 飞吸 lues the fírst ur飞it that she obtaíns at Vi and the second
unit at Wit where Ví > Wí > O. Each bidder subITÚts two bids; the two highest
bids wìn. Retain the tie-breaking rule in the tex t. Show that in discriminatoηland
uηiform-prìce auctions , player i's action of bidding Vi and wi does not dominate
a11 her other actions, whereas in a Vickrey auction it does. (In the case of a Vickrey
auction, consider separately the cases in which the other players' bids are such that
player i winsηo units , one u日泣， and two 注目 ts when her bidí3 are Vj and Wi.)
Goods for which the demand exceeds the supply at the going price are sometìmes sold to the people who are willing to wait longest in lü冶. We can model such
situations as multiunit auctions in whìch each person's bid is the amount of time
she is 叭'illing to wai t.
吉 EXERCl SE 90.2 (Waiting in line)
Two h飞mdred pεople are w i1ling to wait in line to
see a movie at a theater whose capacity is one hur飞dred. Denote person i's valuation of the movie ìn excess of the price of admissiOl、 expressed in terms of the
amount of tímξshe is ，气rilling to wait, by 飞 That is, person i's payoff if she waits
for t i units of time is 'V i - ti. Each p时son attaches no value to a second ticket, and
cannot buy tickets for other people. Assume th 3. t vl > V2 > ... > vzoo. Each
person chooses an arrival tíme. If sev台ral people arr Í\'e at the same time , then their
order in line is deterr挝nεd by their index (l ower-numbered people go first). If a
person arri飞'es to find 100 or more people already in 1ine , her payoff is zero. Model
the situation as a variant of a discriminatory multiunit auction , in which each person submits a bid for only one unit, and find its Nash equilibria. (Look at your
answer to Exercise 87 .1 bεfore seeking the Nash equilibria.) Arri飞'al times for people at movies do not in general 吨em to conform with a Nash equilibrium. What
feature missiηg from the model could explain the pattern of arrivals?
o
、"
The next exercise is another application of a multiur飞it auctìon. As in the previous exercise each person 飞Nants to buy only one unit, but in this cas台 the price paid
by the winners is the highest losing bid.
( EXER Cl SE 90.3 (Internet pricing) A proposal to dεal V\恤 co吨estion on electronic
message pathways is that 出 ch rnessage sh<旧ld include a field stating an arnount
of money the sender is ，飞!illing to pay for 出e message to be sen t. Suppose that
duril1吕 some tirne ìnterval , each of n people 飞，\'ants to send onεmessage and the
capacity of the pathway is k messages, with k < n. The k messages whose bids are
highest âre the ones se时， and each of the persons sending these messages pays a
pr附 equal to the (k + l)st 1吨hest bid. Model 出
th白
is 创
s it旧t抗ion asam
古乳H址
i址lt川
血
(Use 白
t h飞 e same tì淀
e-break
际ÍIη19 叭
r u阳l怡
ea
衍s the one in the 怡
tex
对t仁.) Does a pe
时rs即
on'、s 缸
a ctiOI旦.10f
biddi如
ng the value οof her me
臼ss
路
鸣
a
伊
g
ewe
盼
akl马Y doαrr妇
1让inat怡
e a 1l her other actions? (Note
that the auction differs frorn those considered in Exercise 90.1 becausεeach person
submits only one bid. Look at the argurnent in the text that in a second-price
sealed-bid auction a player's actiün of bîdding her 毛址ue weakly domin时es a11 her
other actions.)
3.6 Accident law
91
Lobbying as an auction Variants of the models in this section can be used to understand some situations that are not explicitly auctions. An example , illustrated
in the next exercise , is the competition between groups pressuring a goverrunent
to follow certain policies. This exercise shows also that the outcome of an auction
may depend significantly (and perhaps counterintuitively) on the form the auction
takes.
a EXERCISE 91. 1 (Lobbying as an auction) A government can pursue three policies ,
X , y , or z. The monetary values attached to these policies by two interest groups , A
and B, are given in Figure 91.1. The government chooses a policy in response to the
payments the interest groups make to it. Consider the following two mechanisms
First-price auction Each interest group chooses a policy and an amount of money
it is willing to pa y. The governrnent chooses the policy proposed by the group
willing to pay the mos t. This group makes its payment to the government ,
and the losing group makes no paymen t.
Menu auction Each interest group states , for each policy, the amount it is willing to pay to have the government implement that policy. The government
chooses the policy for which the sum of the payments the groups are willing
to make is the highest , and each group pays the government the amount of
money it is willing to pay for that policy.
1n each case each interest group's payoff is the value it attaches to the policy
implementeà minus the payment it makes. Assume that a tie is broken by the
governmen t' s choosing the policy, among those tied , whose name is first in the
alphabet
xnUOU
AB
foo
PAPA
uu
酌UFb
trk-Tt
JQU
、、
已它
duh
Ehm
TEA-tJ
y
3
-100
z
-100
3
Figure 91.1 The vallles of the interest grollps for the polici巳 5 X , y, and z in Exercise 91.1.
Show that the first-price auction has a Nash equilibrium in which lobby A says
it will pay 103 for y, lobby B says it will pay 103 for z, and the governrnent's revenue is 103. Show that the menu auction has a Nash equilibrium in which lobby A
announces that it will pay 3 for X, 6 for y, and 0 for z, and lobby B announces
that it will pay 3 for X, 0 for y , and 6 for z, and the goverrunent chooses x , obtaining a revenue of 6. (In each case the pair of actions given is in fact the unique
equilibrium.)
3.6 Accident law
3.6.7
Introduction
In some situations , laws influence the participants' payoffs and hence their actions.
For exarnple , a law may provide for the victim of an accident to be cornpensated by
Chapter 3. Nash 辈quilibrìum: lIIustration革
92
a party who was at fault , and the size of the compensation may affect the care that
each party takes. What laws can we expect to produce socìally desirable outcomes?
Agame斗heorεtic analysìs ìs useful in addressing this question.
3.6.2
i game
T加
Consider the interaction between an injurer (player 1) and a victim (player 2). The
vìctìm suffers a loss that depends on the amounts of care taken by both her and
the injurer. (How badly you hurt yourself when you fall down on the sidewalk
in front of my house depends on both how well I have cleared the ice and how
ca 1.'efully you tread.) Denote by Gj the amount of care player i takes, mea如 red
in monetary terms, and by L(肉，电) the loss , also measured ìn monetary terms ,
suffe陀d by the vìctim, as a function of the amounts of care. (In many cases the
victim does not suffer a los5 wìth certaint予 but only with probability less than
one. 1n such ca5es 认ie can interpret L( Gl , a2) as the expected loss-叩the average
loss suffered over many occurrence叫 Assume that L(al , a2) > 0 for all 飞raluεs of
(a1 , G2) , and that more care taken by either player reduces the loss: L ìs decreasing
in a1 for any fixed 飞ralue of 句， and decreasing 如句 for any fixed value of 肉，
A legal rule determines the fraction of the l05s bome by the injurer, as a function
of the amounts of ca陀 taken. De∞te this fraction by p(al ， a 纣 . If p(al , a2) = Ofor
all (叭 ， G2) ， for εxample， the vìctim b仓ars the entire los5 , regardless of how much
care she takes or how little care the injurer takes. At the other extreme , p(al' a2) 口 l
fOl' all (a ]， 时 means that the victim is fully compensated by the injur盯 no matter
hO\\I careless she is or how carefuÌ the Ìr飞jurer is.
lf the amounts of ca 把 are (町 ， tì2) ， then the Ì口jurer bears the cost al of taking
care and suffers the loss of L 但 1 ， a卦， of which she bears thεfractìon p( 肉 ， G2)' Thus
the injur时 's payoff is
一 Gl- p(al ， a2)L(al ， 句) .
Simìlarly, thεvictim's payoff is
币 G2 → (1 … p(al ， a2))L(a }， G2)'
For any given legal rule , embodied in p, we can 1衍。del the interaction between
the injurer and víctim as the follmving strategic game.
PJayers The i叶口陀r and the victim.
Act;ons Th e set of actions of each pl吗rer is the set of possible levels of care
{只ònnegative 口umber啡.
Preferences The injurer's p陀fere口ces are represented by the payoff function
-al … p( 叭，日2)L( 町，也) and the victim's preferences are represented by the
payoff function -a2 - 口一 p(哟，句 ) )L(肉，句)， where a1 í5 tt飞e injurer's level
of care and a2 is thε 飞rìctim's level of care
3.6 Accident law
93
How do the equilíbria of this game depend upon the legal rule? 00 any legal
rules lead to socially desirable equilibrìum outcomes?
1 restrict a tter飞 tion to a class of legal rules known as negligence with contríbutory
Ill'gligel附. (This class was established in the United States in the mid-19th centur予
and prevailed untíl the mid-1970s.) Each rule in this class requires the injm时 to
compensate the victim for a loss if and only if bolh the victim is sufficiently careful
Gnd the i日jurer is sufficiently careless; the required compensatior飞 is the totalloss.
Rules in the dass differ in the standards of care they specify for each party. τhe
rule that specifies the standards oÍ care X1 for the injurer and X2 for the vÌCtim
requires the i吟urer to pay the victim the entire loss L( 肉，句) when G1
X1 (the
injurer is ir飞sufficiently careful) and 的三 X2 (the victim is su日iciently careful) , and
ω2) οof the loss born
nothi吨 。thenvise. That is , u叫 er this rule thε fraction p(a} , 必句
斗jure
盯r is
by the in
p(n1 , 11 2)
(IK 山川Z
o if Gl ? X1 or a2 < X2
Included in this class of rules are those for which X1 is a positive finite nu日1ber
and X2
0 (the ÍI亏 urer has to pay if shεis not sufficiεntly careful , e\'en if the
飞rictim takes no care at a l1), kn缸。wn as rules of pure 11号Iigence， and that for which X1
is infinite and
0 (the injurer has to pay regardless of how ca陀ful she is and
how careless the victim is) , known as the rule of strict liabilìty.
3.6.3 Nash equilibrium
Supposε 附 decide tha t the pair (句 ， â 2 ) of actÎons is socially desirable.
We wish
to a只:swεr the question: Are there valu础。f X1 and
such that the game generated by the rule of negligence with contributory ne在ligence for (X1, X2 \ has
(â 1, â2) as its unique Nash equilibrium? If the answer is affirmative, then, assum附
ing that the solution concept of Nash equilíbrium is appropriate for the sìtuatior飞
快'e are considering, we have found a legal rule that ìnduces the socially desirable
outcome.
Specificall予 suppose that we select as socially desirable the pair (â 1, â2 ) of
actior飞s that maximizes the sum of the playe邸， payoffs. That is ,
、
{句，句) maximizes -al 一的一 L( 町，时
(For some functions L , this pair (句 ， â 2 ) may be a r.εasonable candidate for a socially
desirablεoutcome; in other cases it may induce a very in吨uitable distribution 01
payoff between the players, and thus be an unlikely candidate.)
1 claim that the unique Nash equìlìbrium of the game induced by the legal rule
(句，占2) is 仙 ， â2)' That
of neglìgence with contributory negligence f时间，马)
is , if the standards of εare are equal to tl飞eir socially desirable le飞rels， then these a主e
the levels chosen by an 如jurer and a victirn in the only equilìbrium of 在e game.
The outcorne is 佼lat the injurer pays no cornpensation: her level of care is 句， just
high enough that p 仙，时= O. 人t the 明
Chapter 3. Nash 在quilibrium: lIIustrations
94
O
。1
aJ 一→
O
a2 …•
a2
AAM
l F nu
叫/F
句，/啕
li
，，a
，
毒、、
尺二l， (12)
)
Figure 94.1 The left panel shows the ínj让 reτ's payoff as a function of her le干 el of care a] whel飞 the
victim's level of care !S "2ζ 占2 (see (94 .1 )).τhe ríght pa时必ows the victim's payoff as a function of
her JeveJ of care 112 时 hen the injurer's level of caγe is 111 .1 1 (see (94.2刘华
且igh enough that if the injurer reduces her level of care even slightlμthen she has
to pay fullεompensatìon.
1 first argue that (鸟 ，(1 2) is Ll Nash 叫 uilìbrium of the game, then show that ìt is
the ol1 ìy equilibrium. To show that (t7 1, LÎ2) is a Nash equihbrium, 1 need to show
that the í时旧时 's action t7 j is a best response to the vìctím's action â2 and vice verS Ll
Given that the victim's aεtion is h the injm巳r has to pay compensation if and only if Ll j < â j . 丁hus the i口jurer's payoff ís
Injur盯's action
U 1(町， iî 2 )
(h 三;:
(94.1)
For ilj = t1 1' thìs payoff is …注 j. 开始εtakes more care than 句， she is 认'orse
off, because care ìs costly a公d， beyond âj , does not reduce her lìabílity for
compensation. If she takes less caτe， then , gi飞ren the ViC ìÌ I刀 's level of care ,
she has to pay compensation , and we need toεompare the money saved by
taking less care with the size of the compensatìon. The argum巳nt ìs a li ttle
tricky. Fì rst, by definition ,
(iîj , iî2) maximizes -Ll l - a2 - L( 句，创.
Hence
il l n飞aximizes ← al
t1 2
L( 汀 J ， lh)
(given iîÚ Because iî2 is a constant, it fo11ows that
âj maximìzes -ilJ - L(肉， (1 2)'
But from (94.]) we s巳e that … al ← L( 肉 ， â 2 ) is the iηjurer's payoff U1 (al , â2)
when her actîon is al < â1 and the victim's action is íh We conclude that
the injurer's payoft takes a form like that in the Jeft panel of Figure 94. 1. ln
particuJar, â1 maximizes Uj (al' 削， so that â1 is a best response to â2.
Victim's action Given that the injurer's action Îs â1, the victim nεver receives
compensatio拉丁hus her payoff ìs
u2(â 1, il2) = -a2 - L( t1 1, a2)'
(94.2)
3.6 Accident law
95
队与 can argue as we did for the injurer.
…
By defì州iOI飞机}， â2) maximizes
a1 - a2 - L(a1 , a2) , so
m
v八
究M
qJ
舶
Anu
mZPM
<
nu
<
L(â 1, a2)
a2
M
(given â1 ). Because tÎ} is a constant, ìt follows that
a2 役laxl臼nzεS
-{/2
L( tÎ 1, (12)'
(95.1)
which ÌS the victim's payoff (see (94.2) and the right panel of Figure 94.1).
That is , tÎ 2 maximizes 的(句 ， a2) ， so th拭占 2 is a best response to tÎ}
We conclude that (句，也) is 击时ash equilibrium of the gameinduced by the
legal rule of negligence with contributory negligence when the standards of care
are tÎ l for the injurer and â2 for the victim.
To show tha t (句，占2) is the only Nash equilibrium of the game, fírst consider the
injurer's best response function. Her payoff functio贝岭
nH
叫F如
,
if al < tÎ} and a2 ? tÎ 2
if al ? â1 or a2 < 占2.
、、，，
J，
i t I ( nH-
We can split the analysis into three cases, according to the victirr飞 's level of care
的< tÎ 2:
In this cas巳 the injurer does not have to pay any compensation, regardless of her level of care; her payoff is … al , so tl飞at her best response is I1 J = O.
旬之三岛
In this case the inju陀r's best response is 向， as 1 argued when 1 showed
that (鸟 ， â2) is a Nash equilibrium
a2 > 句
In this case the injur.εr's best response ís at most tÎ l' because her payoff
for larger values of al is equal to -111 , a decreasing function of al'
We conclude that thεinjurer's best response function takes a form líke that shown
in the left panel of Figure 96.1
Now, given that the injurer's best respo口se to any value of a2 is never greater
than tÎ l , in any equilibrium we have al 三句: anypoint (al , 112) atwhich thevictim's
best response function crosses the injurer's best response function must haveβ1 <二
tÎ 1. (Draw a few possible best responsεfunctions for the victim in the left panel of
Figure 96. 1.) We know that the 飞怡im's best response to tÎ 1 is tÎ 2 (because (â 1,tÎ 2 ) is
a 时 ash equilibrium) , so we need to worry only about the victim's best responses to
values of al with al < 乌 (i.e. for cases in which the i叫 urer takes insufficient care).
Let al < tÎ 1 . 古len if the victim takes insufficient care she bears the loss; othe卜
wise she is compensated for the loss , and hence bears only the cost 112 of her taking
care. Thus the victim's payoff is
rsF飞 31 、
句/伽
一一
叫/­
n抖
u ( l nu )
nMhnu
q4qL
……
L(al , a2)
if 112 < tÎ 2
if a2 ~二tÎ 2.
(95.2)
Chapter 3. Nash 在quilibrium: lII ustrations
9岳
?何
牟 ggj
J
冈F伽
门时
a?
。
2172{RU
â1
111 •
。
卢
lÎ 1
、
、
?、 d
111 •
Figur曹雪6.1 丁he players' best response functions under the rule of negligence with contributory neglí
gence when (鸟， X2) 工怡，创 .τhe injurer'远拢到 response function b1 is in the left paneL The \'ictím's
best response functíon b2 ís ìηthe right pan日1 仔he positíon of the víctim's best rεsponse fur比tion for
11[ > ÎÌ 1 is not sígnìfic在nt and ís not determined in the tex t.}
Now, by (95 .1) the level of care lÎ 2 maximizes 一句
-112 … L( lÎ l' 112) 三-lÎ2
L( lÎ 1 ， (/2) , so that
L( â 1,lÎ 2) for all 112
Further, the loss is nonnegati\吃 SO -lÎ 2- L(lÎ 1 ，lÎ2) 三
112 - LUh ， 旬)三
for all 112.
W们onclude that
(96 .1)
Finally, the loss inζ陀ases as the injurer takes less care, so that given a1 < 向 we
have L( 1l 1, (/2) > L( lÎ 1,1l 2) foralla2' Thus -112 -- L( 1l 1,1l 2) < -(/2ι( tÎ 1 ，1l 2) for a11
(/2 , and hence, llsing (96 .1), we condude that
-(/2- L((/] ,
<
…
fh for all 112
From (95.2) it follows that the 飞rictim's best response to any a] < lÎ 1 is 旬， asshown
in the right panel of Figure 96 .1.
Combini鸣 the t叭?叫)est response functio邸， we see that (lÎ 1, ÍÎ2) , the pair of levω
els of care that maximizes the sum of the players' payoffs , is th巳 unique Nash e弓ui­
librium of the game.τhat is , the rule of negligenεe with contributory n鸣ligence
for standards of Ci:1rεeqllal to lÎ 1 a1飞d 占2 induces the players to choose these le飞rels
oI care. If legislators can determinεthe values of lÎ 1 and 缸， then by writing thes巳
levels into law they 飞代lill induce a gam已 that has as its unigue Nash equilibrium the
socially optimal actions.
Other standards also i叫uce a pair of levels of care equal to (ι 1Î 2 ) , 部 you are
asked to show in the following exercise.
@ EXERCISE 96.2 (Alternative standards of care under negligence wìth contrìbutory
negligence) Show that 仙，lÎ 2) is the unique Nash equilibrium for the rule of negligence with contributory negligence for any value of (X 1, X2 ) for which either
X1
lÎ 1 and X2 :S: lÎ 2 (including the pu陀 neglig巳nce case of X2 工 0)， 01' X 1 之九f
and X2 = lÎ 2 for suffìciently large M. (Use the lines of argument in the tex t.)
97
Notes
? EXERCISE 97 .1 (Eguilibrium under strict liability) Study the Nash 叫出librium
(equilibria?) of the game studied in the text under the rule of strict liability, in
which X} is infinite and X2 0 (i .e. the injurer is liable for the loss no matter how
careful she is and how careless the victim is). How are the equilibrium actions
related to â} ar飞d
Notes
The model in Section 3.1 was developed by Cournot (1838) ‘在1e model in Section 3.2 is widely creditcd to Bertrand (1 883). The box on page 69 is based on
Leonard (19q4) and Magnan de Bornier (1992). The models are discussed in more
detail by Sh在piro (1989).
τhe model in Section 3.3 is due to 日otelling (1929) (though the focus of his P公
per is a model ìn which the players are fi口ns that choosεnot only locatìons, but also
prices). Downs (1957 , especially Chapter 8) popularized Hotelling's model , using
lt to ga如 ìnsights about electoral competitíon Shepsle (1991) and Osborne (1995)
survεy work in the field.
The War 吃f Attritio l1 studied in Section 3.4 is due to Maynard Smith (1974); it is
a variant of the Ðollar A lI ctio /1 presented by Shubik (1971) (s时 Example 175 .1).
Vickrey (1961) initiatεd the formal modeling of auctions, as studied in Sec俨
tìon
The literature is surv吗red by Wils∞ (1992). 非le box on page 81 dravvs on
丑erodotus' Histories (Book 1, paragraph 196; s假， for example , Herodotus 1998 ， 8哟，
Langdon (1994) , Cassady (1967, Chapter 坷， Shubik (1983) , Andreau (1999 , 38~39) ，
the wεbsite www.eBay.ω风 Roth and Ocker飞 fels (2002) , ar飞d personal correspon棚
denc台 with Robin G. Osborne (on ancient Greece and 丑。me) and John H. Munro
(on medieval Europe).
The model of accident law in Section 3.6 originated with Brown (1973) and
Diamond (1974); the result about r飞egligence with contr和 utory negligence is due to
Brown (1973 / 340-341). The literature is surveyed by Benoît and Kornhauser (2002).
Novshek and Sonnenschein (1978) study, in a general setting, the issue addressed in Exercise 62 .1. A brief summary of the early work on common prop白ty
is given in the Notes to Chapter 2. The idea of the tie-breaking rule being determined by thεequilìbrium， used in Exercises 68.2 and 69.1 , is due to Simon and
Zame (1990).τhe result in Exercise 75.1 is due to Wittman (1977). Exercise 75.2
is based on Osborne and Slì飞rinski (1996). 币1e notion of a Condorcet winner de唰
fined in Ex时ci忧 75.3 is associatεd with Marie-Jean-Antoine-Nicolas de Caritat,
marquis de Condorcet (1743-94), an early studεnt of voting proεedures. 丁hegame
in Exεrcise '80.1 is a variant of a game s协died by Bla.:::kwε11 and Girschick (1954,
Example 5 in Chapter 2). It is an example of a noisy duel (which models the situation of duelists, each of whom chooses when to fire a single bullet, which her
opponεnt hears, as she gradually approaches h衍 rival). Duels were first mo
‘
98
Chapter 3. Nash Equilibrium: lII ustration辈
situation consid台red in Exercise 90.2, in which people decide whεn to join a queue,
Ís studied by Holt and Sherman (1982). Exercìse 90 .3 怒
i s bas恍
ed on Ma
配cKi淀
e
and Vari以a缸
an (1 995) .
Mixed Strategy Equilibrium
4.1
Introduction 99
4.2
Strategic games ìn which players may randomize 106
4.3
Mixεd strategy Nash equilìbrium 107
4 .4
Domìn 吕ted actions 120
4.5
Pure equilíbria when randomization is allowed 122
4.6
IlI ustration: expert diagnosis 123
4.7
Eq Ul librium in a single population 128
4.8
IIlustratìon: reporting a crime 131
4.9 The formation of players' beliefs 134
4.10 Extension: finding all mixed strategy Nash equilibria 137
4.11 Extension: games ìn which each player has a
continuum of actions 142
4.12 Appendix: representing preferences by
expected payo仔s 146
Prerequìsite: Chapter 2
4.1
Introduction
4.1.1
Stochastic s伦adystates
NASH EQUILIBRIUM of a strategic game is an action profile in which every
player's action is optimal given every other player's action (Definitior飞 23.1).
Such an action pr。在le cürresponds to a steady state of the idealized situation in
which for each play盯 in the game there is a population of individuals, and whene飞咽 the gamεis played , üne player is drawηrandomly from each population (see
Section 2.6). 1n a steady state, every play缸 's beha飞Tior is the same whenever she
plays the game, and no player wishes toζhange her beha叫时~ knowing (from her
experience) the other players' behavior. 1n a steady state in wmch each player's
"behavior" ís simply an action and witmn each populatior飞 a11 players choose the
same action, the outcome of every play of the game is the same Nash 叫出librium.
More general notions of a steady state allow the players' choices to vary, as
long as the patterηof choices remainsωnstant. For example, different members
of a given population may choosεdifferent actions, each player choosing the same
action whenever she plays the game. Or each individual may, on each occasion
she plays the game, choose her action probabilistically according to the sar肘， u汗
char飞起ing distribution. These two more general notions of a steady state are equiv-
Chapt衍 4.
100
Mbc:ed Strategy 军quilibrium
alen t: a steady state of the 在rst type i口 which the fraction p of the population representìng player í chooses the action a corrεsponds to a st臼 dy state of the seεond
type in which each member of the population representing player i chooses a with
probabílity p. In both cases, in each play of the game the probability that the individual in the role of play衍 i chooses a is p. Both these notions of steady state are
modeled by a mixed strategy N剖h equilibrium, a generalization of the notion of
Nash equilibrium. For expository convenience, in most of this chapter 1 interpriεt
such an equilìbrium as a model of the second type of steady state, in which each
player chooses 乞er actions probabilistically; such a steady state is called stochastic
("i nvol vir飞 g probability'丁
4.1 .2 Example: Matching Pennies
An analysis of the game Matching Pennies (Examplε19.1) illustrates the idea of
a stochastic steady stat廷. My discussion focuses on the outcomes of this game ,
gl飞'en in Fìgure 100 .1, rather than payoffs that represent the players' preferences ,
as before.
Head
Head
Tail
$1 ， 一 $1
…
$1 , $1
Tail
… $1 ，
$1
$1 ，一 $1
Figure 100.1 The outcom吉S of Matclúng PCllnies.
Aswesaw pre飞如usly， this game has no Nash equìlibrium: no pair of actions is
compatible with a steady state in which each player's action is the sarr飞e whenever
the garr飞e is played. 1 cla幻ÍI泣1飞、， howe队飞ver乙~ that the gam恕e has a st约ochastí比c sb怡eady st怡at怆ein1
which ea挝chpl怡ayer cho
∞
osese怯acho
时f门he们
r川act
柑巾
t挝io肌，yi
加
江thp时a拍bility
剧圳
τoe归b
悦li始sh 白
附1lt， 1 need to 吨ue that if player 2ωoses 台achofher a伽ns with probability ~，
阳1 player 1 optimally chooses each of her actions with probabilìty , and vice
versa.
i
i
S叩pos使刷衍材t血
ha挝tp
乒laye旺
ω让
r
2 ω os优e臼
阳se
俯ach ofh附
era挝ctions
础
叭
w ìt白
hp
严rob
阳怡ab
说圳圳?让训州
i过l山i时 If playe衍r1
εch旧
00
仍
se
部s He，
扭
tαld 叭
w i技t仗
迅h probability p and 茹
1b
挝挝
i辽1 with probabi川li让ty1 一 p, then each out-
i
刷过li泣tY p a 叫 each 0址吠C∞om
丑1
come (He叫 Head) and (Head , Tai l) occurs with proba抽bi
(阳药础
i汗!， 怆
H t硝 扭d (写
TaiI， 丑i
Ta 的
i斗) 创
0 Cα
∞创
ζ轧
cur
盯
u臼rs
臼s 叭t出
11 probabilìty 抨i (口1- ，川斗牛. τ
布
hu
旧
础s 阳
u
th
怡ep
严严roba刷
bil均
i技t咛
均
Y
tl刚 the outcome is eith在 (Head， Head) or (Ta汀，布的， in which case player 1 gains
$1 , is jp 卡;但…抖， which is 吨ual to
In the other two outcomes, (Head , Tail)
and (Tail , Head) , she loses $1 , so the probability of her losing $1 is also !‘ In par俨
ticular, thεprobability distribution over outcomes is independent of p! 丁nus every
value of p is optima l. In particular, player 1 can do no better than choose Head with
A similar analysis shows that player 2
probability and Tail with probability
optimally choo附 e础 action with probability when play臼 1 does so. We corν
clude that the game has a stochastic steady state in whichωch player chooses each
aωn with p时ability
!.
i
i.
i
!
F
4.1 Introduction
101
1 further claim that, under a reasonable assumption on the players' pref台i rences，
the game has no other steady state. This assumption is that each player wants the
probability of her gaining $1 to be as large as possible. More precisely, if p > 号， then
each player prefers to gain $1 with probabílity p and los对 $1 with probability 1 - P
than to gain 革1 with probability 可 and lose $1 飞气rìth probability 1 子
To show that under this assumption there îs no steady state in which the probabìlity of each 1均rer's choosing Head is dîfferent from !' denote the probabìlity
with which player 2 chooses Head by q (so that she chooses 百 il with probabìlíty 1 一书. If player 1 chooses Head with probability p, then she gaîns $1 with probabi 1ity pq 个 (1 纠口一的 (the probab i1ity that the outcome is either (Hεéad， Head的)
创r (7
O
刊
ài♂1， 丑
7ài均.
+ p(2q - 1) and the second 沁i s 叫
e qual to q 才- p(l … 2q). Thus if
号< ! (player 2 chooses Head wîth probab Ui ty less than
the fi时 probabìlîty is
decreasing ìn p and the seconà ìs increasîng in p, so that the lower is p, the better
is the outcome for player 1; the value of p 也at induces the best probabilìty distri跚
bution over outcomes for player 1 is O. That is , íf player 2 chooses Head with prob自
由ility less than !' then the u均uely be叫olicy for player 1 is to choose Taíl with
certainty. A similar argument sho马川 that îf player 2 chooses Head with probability
grea怡r than !' the uniq时 ybe句。licy for player 1 is to choo附 Head with certainty人
Now, if player 1 chooses an action with certaînty, a sîmilar analysis leads to the
conclusion that the op垣mal policy of player 2 必 to choose an action wîth certainty
(Head if player 1 chooses Tail and Tail if player 1 chooses Head).
We conclude that there îs no st讼eady sta挝te 泣m飞 which the probabi拄1i让ty that player 2
chοose仍s Head di证ffeε衍仍
r陀sf
时丘古附
衍I放n!.
O
舶
A叩
s归
t恒
at祀
e 川
i nw
咐
hi协
ch 脂
th
挝
ep
严ro
伪
ba
巾bi
州山
lit々
yt由
ha
况tp
卢
lay
阳
盯r 1 chooses Head differs 仕的 3· ITmin
e
the only stochastic 阳ady 如 te each player Ch005εs each action 时h probability ~.
As dîscussed in thεopening sectîon (4 .1 .1), the stable pattem of behavior we
have found can bεaltematively interpreted as a steady state in which no player
randomizes. Instead , half the players in thεpopulation of in岳飞Jiduals who take
the role of player 1 in the 伊me choose Head whenever they play the game and half
of them choosε Tail whenever thεy play the game; similarly half of those who take
the role of player 2 choose Head and half choos
i让ty 总
i 5e
吨气ual tο1 一 q
4. 1.3
!),
Genera/i芝加9 the analysis: expected payo疗5
The fact that Matchi号 Pennies has only two outcomes for each player (gain $1 , lose
$1) makes the analysis of a stod飞船tic steady state particularly simple because it
allows us to deduc已 under a weak assurnption, the players' preferences regarding
Chapter 4. Mixed Strategy Equilibrium
102
lotteries (probabilìty distributions) over outcomes from their preferences regarding
deterministic outcomes (outcom臼吐lat occur with certainty). If a player prefers
the deterministic outcome a to the d社衍ministic outcomε 丸 it is very plausible that
if P > 号， then she prefers the lottery 担 which a occurS with probability p (and b
occurs with probability 1 一片 to the lottery in which a occurs with probability 号
(and b OCCurS with probability 1 …叭.
In a game with more than two outcomes for som心 player， we caru飞ot extrapφ
late 如 this way from preferences regarding deterministic outcomes to pref扭扭ces
regarding lotteries over outcomes. Supposε! for example, that a game has thrεe
possible outcomes, a, b, and c, and that a play时 prefers a to b to c. Does she prefer
the deterministic outcome b to the lottεry ín which a and c each occur with prob站ility or vice versa?τhe information abOL的er prefe陀附s over deterministic
outcomes gives us no clue 在bout the ans'v年时 to this question. She may prefer b to
the lottery in which a and c each 0削r with probability ~， or she may p时e时r 由
i如
ot位ter巧yγ tωob
仇; both pr
拮
ef伦
er
陀
ences are ∞
cζon怒硝$纣i始st怡
ε nt with h盯 pr陀
εf位
erηri凶
ng a to b 抬
toc誓 Tostudy
her behavior when she is faced with choices between lotteries, we need to add to
the model a dεscription of her preferences regarding lotteries over outcomes.
A standard assumption in game theory restricts attention to preferences regarding lotteries over outcomes that may be represented by the expected value of a pay。ff function over deterministic outcomes. (See Section 17.6.3 if you are unfamiliar
with th已 notion of 、xpectεd 飞lalue".) That is, for every player i there is a payoff
function 11; with th泛扣'operty that player i prefers one lottery over out∞mes to another if and only if, according to ll; , the expected value of the first lottεryexceeds
the expected value of the second lottery.
For example , suppose that there are th1程。utcomes， a, b, and c, and lottery P yields a with probab i1i ty ].1 11 , b with probability Pb , and c wit1飞 probabil­
ity pc , whereas lottery Q yields these three outcomεs with probabilities 如 I qb ,
and 华. Then the assumption is that for each player i thεre are 川 mbers 叫均 F
tli( 的， and 以 c) such that player i prefers lottery P to lottery 苟延 and only if
PIl Ui(a) + P/l !lì( 件十 pcUi(C) > 号。 tli(a) 十 qbUi(b) 十号cUi(C) ， (1 discuss the represent在 tion of preferences by the expected 飞'alue of a payoff functi011 in more detail in
Secti
!'
4.1 I码在roduction
103
terìes over outcomes required for them to bεrepresented by the expected value
of a payoff function. 。巾 not discuss these restrictìons, but the box at the end
of this section gives an exarτlple of preferences that víolate thern.) Nevertheless,
we obtain many insights from modeJs that assume that preferences take this foτm;
following standard game theory (and standard economic theorY)t 1 maìntain the
assumption throughout the book.
The assumption that a player's preferences are represent号d by the expected
value of a payoff function does not restrìct her attitudes to rìs k: a person whose
preferences are represented by such a function may have an arbitrarily strong
liking or dislike for rísk.
Suppose , for example , that a, b, and c are three outcomes, and a person prefers
a to b to c. If the person is very averse to risky outcomes, then she prefers to obtain
b for sure rather than to face the Iottery 如 which a occurs with probability p and c
occurs with probability 1 一 pJ ev巳n if Pis relatively large. Such prεferences rnay be
represented by the expected value of a payoff function u for which u(α) is cIos毛毛 to
u( 抖， which is much larger than u(c). For a case in which a, b, and c are nurnbers ,
S让ch payoffs are illustrated in the left panel of Figure 103 .1.
If the pεrson is not at all averse to risky outcomes, then shεprefers the lotte巧
to the certain outcome b, even if p is relatively small. Such preferences are represented by thεexpected value of a payoff function u for which u(a) is much larger
than 叫的， which ìs close to u(c). For a case ìn which a, b and c are numbers, such
pay。在s are ìllustrated in the right panel of Figure 103 .1. If u(a) = 10, u 圳= 9,
and u(c) = 0, for example , then the person pr'εfers the certain outcome b to any
10忱的 betwee川 and c that yields a with probability less thal飞岳阳 ifu(a) 10 ,
町的口 1 ， and u(斗= 0, she prefers any lottery between r. and c that yields a with
p时ability grea阳阳1 击 to the certain 0忧。meb
Suppose that the outcomes a l'e amounts of money, and a person's preferences
are represented by the expected value of a payoff functìon in which the payoff of
each outcome is equal to the amount of money involved. Then we say the person is
risk l1 eutral. Such a person compares lotteries according to the expected arnount of
J
u(a)
u(b)
飞)
一，ι
，l，
飞
h、
u pb
•
u(a)
•
•
•
•
C
b
G
C
告
G
Figure 103.1 The BemoulH payoffs for the outcomes a, b, and c for persons who prefer a to b to c and
very averse to 始终 (left panel) and not at all averse to risk (right panel) 。
在re
ζhapter 4.
104
Mixed Strategy 主quilibrium
money involved. (For example , she is indifferent between receiving $100 for sure
and the 10阳y that yields $0 with probability 是 and $1000 with prob拍ility 击)
On the one hand , the fact that pωple buy insurance suggests 出at in some circur主l­
stances preferences are risk averse: people prefer to obtaín $z with certainty than
to receive the outcome of a lottery that yíelds $z on average. On the other hand ,
the fact that people buy lottery tickets that pay, on average , much less than their
purcha黯 prìce， suggests that in other circumstances preferences are risk prt户'对ng.
ln both cases, preferences over lotteries are not represented by εxpected monetary
values, though they still may be represented by the expected value of a payoff function (in whìch the payoffs to outcome are different from the monetary 飞ralues of the
outcomes).
A到y given preferences over det时ministic outcomes are represented by many
different payoff functions (see Section 1. 2.2) 拿 The same is tnLe of preferences over
lott创业s; the relation between payoff functìons whose expected 飞ralues represent
the same preferences is discussed in Section 4.12.2 iηthe appendix to this chapter.
ln particul地 we mav choosεarbitrary payoffs for the outcomes that are best and
worst according to the preferences , as long as the payoff to the best outcome excεeds thεpayoff to the worst outcome. Suppose , for example , that there are three
outcomes, a, b, and c, and a person prefers a to b to c, and is indifferent 乞etween
b and the lottery tha t 内lds a with probability and c with p时ability
宫廷n
we may choose u(a) = 3 and u(c) = 1, in whìch case u 例 :2; 。毛 for example ,
we may choose u( 叫= 10 and u(c) 口。， 1只 which case 叫 b)=5， oru 但) := 1 and
叫 c) = … 1, in whìch case 叫 b) = 0
i
i
SOME EV lD ENCE ON 在 XPECTED 1'1\ YOFF FUNCTIONS
Consider the following two lott凹 ies (the first of which is, in fact , deterministic):
Lottery 1γ'o u receive $2 million with certainty.
Lottery 2 You receive $10 million with probability 0 .1，辑部illion with probabil翩
ity 0.89 , and nothing with probabìlity 0.0 1.
Which do you pref，εr? Now consider two more lotteri部:
Lottery 3 You receive $2 million with probability 0.1 1 and nothing witl1 probability 0 .89.
Lottery 4 油u receive $10 million with probabilìty 0.1 and nothing with probabilityO.9
队fhich do you prefer? A significant fraction of εxpenm岱飞tal subjects say they pre-
fer lott衍Y 1 to lottery 2, and lottery 4 to lottery 3. (See, for example, Conlisk 1989
and Camerer 1995 , 622--6 23.)
These preferences cannot be represented by an expectεd payoff function! If
they could be there would exist a payoff function u for which the expected payoff
i
碍.1
Introduction
105
of lottery 1 exce巴ds tha t of 10ttery 2:
μ(2)
0 .1 μ(10) 个 0.89 叫 2) 个 0.01 叫 0) ，
where the amounts of money are expressed ìηmillions. Subtractìng 0.89 叫 2)and
adding 0.89 叫。) to each side we obtaìn
0. l1 u(2) 十 0.89u(0) > O.l u( lO) 个 0.9u(0)
But this inequality says that the expected payoff of lottery 3 exceeds that of 10ttery 4! Thus preferences represented by an expected payoff function that yield a
preference for lottery 1 over lottery 2 must also yield a preference for lottery 3 over
lottery 4
Prefer已nces represented by the expected valuεof a payoff function are, ho认儿
ever, consistent with a perso日 's being indifferent between lotteries 1 and 2, and
betw时n lotteriεs 3 and 4. Suppose we assume that when a person is almost indiHerent between two lotteries, she may make a μmistake"τhen a person's ex响
pressed preference for lottery 1 over lottery 2 and for lottery 4 over lottery 3 is not
directJy inconsistent with her preferences' being represented by the expected va1ue
of a payoff functio日如 which she is almost indifferent bet飞把en lotteries 1 and 2 and
怡tween lotteries 3 and 生狂， however, we add the assumption that mistakes are
di5tributed symmetrically, then the fr仅luency with which people expre5s a prefer翩
ence for lottery 2 over lottery 1 and for lott心ry 4 over lottery 3 (also inconsistent
with preferences reprεsented by the expected value of a payoff function) should be
similar to that with which people express a preference for lottery 1 over lottery 2
and for lottery 3 over lottery 4. In fact , however, the second pattern is significantly
more common than the first (Conlisk 1989) , 50 that a more significant modification
of the theory is needed to explain the ob5ervations
A lìmitation of theεvidence is that it is based on thεprefe陀nces expressed
by people faced with hypothetical choices; understandably (gi飞'en the amounts of
money involvε码，口o experiment has been run i贝 which subjects were paid according to the lotteries they chose! Experiments with stakes consistent with normal
research budgets show few choices inconsistent with preferences represented by
the expected value of a payoH functìon (Con1isk 1989). This evidenc巳， however,
does not contradict thεevidence based on hypothetical choices with large stakes:
with larg时 stakes subjects might make choices in line with the prefεrences they
express when asked about hypothetical choices.
In summ盯y， the evidence for an inconsistency with p陀ferences compatible
with an expected payoff function is, at a mìnìmuffi, suggestive. It has spurred
the development of alternative theories. Nevertheless, the vast majority of models i
Chapter 4. Mixed Strategy Equilibrium
106
4.2 Strategic games in which players may randomize
τb study stochastic steady states , we extend the notion of a strategic game given
in Definition 13.1 by endowìng each player with vNM prefe陀nces about lotteries
over the set of action profiles.
j rences)
D笠 FINITION 106.1 (Strategíc game with vNM pr拚
A strategic game (with
vNM preferences) consists of
• a set of players
• for each play衍， a set of actions
• for each player, preferences regarding lotteries over action pro fi1 es that may
be represented by the expected value of a ("Bernoulli") payoff functicn over
action profiles.
A two-player strategic game with vNM preferences in which each player has
finitely many actions may be presented in a table like those in Chapter 2. Such
a table ìooks exactly tl飞且e same as it did before, though the interpretation of the
numbers in the boxes is different. ln Chapter 2 these numbers are values of payoff
functions tha t reprεsent the players' preferences over deterministic outcomes; here
thεy are the values of 拮ernoulli) payoff functions whoseεxpected \'alues represent
the players' prefereηces over lotteries.
Given the change in the interpretation of thεpayoffs， two tables that represent
the same strategic game wíth ordinal preferences no longer necessarily represent
the same strategic game with vNM preferences. For exarnple, the two tables in
Figure 107.1 represent the same garne with ordinal preferences… namely the Pris
0 l1 er's Dílemma (Sectio口 2 号公 ln both cases the best outcome for each player ís that
in which she chooses F and the other player chooses ♀ the next bεst outcome is
(Q , Q) , then comes (F , F) , and the worst outcome is that in which she chooses Q
and the other player chooses F. However, the tables represent d阶rel1 t stra校
games with vNM preferences. For example, in the table on thεleft， p la yer l's pay
Iη?抒F 附
off to (Q , Q) is the sa仰
proba抬b
讪
刷i让li让ty and (F, 盯F) with pro
协babi出
i丑i tW
y
i七扫扫
μ叫
l1 巳， Q) + ←
i i叫
t勺1川(σF， 川F) 二 .;3 十 j 1 工
2口 M
均1 (Q , Q)川)， whereas in the table on the right, her payoff to (Q , Q) is grea 加 than
herexpe必d payoff to 协 lottery (3 >
料~ .1). Thus thεlef山and table rep叶
sents a situation in which playεr 1 is indifferent be队leen the deterministic outcome
(Q , Q) and the lottery in wh灿 (F， Q) occurs with probability ~ and 在， F) occurs
with pr抽油ilìty In 出
th飞e 咆
ri gh山ar时怡协
ble， r阳e盯
附
veer川hep
附陀fers
川
t出
hede吐te衍ηrm
m
妓
m
吉哀
飞
outcome (Q , Q) 怕
t o the 沁
1 0 ÌÌ!饺
ery
予.
To show, as in this example, that two tables represent different strategic games
wit
i
i(
i
i.
i
4.3 Mixed strategy Nash equilibrium
Q
F
Q
2, 2
3, 0
107
F
。， 3
1, 1
Q
F
F
Q
3, 3
0，岳
岳， 0
1, 1
Figure 107.1 Two tables that repre始时 the same strategíc game wìth ordinal preferences but different
strategic games with vNM preferences
ences over determimstic outcomes are the same as they are in BoS (Example 18.2),
and their preferences over lotteries satisfy the following condition. Each player
is indifferent betwεen (i) going to hεr less preferred concert in the company of
the other player, and (ii) the lottery i川hich with probabìlìty she and the other
player go to different concerts and w协 probabìlity ~ they both go to her morε
preferred concert. Ðo the same in the case that each player ís indifferer飞t between
(i) goíng to h在 less preferred concert in the company of the other playεr and (i i) the
!
$
lot忱t衍叫叮yin wl刘飞甘ì出
ch川
川
W
川
wV川i让thp玄roba拮b
切剃
圳
i乱lity ~ 由
s hea叫 t白
heot白
he盯rplaye盯rgot如时
叫
O
di证ffe盯
m
阳附
ren
创I川
1让tc∞
O挝附e
时玄竹圳
硝硝
t沁
arη飞 dw
阳it白
hpro
巾
协
baa均bil
刷刷
芷白
i挂l咛叶i 白
t h飞e叮
y巾bo
时thgo 的e盯f门mo
each player's I当ayoff to the outcome that she least pre岳阳叫ual to 0 and her payoff
to the outcome that she most prefers 吨以al to 2.)
Despite the importance of saying how the numbers in a payoff table should be
interpreted, users of game theo巧 sometimes fail to m及ke the interpretation clear.
斗叽飞en one iηterprets discussions of Nash equilibrium in the literature, a reaSOI飞w
ably safe assumption is that if the players are not allowed to choose their actions
randomly, thε口 the numbers ín payoff tables are payoffs that represent the players' ordinal preferences, whereas if the players are allowed to randomize, then
the numbers are payoffs whose expected 飞世ues represe挝 the players' prefiεrences
regarding lotteries over outcon飞es.
4.3
Mixed strategy Nash equilibrium
4.3.7 Mixed strategies
In the generalization of the notion of Nash equilibrium that models a stochastic
steady state of a strategic game with vNM preferer飞ces， we allow each player to
choose a probability distribution over her set of actìons rather than restricting her
to choose a single deterministic action. We refer to such a probability distribution
as a mixed strategy.
气 DE FlNITION 107.1 (Mixed strategy) A mixed strategy of a player in a strategic game
is a probabilîty distribution over the player' s actions.
1 usually use αto denote a profile of mixed strategies; 叫时 is the probab在ty
assigned by player j's mixed strategy 问 to her action ai. To specify a mixed strategy
of player i we need to give the probabìli句r it assigns to each of player i's actions.
For exa时le， the st附gy of player 1 in 协始ing Pennies that 路signs probab血咛 i
to each action is the strategy 叫 for which tX l (Head) 口! andα1(百il) 口 i. Becau
Chapter 4. Mixed 5trategy Eq 饭ilibrium
108
this way of describìng a IIÙxed strategy Îs cumbersome, 1 often use a shorthand
for a game that ìs presented in a table like those ín Figure 107.1: 1 write a IIÙxed
strategy as a
of probabilities, one for each actìon, in the order the actíons are given
Ìn thc tabfc. For exa时le， the mixed 蛇在校gy , ~) for player 1 in either of the
games ìn Figure 107.1 a仰伊s probability ~ to Q and probabìlity ~ to F
A mixεd strategy may assign probabìlity 1 to a single action: by allowing a
player to choose probability distributions , we do not prohibit her from choosto such a mixed strategy as a pure strategy.
ing deterministic actions. We
Player i's choosing thεpure strategy that assigns probability 1 to the actîon Gi is
equivalent to her simply choosing the action Gj , and 1 denote this strategy simply
by Gi.
q
4.3.2 Equilibrium
The notio到 of equilibrium th叫 we study is called "mixed strategy r\Jash equilibrium". The idea behind it is the same as the idea behind the notion of Nash equilibrium for a game with ordinal preferences: a mixed strategy Nash equilibrium is
a mixed strategy profile o:.~ with the property that no player i has a IIÙxed strategy
α i such tha t she pr时εrs the lottery over outcomes generated by the strabεgy profile (Ai， A~i) to the ìottery over outcomes generated by the strategy profile 0:.气 τhe
followi在g definition states this conditíon using payoff functions whose expected
values rep 1'esent the players' preferences
‘
‘
DEF!NITIO 只 108.1 (Mixcd strategy Nash 叫 uìlibrill lì1 o[ strategic game wÎth ♂NM pre[-
ere l1 ces) The mixed strategy profile 扩 in a strategic game with vNM preferences is
a (mixed strategy) Nash equilibrium if, fo 1' each player i and every mixed strategy
Cii of player Î, the expectεd payoff to player i of i飞;， is at least as large as the expecbεd
payoff to player i of (酌 ， LY* accordi吨 to a payoff function whose expected value
represents playεr i's preferences over lotteries. Equivalently, for each playe1' i,
Ui(ll') 主 Ui(ll川 '~i) for every mixed strategy iti of player i,
(108.2)
where Uj 位 ) is player i's expected payoff to th台 mixed stra tegy profìle 儿
The techníq过e of constructing the players' best respo日se func tÎ olls (Section 2.8) ,
useful in fjnding Nash equilibria of some strategic games with ordinal prefereηces，
is useful too in finding mixed strategy Nash equilibria of some strategic games
with\个JM preferences， εspecially ve 1'y simple ones. 1 discuss 出is technique in the
next section. ln Section 4.3.4 1 discuss a charaεterization of mixed st1' ategy Nash
叫uilibrium that is an invaluable tool for studying the equilìbria of any garne.
4.3.3 Best r在sponse functions
General deftnitiol1 人s before, 1 denote player i's best response functioll by Bí. For a
B町
i (忡
aι
-i) 始
ís 由
th飞e set of playe
盯rl印，、sb
快
白S纣ta
e
缸ct丘ion
strategic gam ε with ordinal p 1'eferences , 玩
飞w
厅11挝
1冶
en the 且
1 is纣t of the ot出
he
创r‘ P
抖la
玛严
y
咛f吧
时吉郑s' actions 始
e
isαι
…川i. For a strategic 伊那e with vNM
4.3 Mixed strat暗啻 y Nash equilibrium
109
prefεrences， Bi(tLí) is the set of pl在yer i's best mixed strategies when the lìst of
the other players' mixed strategies is rx-i. From the definition of a mixed strategy
equilibrium, a profile 旷 of mixed strategîes is a mixed strategy I南sh equ í1ibrium
if and only if every player's mixed strategy is a best responsεto the other playεrs '
mixed strategies (d. Proposition 36.1):
the 11l ixed strategy pri哺le rx* is a mixed strategy Nash equilibriu 11l 扩 and
only 扩巧巧 in B i (ιi) for every player i.
Two-pl呼er two-actíon games The analysis of Matching Pennies in Section 4. 1.2shows
that each play凹 's set of best respons础 to a mixed strategy of the other player is
either a single pure strategy or the set of all mixed strategies. (For example, if
player 2's mixed 阳tegya臼igns probab i1ity less than ~ to Head , then playεr l's
unique best response is the pure stratεgy Tail , 迂
i fp
乒la
吗y盯 2's mixed st壮rat饺
eg
部yassigns
伊roba
P
拍bil
刷
日副
i垃it忖
y great怡
川
e
r川t什
川h阳
蜘f怆e咯gyH
咱 抽
f如叫， and if player 2's mixed 如tegy 部signs probabìlity ~ to Head， 挝n
a11 of player l's mix叫 strategies are best responses.)
丁he character of each play白 's set of best responses in any two-pl町er game in
whid飞 each player has two actions is simil盯 to the character of each play时 's set of
best responses i我 Matching Pe l1 nies: it consists either of a single pure strategy or of
a11 mixed strategies. The reason lies in the form of the payoff functions.
Consider a tvγo-player game in which each player has two actìons , T and B for
player 1 and L and R for player 2. Denote by Ilj , foτ1 ， 2, a Bernoulli payoff
function for player í. (That is，问 is a payoff function over action paìrs whose expected value represents player i's prefereI飞ces regardiI飞g lotteries over action pairs.)
Player l's mixed strategy α1 assigns probability 问{盯 to her action T and probability 问 (B) to her action B (with 问 (T) + 叫 B)
1) ‘ Forco阳 eniencε， let p 口创 (T) ，
so that lX} (B) = 1 - p. Siinilarly, denote thεprobability 的(斗 that player 2's mixed
strategy assigns to L by q, so that 句 (R) 口 1 -q.
We takεthe players' choic仓s to be independent, so that when the players use
the mixed strategies lX} and 旬， the probability of any action pair (a J, aJ is the
product of the probability player l's mixed strategy assigns to a} ar飞d the probabilíty player 2's mixed strategy assigns to 句. (Se巳 Section 17.6.2 in the mathematical appendix if you are not fam i1iar with the idea of independence.) 古lUS
the probabi1i ty distribution generated by the mixed st红rat怡e鸣gypa♂ir (iX向1 ， α句2) 0飞v哈啊
盯r 也
e
the
f岛
ou
町rp
严
os部si凶
悦〉垃le outcomes of the game has tl怆飞冶
b
e扣
fiorm given in飞 Figure 109
织川
.1: (T , 斗
L) 仪
0 c­
cu
眩f骂swit白
hp
严roba
协bi
趴丑li技ty 问
pq
华
问， (T , R) occurs with probability p(l … 刑 , (B , L) occurswith
probability (1 - P)q, and (B , R) occurs with probability (1 - p) (1 - q).
L(号}
T (p)
B (1 - p)
R(l … q)
pq
p(l 二 q)
(1 - p)q
(1 … p) (1 … q)
Figure 109.1τ且e probabílities of the four outcomes in a tw分player two-action strategíc g缸到ewhen
player 1's mixed strategy 险 (p， l p)andplayer2'smix硝 strategy is (q , 1 一的.
Chapter 4. Mixed Strategy 军quilibrium
110
From this probability distribution we see that player 1's expected payoff to the
mixed strategy pair (的，在2) ìs
pq'U 过了， L)÷pO …的 'ul(T， R) 十 (1 … p)q 叫 (B ， L) 十 (l-p)(l 一号 )， ul(B， R) ，
which we can alternatively write as
p [q . Ul ( 了， L) + (1 …吁 )， u1(T， R 月十 (1
p)[q ， ul(B， L) 十 (1 - q) . Ul (B , R)].
The fìrst term in square brackets is player l's expected payoff when she u把sap μ re
strategythat assigns prob拍i1i ty 1 to T ar飞d player 2 uses her mixed strategy 的; the
second term iηsquare brackets is player l's expected payoff when she uses a pure
strategy that assìgns probability 1 to B and player 2 uses her mixed strategy 的. Denote these two e叩ected p在yoffs E1(了，的) and E1(扎的). Then player 1's expec枪d
payoff to the mixed strategy pair (町，时 lS
pEl ( 了，的)十 (1- p)E 1 ( 扎的)
That is , player 1's expected payoff to the mixed strat吃y pair (例，问) is a weighted
average of heτexpected payoffs to T and B when player 2 uses the mixed strategy α2 ， with weights equal to the probabìlities assigned to T and B by 夜 1 ，
In particular, player 1 、 expected payoff, given player 2's mixed strategy, is a
linear function of p-when plotted in a graph, it Îs a straight line. 1 A case ín which
El (丁Fα2) > E1 (B , tX 2) isìll邸trated in Fig 时e 110.1
CVEXE只 CISE 110 .1 (Expected payoffs) Construct diagrams like Figure 110.1 for BoS
(Fi gure 19.1) and the game in tl飞e right p在nel of Figure 2 1. 1 (in each case t陀ating
the numbers in the tables as Bernoulli payoffs). In each diagram, plot player 1γ's
expected payoff as a function of thεprobabìlìty p 吐
ti飞at she assi吝ns tοh飞er top action
i阳
叫叫
n
加\.掏
tl
an飞d 1.'
?jUL
γmA
hd
CX
JLO-L
飞
1jpa
VJE
,
AM
sy
(T， 的)
o
f?Z
F45a
pE 1(T， æ纣+ (1- p)E1(B , lt2)
E1 (扫， α2)
。
p
p• 1
Figure 110.1 Player 1'5 expected payoff 在s a function of the proba七 ìlìty P she assigns to T in the game
ìn which her i\ ctions are T and B, when player 2's mìxed 供应饺gy lSα2 and E](T， 的) > E](B ， 均)
1 See Section 17.3 (in p在 rti∞ larFi铲II在 496 ， 2) foτmy usage of the term "Iinei汀'\
4.3 Mixed strategy Nash equilibrium
111
A signìficant implication of the linearíty of player 1's expected payoff is that
there are three possibilíties for her best responsεto a given mixed strategy of
player 2:
• player 1's lI nique best response is the p旧e strategy T (ìf E1 (T , a: 2) > ε1 (B ， α2) ,
as ín Figure 110.1)
• player 1's unique best response is the p山e strategy B (ìf E1 (B ， α2) > E1 ( 丁，我2) ,
iηwhich case the line reprεsenting player 1's expected payoff as a function
of p in the analogue of Fì gure 110.1 slopes down)
• a11 mixed strategies of player 1 yield thεsame expected payoff, and hence
a11 are best responses (if E1 (T， 均) =问( B，在 2) ， in which case the line rep翩
resenting plaýer 1's expected payoff as a function of p in the analogue of
Figure 110.1 is horizontal) 拖
In particular, a mi民d strategy (p , 1 ~ p) for which 0 < P < 1 is never the unique
best response;εither it is not a best resp∞ se or all mixed strategies are best re拍
sponses.
CD EXERCISE 11 1.1 (日es i: responses) For each game and each value of q in Exercìse
110.1 , use the graphs yOll drew in that exercise to find player 1's set of best re翩
sponses.
Examp[e: Maft为 ing Pennies The argument in Section 4. 1. 2 establìshes that ^如 tch­
ing Pennies has a u口ique mixed strategy Nash equilibrium, in which each play时 's
mixed strategy as吨ns probability ~ to Head and p时ability i to Tai l. 1 now
descríbe an alternative rOllte to this conclusion that uses the method described
in Section 2.8 .3, which involves explicitly constructing the players' best response
functions; 往前 method may be used in other games.
Represe民t each player's preferences by the expected 飞吼ue of a payoff function
that assigns the payoff 1 to a gain of $1 and the payoff -1 to a loss of $1. τhe
resulting strategic game with vNM preferences is shown i抗日gure 11 1.1.
Head
Tail
Head
1, …1
Taìl
…1, 1
1, 1
1 ，一 1
…
Figure 111.1 Matching Pennìes.
Denote by P the probability that player 1's mixed strategy assigns to Head , and
by 号 the probabìlity that play在 2's mixed strategy assigns to Hcad. 丁hen， given
player 2's mixed strategy, player 1's expected payoff to the pure strategy Head is
号 .1 十(1一听. (-1)
2q … 1
and her expected payoff to Tail Ìs
q. ( 一片+ (1 一的 .1
1 - 2q.
Chapt带r 4.
112
Mixed Strategy 在quilibrium
B2
2
B1
1
2
O
1
p•
Figure 112.1 The p!ayers' best response functio日s in Matchiìlg Peníúes (Figure 111 .1) wl飞en randomízation is allowed. T头εprobab i!i ties assig口ed by players 1 and 2 to Hcad are p and q, respectí飞 ely. The best
response function of play巳 r 1 is black and that of player 2 ìs gray 了he disk índícates the uníque Nash
equ iJí brium.
!'
τhus if 号<
then p1ayer1 's expec 时 payoff to Tail exceeds her expected payoff to
Head , and hence exceeds a1so her expected payoff to every mixed strategy that assigns a positive probability to Head. (Recall the discussion in the previous sectioη)
Simi1arly, if q > ~， the内时以pected payoff to Head exceeds her expe配d payoff
to Tail , and hence exceeds her expected payoff to êvery mixed strategy that assigns
a positive probability to Tai l. If q 口 j ，阳
mixed 刽
s t仕
巾rat饺￠鸣
♂ieωs， yie1d the same expected payoff. We conclude that p1ayer 1'5 best
g
responses to player 2's strategy are her mixed strategy that assigns probability 0 to
Head if q < 怡h飞忧e町r门mix
a1丑1 her mixed strategies if q
~. That is, denot吨 by B] (q) 也esεt of probab i1i ties
player 1 assigns to Head in best responses to q, we have
!'
Bl(q)
=
( {O}
if q < ~
~ {p:O:; P :S: 1} if 号 =i
l{l}
ifq>!
!'
The best respon使 function of player 2 is 位邵阳': B2 (p ì = {1} if P <
B2 (p) =
柯: 0 :;号< 1} if P 工 i ， and B2 (p) = {O} if P >
Both players' best response
functions are illustrated in Figure 112 .1 (the best 陀sponse fur飞往ion of player 1 is
black and that of player 2 is gray).
丁i七飞es使
εt of mixed strategy Na
附sh 叫
e qui山
lìbri总
a of 位
t he gam
双le cor玄肥
e5ponds (怡as befo
仅Iτ‘e它
to 枝
the set of in飞terseζt挝lOn飞s of th飞 e best ，
re
陀sponse fur丑1ctior口飞s in this 白
figure; 飞认
A营它
e see that
t出
he
仔re 岭
i s one intersection , cori-esponding to the equilibrium we found previously, in
which each player assigns p灿拎出ty to Head
Matching Pennies has no Nash 叫出librium if the players are not allowed to
randomiz已If a game has a N岛h 叫uilibrium when randomization is not aUowed ,
is it possib1e that it has additional equilìbria whεn randomization is allowed? The
following example shows that the answer is positive.
i.
!
4 .3 Mixed strategy Nash equilibrium
113
IFF
IJF
nU
「L
叮ιnu
nDQU 11nUnu--
Figure 113.1 A version of the game BoS with vNM preferences
Example: Bo5 Consider the two-player game with vNM preferences in which the
players' preferences over deterministic action profiles are the same as in Bo5 and
their preferences over lotteries are represented by the expected value of the payoff
functions specified in Figure 113. 1. What are the mixed strategy equilibria of this
game?
First construct player l's best response function. Suppose that player 2 assigns
probabil均 q to B. Then player l's expected payoff to B is 2. q + 0 . (1
q) = 2q
and her expected payoff to 5 is 0 . q + 1 . (1 - q) 二 1 - q. Thus if 2q > 1 - q,
or q > ~， then her unique best response is B, while if q < ~，阳n her un
时igue b创
respons优阳
ei妇5 5 IHt q = ii
μ
扫
，λ吐阳}
tl
yield the same expe凹ct怡ed pa叮yoff扫s， so that every mixed strategy is a best response.
In summary, player l's best response function is
•
1-31
rTI
对J1-3
rT且
aJ
‘》，
、a
rJl 、
、
厅时， nu
，厅时，
nU <- 4'Lt
7 <- 1i
trJ
、
，L
、 rJl
<=>
1·1·1
rT且
f
nuut-、a‘》，
aJ
/llJ、 tt 飞
门时，
、le/
日υ
(
1
Similarly \ve can find player 2's best response function. The best response functions of both players are shown in Figure 113.2.
Weseetl时 the game has three mixed strategy Nash equilibria , in which (p , q) =
(0， 0) ，作，幻， and (1 , 1). The first and third equilibria correspond to the Nash equilibria of the ordinal version of the game when the players were not allowed to
randomize (Section 2 .7 .2). Th e second equilibrium is new. In this equilibrium each
q
1
3
O
Figure 113.2 The players' best response functions in BoS (Figure 113.1) when randomization is allowed
The probabilities assigned by players 1 and 2 to B are p and q, respectivel予 The best response function
of player 1 is black and that of player 2 is gray.ηle disks indicate the Nash equilibria (two pure , one
mixed).
Chapter4. Mixed Strategy Equilibrium
114
player chooses both B and 5 with positive probability (so that each of the four
outcomes (B , 的 ， (B ， 的 ， (5 ， 剖， and (5 , 5) occurs with positive probability).
( EXERCISE 114.1 (Mixed strategy 叫uilibria of Hmvk-Dove) Consider the two-player
game with vNM preferences in which thεplayers' priε feriεnces over detεrministic
action prof过εs are thεsame as in Hawk… Dove (Exercisε3 1. 2) and their preferεnces
over lotteries satisfy the following two conditions. Each playe盯r 沁
i s in飞dif打ferent betw
附een飞 (件
i污) 泊
th伫eo忧ome(但
Pa岱妇i衍Uε乙， Passi扣U时E叶) 肌
an
n(削i斗) 白
thει10
仗tt怡er可
yt出
ha挝ta剧$部
ssi
创咔
ig川 proba拍
bilit
让ty!
Eωo (μ
Ag
挝
S陀附衍附
si阳
ve♂， Agg
伊
沱E
r
臼削5必
附仰
S抑
妇i
s挝i飞ve and the other player is passive; each player is indifferent also between (i) the
outcome in which she is passive and the other player is aggressive and (ii) the
lottery that assigns probability 卡。 the outcome (Aggressi凯 AggresslVe) and probability ~ to the outcome (但(但
Pas必si盹 P
阳d附础衍仰仰仰
1切{阳切时
叫
ve
j淤E叶). 旦
Find payo
挝ff忽sw
呐ho使 e帜
χ?
阿
e杭ct饺eωd valu
陀pr陀
es挝
en
挝t these 严
p refe
衍re丑ξεs (take 础
e ach飞 playe
仔r's payof赶H始
o (Aggre
惚♂55纣i切， Aggn陀ε5必sì沁
臼ε叶)
U
to b沱伫 Oa衍nd each playε衍玄 's payoff to the outcome in 认'hich she is passive and the
other player is aggressive to bε1). Fír飞d 出e mìxed strategy Nash 叫uilibrium of
the resultíηg strategic game
Both Matching Pe l1 nies and BoS have finitely many mixed strategy Nash 叫Ul­
libria: the players' best response functions inìersect at a finite number of points
(one for λ1atchìl1g Pell11ics , three for BoS). On e of the gamεs in the next exercíse has
a continuum of mixed strategy Nash equilibria bεcause segments of the players'
best response functions coincide.
? EXERCISE 114.2 (Games with mixed strategy 叫uilibria) Find all the mixed strategy
Nash equilibria of the strategic games in Fig飞ue 114. 1.
? EXERCISE 114.3 (Aωordination game) 丁'vVO people ("an perform a task if, and only
if, they both exert effor t. They are both better off if they both exert effort and
perform the task than if neither exerts effort (and nothing is accomplished); the
worst outcome for each person is that shεexεrts effort and the other person does
not (in which case again nothing is accomplished). Specifically! the players' p玄efer­
ences are reprεsented by the expeεted value of tl飞e payoff functions in Figure 115.1 ,
where c is a positive r飞umber less th.an 1 that can be i民 lerpreted as the cost of exerting effor t. Find a11 the mixed strategy Nash 心quilibria of this game. How do the
εquilibria change as c increases? Explain the reasons for the changes.
苦 EXERCISE 114.4 (Swimming with sharks)
You and a friend are spending two days
at the beach; you both enjoy swimming. Each of you belìeves that with probabil-
L R
T 6, 0 0, 6
B
3, 2
事igure
6, 。
L R
T 0, 1 0, 2
B
2, 2
0, 1
114.1 Two strategic games with vNM preferences.
4 .3 Mixed strategy ~唱挝 h equílibrium
115
No effort
No 吃。ort
Effort
Figur窑 115.1
叨M
。， 0
0，一 c
-c , O
1 - c. 1 … C
The coordination game í 只在X巳 rcise 114.3.
ity ïτthe water is infested with sharks. If sharks are prεsent， any swimmer will
surely be attacked. Each of you has preferences represented by the expected vaJue
of a payoff function that assigns … c to being attacked by a shark, 0 to sitting on the
beach, and 1 to a day's worth of undisturbed swimrning (where c > O!) 传If a swimmer ìs attacked by sharks on the first day, then you both deduce that a swimmer
will surely be attacked the r飞extda予， and hencεdo not go swimrning the next day. If
at least one of you swìms on the fìrst day and is 罚。t attacked , th心犯 you both know
that thεwater is shark-free. If neither of you swims on the first da y, each of you retair飞s the belief that the probability of the water's being infested is 几 and henceon
the secor飞d day swims ifπc 十 l … π> 0 and sits on the beach if -7[C 十 1 一 π< 0,
thus receivi吨 an expected payoff of max { -1[C 个 I 一凡的. Model thìs situation as
a strategic game in which you and your friend each decide whether to go swimming on your first day at the beach. 丘， for example , you go swimming on the first
day, you (and your friend , if s1飞e goes swimming) are attacked witl飞 probability 汀，
in whïch case you stay out of the water on the second day; you (and your friend , if
she goes swirnming) s\-',tim undisturbed with probability 1 …凡 in which case you
swim on the second day. Thus your expected payoff if you swim on the first day
is 汀(一 C -:- 0) 十 (1 一汀)(1 斗 1) 二一汀C 十 2(1 一吟， independe时 of your 仕
frie
削 巳 r叫 's
act悦Ìm抗飞. Find the mixed strategy Nash equilibria of the game (depending on C and
叫‘ Does the existence of a friend make it more or less likely that you decide to go
swimming 0扫 the first day? (Per飞guins diving into water where seals may lurk are
5正)metimes said to face the same dilemma; Court (1 996) argues 出at they do not.)
4.3.4
A useful characterization of mixed strategy Nash equìlibrium
The method we have used 50 f盯 to study the set of mixed strategy Nash equi嗣
libria of a game involves constructing the pJayers' best response functions. Th is
method is useful in simple games , but is of lìmited use in more complicated ones.
: 我ow present a characterization of mixed strategy Nash equilibrium that ìs invalu
able in the study of general games. The characterìzatìon gives us an easy way to
check whεther a mixed strategy profìle is an equilibrìum; it is also the basis of a
procedure (described in Section 4.10) for finding a11 equilibria of a game.
百le key point is an observation made in Section 4.3 .3 for two-player two-action
games: a player's expected payoff to a mixed strategy pro自le is a weighted average
of her expected payoffs to her pure strategies, where the weight attached to each
pure strategy is the probability assigned to that strategy by the player's mixed
s位挝吗y. This property holds for any game (with any number of players) in which
Chapter 4. Mixed Strategy Equilibrium
116
each player has fuùtely many actìons. We can state it morεprecisely as follows.
A player's expected pay吃ff to the mixed strate，轩I profile αis a weighted average ofher expected pay~庐 to all mixed strategy profiles Of the type (aj , a 斤，
where the weight attached to (ait CCj) is the proba /J ílíty αj(αì) assigned to
aì by player i 云 mixed strategy 仅/.
(116 .1 )
Symbolica11y we have
民 (α) 工 z αí(aj )E í (叭， αd ，
aiEAi
where Ai is play时 i's set of actions (pure strategies) and E j 归川 í) is (as before)
her expected payoff when she uses the pure strategy that assigns probability 1 to
Gi and every other player j uses her mixed strategy aj' 在出 the end of Section 17.2
in the appendix on mathematics for an explanation of the L notation.)
This property leads to a useful characterization of mixed strategy Nash equilibrium. Lεt 扩 be a mixed strategy Nash e气uilibrium and denote by Ei player ì' s
expected payoff in the equilìbrium (i.e.
= U i ( 夜守. Because a* is an e龟uilibrium，
player i's expected payoft given α 二， to a11 her strategies, including a11 her pure
strategies, is at most 号. Now, by (1 16. 号，
is a weighted average of play但 i's exω
pected p巧Toffs to the pure strategies to whichα; assigns po
侃
公it加
s
i如
刊
V
叩
e
严ro
p
忡
ba
抽
bi山
i且1盯
t悖
予.1
y
刊
b挝
丑
T
u
G
player i's 拟
e xpe
配ct怡
ed payoffs to these pure
仍st饺f盹
at!怆较鸣
gies are a11 吨e〈吨t飞ua
丛
l让1 始
tO εj. (If any we
时Y陀
ε
sma
叫
a11e
肝r， 白
thε
岱只 the weighted avε
仅rage would be smaller.) We conclude that the expected payoff to each action to which 可 assigns positive prol灿ility is 可， and 阳
the
e
expe
哎C
吐
创t怡
ed payof好ft怡
Oε
衍
very ot血
he
盯ra
配
actio
∞只 i必sa
挝tmoωst 可
E; <
C: O
仪r只
∞
m
叭飞v沱
e脱1与y
予协圳
川"if 白
th刚
ese
仅cor只
M
飞叫
di出
tion
are sa挝t拍i岭sf良汪扫eωd for every play
归
er川i， then 夜扩车 is a mixed strategy Nash 叫 uilibrium: the
expected payoff to aj is 町， and the expected payoff to a叫 other mixed strategy 忍
at most Ej , because by (116.1) it is a weighted average of
and numbers that arε
at most EiThis argument establishes thεfollowìng resul t.
Ei
• PROPOS ITl ON 116.2 (Characterization of mixed strategy Nash equilibrìum of finite
game) A mixed strategy plψile a* in a strategic game with vNM 户r呐rences in which
each player has 卢时tely many actíons is a mixed strategy Nash equílibriwn 扩 a l1d only 手
fo川即11 player Î,
o tl1ee叩ected
payojf, given α 二 I to every actíon to 时始可 assigl1s positive pro /Jabilíty
is the same
• 衍
t如
hZ衍ε r仰
巧哼
X
严
p7沱川
ec
阳川
cted 阳
pay吃舒
of
在
J;， 伊
g i泛例
v扣?巧
C
at most 如
th
衍ee呻
巧叩
X
p勿附时
ω
eccted p(哼?叩
阿
y
呀
q
庐to a/ηlya
F
ω
ctiOI1
仰
11 沁
t8 切阶硝
hi协
ch α
叮7 aω55句垃伊7η1S 妇
PO
部仰茹幻
sit
往tive
臼ε p
仰
1'0
ηb
加
础b
G
挝
刷
il
江li均守协.
Eac l1 player's expected payojf in an e号uilibrium is hcr e习1ected payojf to any 吃f l1 er actio l1 S
that she uses with positive pro /J ability.
Thεsìgnificance of this result is that it gives conditions for a mixed strategy
Nash equilibrium in terms of each player's expected payoffs only to her pure strate树
gies. For games in which each player has finitely many actìons, it allows us easily
4.3 Mi革ed strategy Nash equilibrium
117
L (0)
c (~)
', 2
3, 3
0, .
5, 1
T (~)
M (0)
B(!)
', 4
及中
1, 1
2, .
0,7
Figure 117.1 A partially specifíed strategic game，让 lustrating a method of checking whether a mixed
strategy profile is a n1 ixed strategy Nash equilibrium. The dots indicate irrelevant p吕 yoffs.
to check whether a mixed strategy profile is an equilíbrium. For example, in BoS
(Section 4.3.3) t怡 $ tfa te g y p a ir (
r j )) is a m i奴X珩e时
ds纣tra
时t怡eg
盯yNa挝sheq
华u
臼
科il垃i池
ibri
初
b阳ri
挝
riu
Gi ) i
F
F {
beca附 gi础
bothequa
时
a1 to ~， and given player l's 处时εgy (~， ~)， playe啊rι2's
剖 expe时ct
剧阳
e仪E忖 ayoffs to
E and S a
Thenextεxarnple 沁
i s s1i培
ghtly ffiηorεcom
孔1plic
臼
at松
εd.
Ex 人 MPLE 117.1 (Checking whether a rnixεd strategy profile is a mixed stratεgy
Nash 叫uilibriurn)
r c1 airn that for the game in Figu陀 117.1 (in which the dots
indicate irrelevant payoffs) , the indicated pair of strategies, (~， O，!) for player 1
and (O， j ，~) for play时 2， is a rnixed stratεgy Nash equilibriurn. 丁o veri行 this
clairn, it suffices, by Proposition 116.2 , to s阳dy each player's expected payoffs to
her three pure strategies. For player 1 these p及yoffs are
T: 1.3+~.1 口 i
M:1.0+~.2
~
B :!5 个 1023
P1ayer l's rnixed strategy assigns positive probabílity to T and B and probability
zero to M , sωot址h隙e two con飞di投ti妇
ons
飞毡s in Propos弘it位ior公1 116.2 are satisfied for p1a 归
ye
盯r川1. τhε
ex职
pe倪ct
创制 pa町yoff
旺f 怡
toe倔ach of印
p1ay
归
r吧er川2冲1主阳t川咱i阴
e臼si岱s ~ (~ .2 十 !.4=i. 3 十 i
硝
w盯 ond
dit悦io川n Propo剑orη1 116.2 are s 挝
1 = ~.1 十 i 7 且;引)， so the t 认
for her.
亨rhich飞 s1趴le uses 飞i\机飞
Note that the expected payoff to player 2's action L, 飞w
岛剖
泊
日
a i l it忖y zero, is the same as the expected payoff to her other two actions. This equa1ity is consist但1Ì with Proposition 116.2, the second part of which requires only that
the expected payoffs to actior飞 s used with probability zero beη o gl'eater than the ex
pected payoffs to actions used with positive probability (not that they necessarily
be 1ess). Note also 仙 at the fact that player 2'sεxpected payoff to L is the sarne as
her expected payoffs to C and R does not irnp1y that the garne has a rnixed strategy
Nash equilìbriurn i口 which p1ayer 2 uses L wíth positive probabìlìty-it rnay, or it
rnay not, depending on the unspecified payoffs.
@) EXER Cl S 廷 117.2 (Choosing nurnbers) P1ayers 1 and 2 eachchoose a positive ìnteger
up to K. If the players choose the sarne nurnber, then player 2 pays $1 to player 1;
otherwise no payrnent is rnade. Each play时 's preferences are represented by her
expected rnonetary payoff.
Chapter 4. Mixed Strategy 在.quilibrium
118
a. Show that the game has a mìxed strategy Nash equilibrium in whìch each
player chooses each positive integer up to K with probability 1/ K.
b. (Mo陀 difficult.) Show that the game has no other mixed strategy Nash equi
libria. (Deduce from the fact that player 1 assigns positi飞'e prob创Jility to
some action k that player 2 must do so; then look at the implied restriction
on player l's 吨uilibrium strategy.)
?主 XERCISE 118.1 (Silverman's game)
Each of 队10 players chooses a positive integer. If player i' s integer is greater than player j' s integer and less than three times
this integer, then player j pays $1 to player í. If player i's integer is at 1εast three
times pl击yer j' s integer, then player i pays $1 to player j. If the integers are equa l,
no payment is made. 在ach play役 's preferences are represented by her expected
mor飞etary payoff. Show that the game has no Nash 均uilibrium in pure strate翩
gies and that the pair of mixed stratεgies in which each player chooses 1, 2, ar叭飞d5
e础 wi让th p时a站bil
刷圳
山
l且it可
y ~ is a 瞅ε
仅们d
付忖:l
仗
st阳附峙yN础 叩ilibr挝ium. (In 缸f忧阳&缸
act
ωt
仗吨
仁
t且
， t忱
tlh白怡
飞
m
饥飞i拉
Xεd st红rat，e怡gie
仍s 忌
i st位
h飞eur川
1让ique mi以
xed strategy l'队
N
、.Jas仙
h equilibrium.) (You can飞not appeal to Proposition 116.2 because the number of actìons of each play时 is not finite.
Hm八rever， you can use the argument for the μif" part of this result.)
?! EXERCISE 118 .2 (Voter participation) Consider the game of voter participatio到 in
Exercise 34.2. Assume that 2 < k <二 m and that each player'g preferences are
represented by the expectation of her payoffs given in Exercise 34 .2. Show that
there is a valuεof p between 0 and 1 such that the game has a mixed strategy Nash
equilibriuI及 in whìch every supporter of candidate A votes with probability [1, k
supporters of candidate B 飞rote with certai挝予 and the remaining m … k supporters
of candidate B abstain. How do the probab iJ ity p that a supporter of candidate A
votes and the expected number of voters ("turnou t") d叩end upon C? (Note that if
every supporter of candidate A votes with probability [1, then the probability that
exactly k … 1 of then飞 vote is kpk-l (1 - P).)
苦 Ex主 RCISE 118 雹 3 (Defending territory) 巳eneral A is def，εnding territory accessible
by two mountain passes against an attack by General B. Genèral A has three di唰
vìsions at her disposal, and general B has two divisions Each general allocates
her dìvisions between the two passes. General A ，叫ns the battle at a pass if and
only ìf she assigns at least as m3只 y divisions to the pass as does General B; she
successfully defends her territory if and only if she wins the b在ttle at both passes.
Formulate this situation as a strategic game and find a11 its mixεd strategy 叫uilib­
ria. (First argue that in every equilibrìum B assigns probability zero to the action
of allocating one division to each pass. Then argue that in any equilibrium she
assigns probability to each of her other actions. Finally, find A's 叫uilibrium
strategies.) In an 叫uìlibrium， do the gεnerals concentratεa11 theír forces at one
pass , or spr巳ad them out?
‘
i
人n implkation of Proposition 116.2 is that a nondegenerate mixed strategy
叫uilibrium (a mixed strategy 问江山brium that is not also a pure strategy 叫出偏
librium) is never a s衍'ict 队Jash equ iJibrium: every player whose mixed strategy
4.3 Mixed strategy Nash equilibrium
119
assigns positive probability to more than one action ìs indifferent between her
equilibrium mixed strategy and evεry actìon to whìch this mixed strategy assigns
positive probability.
Any equilíbrium (in mixεd strategies or 110导 that is not strict has less appeal
than a strict 叫uilíbrium because some (or a l1) of the players lack a positive in
centive to choose their equilibrium strategies, given the other plélyers' behavior.
There is no reason for them not to choose their 吨uìlíbrium strategies, but at the
same time there is no reasor飞 for them not to choose another四 strategy 仙at is equally
good. Many pure strategy equìlíbria -especially in complex games-are also not
strict, but among mixed strategy equilibria the problem is pervasive.
Given that in a mixed strategy equilibrium no pJayer has a positi陀 incentive to
choose her 仅juilibrj um stra tεgy， what determines ho飞IV she randomizes in equilibrium? From the examples studied in Section 4.3.3 (e.g. Matchí可 Pennies and BoS)
we see that a play仔 's equílìbrium mixed strategy in a twφplayer game keeps the
other play盯 indifferent betw时nasεt of her actìons , so that she is willing to ran协
domize. In the mixed strategy equilibrium of BoS , for example , player 1 chooses
B with probabilì守 so 阳 P
抖laye凹r 2 怜
js 拉
i 只瞅
出if赶fe缸rer
d
创r川 et认
w
飞/叫lli扫
l口19 旬
t o choose each w Íth pos剑itivεprobabilíty. Note, however, that the theory
is not that the pl冯rers cor飞 scìously choose their strategies with this goal in mind!
Rather, tÌle conditions for equilibrium are designeà to ensure that it is consistent
wi灿灿dy sta比 ln BoS , for example, if player 1 ω。附 B wi的robability
and player 2 chooses B 认
川y吐州
W
i让世怕
t
命
C
hang
伊
e her action. We have not yet studied how a steady state might come about,
but have rather simply looked for strategy profilesωnsistent with steady states. In
Section 4.9 1 briefly discuss some theories of how a steady state might be reached.
i
1
4.3 .5 Existence of equilibrium in 如 itegames
Every game we have examined has at least one mixed strategy 时ash equilibrium.
In fact , every gam舍 in which each player has 卢η ítely many actions has at least one
such equilibrium.
瓢 PROPOSITION 119.1 (Existence of llÚ xed strategy Nash equilibrium in 负nitegames)
Evcry stmtegic game wi恼怒NMpl呐rences in which e，町h player has卢nítely many actions
lws a míxcd stratcgy Nash 叫uilibriu l11.
Thi s result is or no help in 卢nding equilibria. 苔ut it 1S a useful fac t: your quest
foran 吨uilibrium of a game in wmch each player has finitely many actions in pri叭"
ciple may succeed! r如 te that the 自niter飞ess of the numb时 of actìons of each player
is 0到ly sufficient for the existence of an equilibrium, not necessary; many games in
which the players have infinitely many actions possess mixed strategy Nash equilibria. Note a150 that a play盯 's strategy in a mixed strategy Nash equilibrium may
assign probability 1 to a single action; if every play时 '5 S位ategy does so, then the
equilibrium corresponds to a ("pure strategy") equilibrium of the aS50ciated game
with ordinal preferences母 Rεlati飞rely advanced mathematical tools are needed to
prove the result; see , forεxample， Osborr飞eand 1之ubinstein (1994, 19-20).
120
4.4
Chapt告r 4.
Mixed Str矗tegy 主quilibrium
Dominated actions
In a stratεgic game with ordinal pref，臼εnces， one action of a player strictly dominates another action if it is superior, no matter what the other players do (see Definiti∞ 45.1). 1n a game with vNM prεferences in which players may randomize,
we extend this definition to allow an action to be dominated by a mixed strategy.
DEFINITION 120.1 (Strict dominatí01z) In a strategic game with vNM preferences,
player i's mixed strategy 叫"自tly domir时es her actio叫 if
Uí (αí ， Lí) > ui(a: ， a~í) fo川very list a-i of the other players' actions,
v\'here ui is a payoff functio只飞I\!hose expected val往台 represents player i's preferences over lotteries and Uj (钓，札 i) is player i's expected payoff under ui when she
uses the mixed strate宵夜i and the actions chosen by the other players are gi飞renby
a-í 飞Ve say that the action a; is stric t1 y dominated
Figure 120 .1 (in which only player 1 's payoffs are given) shows that an action
not strictly dominated by any pure strategy (i.e. is not strictly dO llÙnated in the
sense of Definition 45.1) may be strictly dominated by a mixed strategy. The action
T of player 1 is not strictly (or weakly) dominated by either M or B, but 议
it 沁
i s strictly
巾ominat怡eà byt由
d
hemi以肥
x<ed strat怡e咆g
吕
咱 tha
瞅ta削ss
附鸣阳 P
严ro
伪
ba站b
切
b圳
il
注茹
1丑it号y to M and pro己ability!
to B, because if player 2 chooses L, then the mixεd strategy yields playεr 1 the
payoff of 2, whereas the action T yields her the payoff of 1, and if player 2 chooses
R, then the n以ed stra均r yields player 1 the payoff of ~， whereas the action T
yields her the payoff of 1.
i
(
120.2 (Strictly dominating mixed strategies) 1n Figure 120 .1, the mixed
阳t忧e咆
即
g
y 出阳a批ta邵5臼
附s咆nsp时a抽bil
圳
阳
lit
让町
t句川
yr斗i 川1a
飞 a叫 p
伊ro
忡ba拍bil
刷削
山
li叫 川 iβS 川 t仙
he川
O吻 mÌ)祀
strategy that 挝
s t忧
trictl马y domina tes T.翩 Fin叫
d all the mi以
xe
时
ιds仗tr阳
at怡
egie
臼st出
ha
挝t do so.
(
120.3 (Strict domination for mixed 阶ategies) Determine whether each
of the following statements is true or false. (a) A mixed strategy that assigns
positive probability to a strictly dominated action is strictly domin挝ed. (的 A
mixed strategy that assigns posìtive probability only to actions that are not strictly
domìnated is not strictly dominated.
In a Nash 吨 uilibrium of a stratεgic game with ordinal prεferences， no player
uses a strictly dominated action (Section 2 织 1).1 now argue that the same is true of
L
R
T
1
1
M
哇。
B 0
3
Figure 120.1 Pbyeτ1 、 payoffs in 在 strategic game wìth vNM p对ferencesτhe actìon T of player 1 is
strictly dominatéd the mìxed 阳tegy that assigns probability 1to 114 and prob抬出 td ω
4.4 Domin垂ted actio我$
121
a mixed strategy Nashεquílibrium of a strategic game wìth Vr也t1 preferiεncεs. ln
fact , 1 argue that a strictly dominated actior飞 is not a b部t response to anyεollection
of mìxed strategìes of the other players.
S叩po削hat player i's action a; is st削ly dominated by her mixed strategy lti
Player i's expected payoff l1 i ( 町， ιi) when she 础es the mixed strategy 此 i and
the other players use the mixed strategies α 寸 is a weíghted average of her payoffs
l1 i ( α 川 i) as a-i varies over a11 the collections of actions for the other players, with
the weight on each Li equal to the probability wìth whìch it occurs when the other
players' mixed strategies are α 一 j. Player i's expected payoH when she uses the
action and the other players use the 川<ed strategies α_ i ìs a similar we胁ted
average; the weights are the same , but the terms take the form Ui(a; , Li) rather
than I1j 悦， ιJτhe fact that a; is strictly domi州ed byαi means that 民 (αi ， a-i) >
Ui 剖， a _;) for eve哼 colJ ection a 叫 of the other players' actions. He配e player í's
εxpected payoff when she uses the mixed strategy lti exceeds her expe吐ed payoff
whenshe us岱 the action , given 札 i. CO附 q 阳飞 tly，
a;
a;
a strict与 dominated actíon 必 nof 剖ed with positive pi始必的y in any 111ixed
strategy Nash equìlibriuln.
Thus when looking for mixed strategy equilibria we can e1iminate from consideration every strictly dominated action.
As before, we can define the notion of weak domination (see Definitíon 46.1).
DE Fl NITION 12 1. 1 (Weak domination) ln a strategic game with vNM preferences,
player i's mixed 由 ategy tr. i weakly dominates her action if
a;
I1j(α 川… 1 )主 Ui(a; ， 必… i) for every list a-i of the other players' action
and
l1i(ai , a_i) > ui(a; , a-i) for some list a… i of the other players' actio毗
where Uj is a payoff function whose expected 飞ralue represents player i's prefere挝es over lotteries and Lli ( 句， ιi) is player i's expected pay性f under ui whe川he
uses the mixed strategy 夜 i and the actions chosen by the other players are giveηby
-Î' We say that the action
is weakly dominated.
a
a;
We saw that a weakly dominated action may be used in a Nash e司uilibrium
(see Figure 47.2). 丁hus a weakly dominated action may be used with posi位ve
probability in a mixed strategy equilibrium, so that we cannot eliminate weak站
dominated actions from consideration when finding rnixed strategy èquilibria.
EXERCISE 121.2 (Eliminating dominated actions when finding equilibria) Find a11
the rnixed strategy Nash equilibria of the game in Figure 122 .1 by first eliminating
any strictly dominated actions and then constructíng the players' best response
阳附加ns.
Chapter 4. Mixed Strategy Equilibrium
122
L
M
R
T 2, 2 0, 3 1, 2
B
3, 1
1, 0
0, 2
Figure 122.1 The strategic game wíth vNM preferences in Exercise 12 1.2.
The fact that a play时 's strategy in a mixed strategy Nash equilibrium may be
weakly dominated raises the question of whether a game necessarily has a mixed
strategy Nashεquilibrium in whichno player's strategy is weakly dominated. The
following result (which is not easy to prove) shows that the ans\早在 is affirmative
for a finite game.
• PROPOSITION 122.1 (Existence of mixed strategy 1飞.Jash卢 equilibrium with no weakly dominated strategies in finite games) Every strategic game wit l1 vNM pr，政 rences
in which each player has 卢 llitely many actions has a mixed strategy N部h equilibrium in
wllich 110 player 亏 strategy is ωeakly dominated
4.5
Pure equilibria when randomiz就ion is allowed
The analysis in SectiOI飞 4 .3.3 shows that the mixed strategy 如ash equilìbria of BoS
in which each player's stratεgy is pure correspond precisely to the Nash equilibria
of the version of the game (considered in Section 2.3) in which the players are 口。t
a l1 0wed to randomize. The same is true for a general game: equilibria when the
players are not allowed to randornize remain equilibria when they are allowed
to rm飞domize， and any pure equilibria t始t exist when thεplayers are allowed to
randomize are equilibria when they are not allowed to randomÌze.
To establish this daim, let N be a set of players and let Aù for each player i , bε
a set of actions. Consider the following two games.
G: the strategic game with ordinal preferences in whìch the set of players is N ,
the set of actions of each player i is AiJ and the preferences of each player i
a 1'e represented by the payoff function Uí
the strategic game with vNM preferences in which the set of players is N , the
set of actions of each player i is Ai , and thεprefereηces of each player i are
represented by the expected value of Ui.
First 1 argue that any Nash equilibrium of G corresponds to a mixεd strategy
Nash equilibrium (iηwhich each player's st1'ategy is pure) of G'. Let 旷 bea 到ash
叫 uilibrium of G, and fo 1' each player i 1校在; be the m附d strategy that assigns
pr出ab i1ity 1 to aj. Since [/* is a Nash equilibrium of G, we know that in
no
player i has an action that}创ds her a payoff higher than does ai when all the 州er
players adhere to l1.' ~i' Thus 旷 satis在es the two conditions in Propositìon 116.2, so
that it is a mixed strategy 叫ui1ibrium of G', establishing the following resulι
麟?汉OPOSITION 122.2 (PU陀 strategy equilibria survive when randomization is al偏
lowed) Let a* be a Nash e号:对ibrium of G and卢)1' each player i let 可 be the mixed strat号Y
4.6 IlI ustration: exp咽 rtdia宰 nosis
123
01 player ì that ass拉ns probabílity one 切 the actio l1 ai. Thel1 旷 ìs a mixed strategy N部h
equìlíbríum of G'.
Next 1 argue that any mìxed strategy Nash equilibrium of G' in which each
player's strategy is pure corresponds to a Nash equilibrium of G. Let 扩 be a
mìxed strategy Nash 以luilibrìum of G' in which every player's míxed strategy
is pure; for each player i, denote by aj the actíon to whichαi assigns probability
one. Then no mixed strategy of player í yìelds her a payoff hìgher than doesα;
when the other players' mixed strategìes are given by a~i' Hence , in particula几
no pure strategy of player i yields her a payoff b咯her than does αi. Thus a* is a
Nash equilibrium of G. In words, if a pure strategy is optìmal for a player when
she 1S allowed to randomize, then ìt remains optìmal when she is prohibited from
raηdomìzing. (More generall1, prohibiting a declsion-maker from taking an action
that is not optimal does not change the set of actions that are optimal.)
瓢 PROPOSITION 123.1 (Pure strategy equilibria sur飞'lve when randomization is pro俨
hibited)ζet a:* be a mixed strate苦y Nash equilìbriu111 01 G' ín which the mixed strategy 01
each playe l' i assÎgns probability one to the sÎngle actioll aj. Then a* is a Nash 叫出li告 rium
01 G
4.6
lII ustration: expert diagnosis
1 sεem to confront the following predicament a11 too frequently. Something about
耳vhich 1am relatively ill informed (my car, my computer, my body) stops workìng
properly. 1 consult an expert, who makes a diagnosis and recornrnends an action.
1 am not sure that the diagnosis is correct… the expert , after a11 , has an interest in
sellìng her services. 1have to decide whether to follow the expε时's advice or to try
to fix the problem myself, put up with it , or consult another exper t.
4.6.1
Model
A simple model that captures the main features of this situation starts with the ass口mption that there are two types of problem , major and mÎnor. Denote the fraction
of problems that are major by 1', and assume that 0 < r < 1. An expεrt knows, on
seeing a problem, whether it is major or mÎl'I or; a consumer knows only the probability r. (The diagnosis is costly neither to the expert nor to the consumer.) An
expert may recommend either a major or a minor repair (regardless of the tr飞le
nature of the problem) , and a consumer may either accept the exper t' s recommerν
dation or seek another remedy. A major repair fixes both a major problem and a
夜unor one.
Assume that a consumer always accepts an exper t's advice to obtain a minor
repair-there is no reason for her to doubt such a dìagnosis-but may either accept or reject advice to obtab a m吗or repai汇 Further assume that an expert always
recornrnends a major repair for a major problem-a minor repair does not 缸 a
major problem, so there is no point in an expert' s recommending one for a major
Chapter 4. Mixed Strategy Equilibrium
124
problem-but may recommend either r，εpair for a minor problem. Suppose that an
expert obtains the same profit J[ > 0 (per unit of time) from selling a minor 距pair
to a cor飞sumer with a mino玄 problem as she does from selling a major repair to a
consumer with a major problem , but obtains the profit 7[1 >πfrom selling a major
repair to a consumer with a minor problem. (The rationale is that in the last case
the expert does not in fact perform a major repa比 at least not in its entirεty.) A
consum时 pays an expert E for a major repair al飞d I < E for a minor one; the cost
> E if her problem
she effecti飞rely bears if she chooses some other remedy is
is major and I' > I if it is minor. (Perhaps she consults other experts before proceeding, or works on the problem herself, in either case spending valuable time.) 1
assume throughout that 1二 > I'.
Under these assumptíons we can model the situatíon as a strategic game in
whìch the expert has two actions (recommend a mínor repaír for a mínor problem;
recommend a major repair for a mínor problem) , and the consumer has two actions (accept the recommendatíon of a major repair; rej仅t the recommendation of
a major 照pair). 1 name the actions as follows.
Expert HO l1 est (recommend a mínor repair for a minor problem and a major repair
for a major problem) and Disho l1 est (recommend a major repair for both types
of problem).
Consumer Accept (buy whatεver repair the expert recommend约 and Reject (buy
a minor repair but seek some other remedy if a major repair is 陀commended)
Assume that each player's preferences are represented by the player's expεcted
monetary payoff. Then the players' payoffs to the four actior飞 paìrs are as follo飞!vs
(the strategìc game ìs gìven ìn Fìgure 125.1).
{丘， A):
With probabiIi ty r the consumer's problem is major, so she pays E, and
with probability 1 - r ìt is min仅二 so she pays 1. Thus her expected payoff ís
-rE 一 (1
1')1.丁he exper t' s profit is 纪
(D , A): The cor飞sumer's payoff is - E. The consumer's problem is major with
probability 1', yielding the expert 汀， and minor with probability 1 人 yíeld­
i吨 the expertπ飞 so that the exper t' s expected payoff is r 汀十 (1
1') π\
(H， 只):
The consumer's cost Îs if her problem is major (in which case she rejects
the expert's advice to get a major repair) and 1 if her problem is minor二 so that
her expected payoff is … rE' - (1 - r)I硝 The expert 伦 tains a payoff only if the
consumer's problem ís minOl~ in which case she gets n; thus her expected
payoff ís (1 - r) 凡
(D , R):
Theconsum时 never accepts the exper t' s ad飞Tice， and thus obtains the ex翩
pected payoff … rE' - (1 … r) I'. τhεexpert does not get any busi附ss， and
thus obtains the payoff of O.
4.6 lII ustration: expert diagnosis
125
Consumer
叫
AVHH
Figur桂 125.1
只可ect (1 一项
7T, -rE - (1 …1" )1
rπ+ (1- r) π 一 ε
飞
户、
ν
-72 'HH
Ja'飞
p
Accept (q)
(Uet
rre
)}
mulnvaHVS
MH(
VA
t
ιrt
F巳
p4e
VA
(1 … r) π，一 rE' -
(1 -- 1" )1
。，一 rE' … (1 - r)1'
A g吕 me between an expert and a (0只sumer with a problem
4.6.2 Nash equilibrium
To find the Nash equilibrìa of the game we can construct the best response functions, as before. Denote by p the probability the expert assigns to H and by q the
probability the consumer assigns to A.
Expert's best respo l1se 卢 II1ction If q 0 (i.e. the consumer chooses 去认妓1 probability onet then the exper t's best response is p = 1 (since (1 …。交> 0). If号= 1
(i.e. the consumer chooses A with probability on功， then the expert's best rεsponse
is P = 0 (since π'> 汀， so that l' 汀十 (1 - r) π， > 7T). For what 飞'alue of q is thε
expert indiffεrentb役W时n H and D? Given q, the exper t' s expected payoff to H is
i严+ (1 一号 )(1 一叶江 and her expected payoff to D is q[r η 十 (1 … 1') π'] ， so she is
índifferent between the two actions if
‘
qπ 十 (1 … q) (J - 1" )π = q[1" π+ (1- ï)n J ]
Upon simplification, this yields q
7τ hτI We con c1 ude that the expe役's bεst
response function takes the form shown in both paηels of Figure 126. 1.
CO i1sumer主 best r.εspmlse ，民mcfiol1 If p
0 (i.e. the expεrt chooses D with probability ∞叶， then the consumer's best respOI飞se depends on the relati飞'e sizes of E and
d 刊 1 - r)1'. If E <刊 1
1' )1二 then the consumeγsb耐 respo附 is q 斗，
whereas if E > 1" E' 十 (1 - 1') 1', then her best response is q 工 O;if E 口十 (1 … r) 1' ,
then she is indifferent betvveen R and A.
If p = 1 (i.e. the expert chooses H with probability one) , then the consumer's
best response ís q = 1 (given E < E').
We conc1 ude that if E < rE' 十 (1 … 1' )1仁 then the consumer' s best response
to every value of p is q
1, as sho飞;\In in the left pa口el of Figure 126.1. If E >
十 (1 叶 r) I' , then the consumer is indifferent behveen A and R if
p[rE + (1- r)I] + (1
p[rE' 个 (1
p)E
l' )I] 十 (l--p)[rE' 十 (1
r)1'] ,
which reduces to
p=
E
In this case the consurner's best response function takes the form shown Ì11 the
right panel of Figure 126. 1.
Chapter 4. Mixed Strategy Equilibrium
126
4132n
时，
咆军主
-1:
Consumer
疗/
Expert
。
疗/7f'
Expert
。
p•
E<
+ (1
•
1')]'
E> rE' 卡。一 r)1'
Figure 126.1 The players' best response functions in the game of expert diagnosis. 了he probability
by the expert to H is p and the probabílity assig扫 ed by the cor飞 sumer to A is q
在 ssigned
E吁lI ilibrill 1n Given the best response functions , if E < rE' 十 (1 r)]' , thenthepair
ofpure strategíes (D , A) is the unique Nash equìlibrium. The condition E < rE' 十
(1 … r)]' says that the cost of a major repair by a口 expert is less than thε expected
εost of an alternative remedy; th巳 only equilibrium yields the dismal outcome for
t1沱江ll1sumer i口 which the expert is always dishonest and the consumer always
accepts her advicε.
If E > rE' 十 (1 - r) [', then the unique equilibrium of the game is in mixed
strategîεs， with (I'， q) 口(1'气 q 汀， where
口yag
*
and
q* 二 J (
In this 吨uîlibrium the expεrt is sometimes honest, son:飞etimes dishonest, and the
consumer sometimes accepts her advice to obtain a major repair and sometimes
ìgnores such ad vice.
As discussed in the introduction to the chapte乙 a mixed strategy equili七rium
can be gi飞'en more than one interpretation as a steady state. In the game we are
studyir吧， and the games studìed earlier in the chapt吼 1 have focused on the Înterpretation in which each player chooses her aεtion randomly, wÌth probabilitìes
give口 by her equilibrium mixed strategy， εvery time she plays thεgame. In the
game of expert diagnosis a different interpretation fits well: among the population of individuals who may play the role of each given player, every individual
chooses the same action whenever she plays the game, but different individuals
choose different actìons; the fractìon of individ uals who choose each action is equal
to the equilibrium probability that that action is used in a mixed strategy equilibrium. Specifically, if E > rE' 十 (1 - r) [', then the fraction p* of experts is honest
(reco血mending mino了 repairs for minor problems) and the fraction 1 … p* is dishonest (recommending major repairs for minor problems) , while the fraction 矿 of
consumers is credulous (accepting any reωmmendatio时 and the fraction 1 -矿 lS
4.6 lII ustratìon: expert dìagnosis
127
wary (accεpting only a recornrnendation of a rnìnor repair). Honest and dishonest
expcrts obtain the sarne expected payofC as do credulous and wary consurners.
CD EXER Cl SE 127.1 (Equilibriurn in the expert diagnosis garne) Find the set of rnixed
strategy Nash equilibria of the gεrnεwhen E 士 rE' + 但…，.) ['.
4丘 3
Propεrties of the mixed strategy Nash equifibrium
Studying how the 叫uilibriurn is affected by changes in the pararneters of the
rnodel helps us understand the nature of the strategic interactior飞 bεtween the
players. 1consíder the effects of three changes.
Suppose that rnajor problerns becorne less comrnon (cars becorne rnore rεli­
able, rnore resources ar，εdevoted to preventive health care). If we rearrange the
expression for p* to
p* = 1
r( E' … E)
(1
r)(E-[吁
we see that p* increases as r decreases (t始 nurnerator of the fraction decreases and
the denorninator increases). Th川 i只 am以ed strategy equílibriurn , the exp衍始 are
zηore honest when rnajor problen飞s are less cornrn∞. Intuitively, if a rnajor problern is less likε单 a consurner has
to lose frorn ìgnoring an expert's advice , so
the probability of an ex如此's being honest has to rise for her advice to be heeded.
τhe vâlue of 旷 is not affected by the change in r: the proba七ìlity of a {丁onsurner's
accepting an expert's advice rernaÍI飞s the sarne when rnajor problerns becorne less
ωrnrnon. Gíven the expert' s beha飞Tio艾， a decrease in r increases thεconsurner's
payoff to rejεcting the expert' s advice rnore than it increases her payoff to accept栅
ing this ad飞rice， so that she prefers to reject thεadvice. But this 归 rtial analysis is
rnisleadìng: in the equilibriurn tha t exìsts after r decreases, the consurner暨 is exactly
as likely to accept the exper t' s advice as she was before the change.
Now suppose that rnajor repairs becorne less expensive relativεto minor ones
(technologìcal advances reduce the cost of cornplex equiprnen斗. We see that p*
decre郁郁 as E decreases (with
a1飞 d [' constant): wher飞 n吨。r repalrs ar号:台ss
costl予 experts are
hones t. As rnajor repairs become less costly, a consurner has
rnore potentìally to lose frorn ignoring an exper t' s advice , so that shεheeds the
advice even if experts are less likely to be hones t.
Finally, suppose that the profit n' frorn an expert' s fixing a rninor problem with
an alleged major repair 妇11s (the governrnent 肥quires experts to return replaced
parts to the consumer, rnaking it more difficult for an expert to fraudulently daim
to have perforrned a major repair). Then q* ìnc陀ases--consurners become less
wary. Experts have less to gain frorn acting disho丑剧tly， so that consurners can be
rnore confident of their ad飞'Ìce.
在肥沃C1 SE 127.2 (Incompetent experts) Consider a (realistic?) variant of t挝 model，
ìn which the experts are not entìrely cornpeten t. Assurne that each expert always
correctly rεcognizes a major problem but correctly recognizes a rnìnor problern
with probability s < 1: with probability 1 - s she rnistakenly thir飞ks that a n飞inor
Chapter 4. Mixed Strategy Equilibrium
128
problern is rnajor, and , if the conswner accepts her advice , perforrns a rnajor repair
and obtains the profit 7r. Maintain the assurnption th必 eachζonsumer believεs
(cor主ectly) that the probabilìty her problern is rnajor is r, ar飞d assurnεthat a εon­
surner who does not give the job of fixing her problern to a只 expert bears the cost
E' if it is rnajor and [' if it is rninor.
Suppose, for exarnpl芒， that 及n expert is honest and a consurner rejects advice to
obtain a rnajor 陀pair. With probability r the consurner's problem is rnajor, so that
the expert recornmends a major repai 乙 which the consumer rejects; the consurner
bears the cost E'. With probability 1 - r the consurner's problem is rninor. 1n this
case with probability s the expert correctly diagnoses it as minor, and the consurner
accepts her advice and pays 1; with probability 1 s the expert diagnoses it as rnajo几 and the consumer rejects her advice and bears the cost ['.丁hus the consurner's
expected payoff i口 thÌs case is … rE' 一 (l- l" )[s/ 十 (1 - s)1'].
Corη1St仕ruct the payoffs for every pai让rof act柱ionsand 在
f in飞dt出
hemixeds
处t主恕
at伦
egye(龟lu
出i­
i吕ibriχ
汀咱
n山
urn
?
Mo
机re wa 主巧
yco
∞
ns
切
umτη1衍$♂
1 Ex 在 RCISE 128.1 (Choosiηg a seller) Each of two sellers has available one indivìsible
unit of a good. Seller 1 posts the price Pl and seller 2 posts the price P2. Each of
two buyers would like to obtain one unit of the good; they simultar冶ously decide
which seller to approach. If both buyers approach the same seller, each trades with
probability ~; the disappointed buyer does 则 sub叫uently have the option to
trade with the other seller. (Th is assurnption rnodels the risk faced by 在 buyer that
a good is sold out when she patronizes a seller with a low price.) Each buy!εr's
preferences are represented by the expected value of a payo仔 function that assigns
the payoff 0 to not trading and the payoff 1 - P to purchasing one unit of the good
at the p对ce p. (Neither buyer val飞础 more than one unit.) For ar飞y pair (扣 ， P2) of
prices with 0 ::二 Pi ::; 1 for i 口 1 ， 2， find the Nash e气uilibria (in pure and in rnixed
strategies) of the strategic 伊rne that rτlOdels this situation. (There are three rnain
cases: P2 < 2Pï - 1, 2Pl … 1 < P2 < ~(1 十 PIJ ， and P2 > ~(1十 Pl ).)
4.7
应quilibrium in a single population
ln Section 2.1 0 1 discussed deterrninistic steady states i到 situations in which the
rnembers of a single population interact. 1 now discuss stochastic steady states in
such situatìons.
First extend the defir飞itions of a syrnrτletric strategic garne and a syrnrnetric
Nashe气功librium (Defínitions 51. 1 and 52.1) to a garne with vNM preferences. Recall th时 a two伽player strategic game with ordinal p陀ferences is symrnetric if each
pl叮er has the same set of actions and each player's evaluation of an outcome depends only on her action and that of her opponent, not on whether she is player 1
or player 2. A symmetric game with vNM preferences satisfies the sarne conditions; its definition differs frorn Definition 5 1. 1 only because a player's evaluation
of an outcome is given by her expected payoff rather than her ordinal preferences.
4.7 Equilibrium in a single population
129
L吃ft
只告ht
L拚
1， 1
R告ht
0, 0
0, 0
1, 1
Figure 129.1 Approaching pedestrians
DEFINIτìON
129.1 (Symmetric tu涉player strat喀ic game with vNM preferences) A
two-player strategic garne with vNM preferεnces is syrnrnetriεif the players' sets
of actions are the sarne and the players' preferences are represeηted by the 似m
pected values of payoff functions U1 and U2 for which ul(a1 ， a2) 冉的怡， ad for
every action pair (al , a2).
A Nash equilibriurn of a strategic garnεwith ordinal preferences in which every player's set of actions is the sarne ìs syrnrnetric ìf a11 players take the sarne
actíon. 丁his notìon of equilibriurn extends naturally to strategic garnes with vNM
preferences. (As before, it does not depend on thεgarne's having only two players,
so 1 define ìt for a garne with any nurnber of players.)
DEFIN lTI ON 129.2 (Symmetríc mixed strategy 1\rash equilibrium) A profile 旷 ofrnixed
strategies in a strategic garne with vNM preferences in which each pla如r has the
sarne set of actions is a symrnetric rnixed strategy Nash equilibrium if it is a rnixed
strategy N革命 equilibrium and 可 is the sarne for every player i
Now consider again the garne of approaching pedestrians (Figure 52 .1, reproduced in Fìgure 129.1) , ìnterpreting the payoff nurnbers as Bernoulli payoffs whose
expected values represent thεplayers' preferences over lotteries. We found that
this game has two dete主rninistìc steady states, corresponding to the two SyIτlrnet­
ric Nash equilibrìa ìn p时e strategìes，在eft， L政) and (Right , R拉ht). The garne also
has a symrnetric rnixed strategy Nash equilibriurn, in which each player assigns
probahility ~ to 问ft and probabílity ~如时h t. τhis 叫uilibriurn corresponds to a
steady state in which half of a11 encounters result in collisions! (Player 1 chooses
Left and player 2 chooses R结ht wíth P时abílity , an句layer 1 chooses 陀1飞引í拚
gh归
t an
t
乒la
P
咛
ye
仔r2c巾
ho
∞
oses Left 叫
w it巾
hp
严ro
巾
ba
拍
bi出
i扫it守
y!.)
1n this example not only is t挝 garne syrnrnetric, but the players' íηterests coincide. The garne ìn Fi gure 129.2 is syrnrnetric , but the players prefer to take díffer偏
ent actions ra ther than t检 same actìons. This garne has no pure syrnrnetríc 吨氓"
libri urn, but has a syrnrnetric rnixεd strategy equilibriurn, ín which each player
chooses each a伪n with probability ~
These exarnples show that a syrnmetric garne may have no syrnrnetric pure
strategy 吨uilibriurn. But both garnes have a syrnmetric mixed strategy Nash
X
Y
X
γ
0, 0
1, 1
1, 1
0, 。
Figure 129.2 人 symmetric game.
Ch矗 pter4. 然 ixed Strat鲁吉Y 主quilibriul桃
130
Stop
COl1tinw?
Stop
CO l1 tÎ l1 ue
1, 1
l 一 ε2
2, 1 一 ε
0，。
Figure 130.1 The game in Excrcise 130.2.
叫uilibriur飞 as does any symmetric game in which each pl问'er has finitely many
actions, by the fo l1 owing result (the proof of which requires relatively advancεd
mathematical tools).
院 P汉OPOSITION 130.1 (Existεnce of symmetric mixεd strategy Nash equilibrium in
symmetric finitεgames)
sy11/1Jætríc strategíc game wíth vNM prcf扫切CCS 111
whích each player ￥ set of actions 必户1ite Iws (/ 5ymmetríc mixed strategy 入iash equilibriU111.
?只CISE 13旦 2 (Approaching
Members of a single population of car drivers
are randomly matched in pairs v\'hen th盯 simultaneously approach intersections
from different directions. 1n each interaction, each dri\'er can either stop or continue. The drivers' preferences are represented by the expected v在 lue of the payoff
functions given in Figure 130.1; the parameter E', 飞扩ith 0 < ε
1, reflects the fact
that each driver dis1ikes beìng the on1y one to stop. Find t挝 symmetric Nash
equilibrium (吨ui1ibri击?) of the g在 me (find both the equìlibrium strate泛ies and t1飞e
equilibrium payoffs).
Now suppose that drivers are (re)educated to feel guilty about choosing CO I1tinuc. Assume that their payoffs when choosing C0 1'1 tin l1 c fall by 0 > 0, so that the
entry (2， 1 … ε) in Figure 130.1 is replaced by (2 _. 6, 1- E'), the entry (1 ~ E', 2) is
replaced by (1 ", 2 - 6) , and the entry (0 , 0) is rep1aced by 卜 ι … ð). Show that
a11 drivers are better off in the 叩mmetric equilibrium of this game than they a陀
in the symmetric equilibrium of the origìnal game. Why is the society better off if
everyone feels guilty about being aí:r 2，reS:SI飞'e? (百leεqui1ibrium of this game, lìke
that of the game of expert diagnosis in Section 4.6, may attractive1y be interpl就ed
as represeI飞ting a steady state in which some members of the popu1ation always
choose ot飞e action and other members a1ways choose the other'action.)
( EXERCISE 130.3 (Bargaini鸣 Pairs of p1ayers from a single popu1ation bargain
over the di飞怡 ion of a pie of size 10. τhe members of a pair simultan仅lU sly make
demands; the possible demands are the nonnega iÌ \'e cven integers up to 10. If the
demands sum to 10 , then each p1ayer receives her demand; if the demands sum
to less than 10, then each player receives her demand plus half of the piεthat re栅
mains after both demands have been satisfied; if the demands sum to more than
10 , then neither player receives any payoff. Find a11 the symmetric mixed strategy
Nash equilibria in whìch each playεr assigns positive probability to at most two
demands. (Many situations in which each player assigns positive probability to
two actions, say a' and a气 can be ruled out as equilibria because when one player
uses such a strategy, some action yields the other p1ayer a payoff higher than does
one or bot1飞 of the actions (/' and a!l.)
4.8 lII ustration: reporting 矗 crime
131
4.8 lII ustration: reporting a crime
A crime is observed by a group of 11 people 刷 Each person would like the police to be
ìnformed but prefers that someone else make the phone call. Specifically, suppose
that each person aUaches the value v to the police being informed ar飞d bears the
cost c îf she makes the phone call , where v > c> O. Then the situation is modεled
by the following strategic game with v时M preferences.
Players The 11 people.
Actio用
Eachplaye白 set of actions is {Call , Don't call}.
Pr，ε ferences
Each player's preferences are reprεsented by the expected value of
a payoff function that assigns 0 to the profile in which no onεcalls， v - c
to any profìle in which she calls , and v to any profi1e in which at least on心
person calls, but she does not.
This game is a variant of the one in Exercise 33.1 , with k 工1. It has n pure Nash
equilibria , in each of which exactly one person calls. (If that person switches to not
calling, her payoff falls from v … c to 0; if aηy other persoηswitches to calling, her
p町off falls from v to v 仰脚 c.) If the members of the group differ i我 somerεspect， then
these asymmetric 叫曰libria may be compelling as steady states. For example , the
social norm in which the oldest person in the group makes the phone call is stable.
If the members of the group either do not diffεr significantly or are not aware
of any differences among themselves-if they are drawn from a single homogeneous population-then there is no way for them to coordinate , and a symmetric
equilibrium, in which every player uses the same strategy, is more compelling.
The game has no symmetric pure Nash eq飞往扯扯ium. (If everyone calls, then
ar飞y person is bett缸 off switching to not calling. If no one calls , then any person is
better off switching to ca l1ing.)
However, it has a symmetric mixed strategy equilibrium in whích each person
calls with positive probabi 1ity less than one. In any such equilìbrium, each perso口 's expected payoff to calling ìs equal to her expected payoff to not ca 1ling. Each
person's payoff to calling is v - c, and her payoff to not ca I1íng is 0 if no one elsε
ω l1s and l' if at least one other person calls, so the equ i1ibrium condition is
v- c
0 . Pr{ no one else ca11s} 十 V . Pr{ at least 0附 other person calls} ,
or
U … c
v (1 … Pr{no 0时 else calls }),
,
or
c / v = Pr{ no one else calls}.
(13 1. 1)
Denote by P the probability with which each person calls. The probabílity that
no one else calls is the probability that every one of the other n … 1 people does not
call , namely (1- py叫. Thus the equilibrium condition is c / v = (1 _ p) n 仆， or
p=l … {ε /V)l/(1叫
132
Chapter 4. Mixed Strategy Equilibrium
This number p is between 0 and 1, so we conclude that tl:飞e game has a u时气ue
symmetrîc mixed strategy equílibrium, in wlüch each person calls wìth probability
l … (C!V)lf何一1) That is , there is a steady state in which whenever a person is
in a group of n people facing the situation modeled by the game, she calls with
probabilìty 1 一 (C/V)l f 阳一门.
How does this 吨uilibrium change as the sìze of the group increases? W毛 see
that as 11 increases, the probability p that any given person calls decreases. (As
11 increases, 1/ (11 一 1) decreases, so that (c!v)lf(n-l) increases.) What about the
probability that at least one person calls? Fix any player Î. Then the event "no one
calls" is the same as the event "i does not call and no one other than i calls". Thus
Pr {no one calls} = Pr {i does not call} . Pr {∞ one else calls }.
(132.1)
Now, the probab i1i ty that any given person calls decreases as n increases , or equiv叩
alently the probabilìty that she does not call increases as 11 increases. Further, from
the equílibrium condition (13 1.日， Pr{∞ oneelse calls} 1S equal to c! v, Índepcndent
of 凡 We conclude that the probability that r飞o one calls il1 creases as 11 increases.
That 始， the larger the group , the less likely the police are informed of the crime!
The condition defining a mixed strategy 吨uilibrium is responsible for this resul t. For any given person to be indifferent between calling and not calling, this
condition r叫uires that the probabilìty that no one else calls be indεpendent of the
si之εof the group. Thus each person's probability of not calling is larger in a larger
group , and hence, by the laws o{ probability reflected in (132.1) , the probabìlity
that no one calls is larger in a larger group
τhe result that the larg衍 the group, the less likely any given person ca l1 s is not
suτprising.τhe result that the larger the group, the less likely at least one person
calls is a more subtle implicatìon of the notion of equ i1ibrium. In a larger group no
indìvidual is any less concerned that the police should be called , but ín a steady
state the beha悦。r of the group drives down the chance that the police are notified
of the crime.
ø EXERCISE 132 .2 (Reporting a crime when the witness巳s are heterogeneous) Consid皂 r a variant of the model studied i只 this section in whichη1 wi打lesses incur the
cost c1 to report the crime , and 112 witnesses incur thεcost c2 , where 0 < c1 <忍，
o < 句 < V , and 111 十 112 = 11. Show that 迂if ζ句1 an飞dc句2 are s飞uf必fi比ci必er时飞哎tl与y clos忧e， 拉
the
枉n1
t出
heg
伊
am
引1e has a mτUl以
xeds
纣tr阳注挝t怆
egyl芮飞\J也
as灿
hequ
旧i出
lib
忱ri妇
umi出
nwhic
凶
chev巳创衍r‘y 飞1\盯、
assigns pos纣it胁
i扒飞re probabi且li让ties to both reporting and not reporting.
? EXERCISE 132.3 (Contributing to a public good) Consider an εxtension of the analysis in this section to the game in Exercise 33.1 for_ k 2: 2. (In this case a player
may contribute even though the good is not pro飞怡ed; thεplayer's payoff in this
case is -c.) Denote by QIl-l, m(P) the probability that exactly m of a group of
n - 1 players contribute when each player contrîbutes with probability p. Wh at
condition must be satisfied by Qn 仆 ，k-l ( 抖 in a symmetric mixed strategy 叫uilib­
rium (in which each player contributes with the same probability)? (When does
4.8 lII ustr，矗tion: reporting a crime
133
a play衍 's contribution make a difference to the outcome?) For the case v
1,
n 骂斗4， k 2, and c = ~扫，白find th盯
eeq
伊山
u归
I川i吕1ibria 帆
eX叩
叫
pl且ici均 (You nee叫
d to 附 the fa削
州
ctt忖叫
t仕t
Q3，1川(1ω
川
川
py斗)=3斗p(口l 一 p)产2， 创
a nd do a bi沃t of 址
a 1gebra.
叫.})
REPORTING A CRIME: SOCIAL PSYCHOLOGY AND GAME T日 EORY
Thirty唰eíght pεople wítnessed the brutal murder of Catherìne ("Kítty") Genovese
over a períod of half an hour in New York City in March 1964. During this period ,
no one signíficantly rεsponded to her screams for help; 丑o one eγen called the
police. Jourr. alists, psychiatrísts, socîologìsts, and others subsequently strugglεd
to understand the wi纹lesses' inaction. Some ascribed it to 叩 athy engendered by
life in a large city: "Indifference to one' s neighbor and his troubles ìs a conditionεd
reflex of life in r也w York as it is in other big cìtìes" (Rosenthal 1964, 81 … 82).
τhe event particularly ìnterested socìal psychologists. It led them to 专ry to u仆
derstand the cîrcumstances und仪飞Nhich a bystander would help someone in trouble. Experiments quickly su苔gested that, contrary to the popular theo叩 people­
even those livìng in large cìtíes-are not 部 general 叩 athetic to others' plights. An
experímental subject who is the lone witness of a person in distress ís very líkely
to try to help. But as the síze of the group of wìtnesses íncreases , there is a declíne
not only ìn the probability that any gì飞'en one of them offers assistance, but also
ín the probab迫lity that at least one of them offers assistanc巳 Social psychologìsts
hypothesize that three factors explaìn these experímental findings喃First， "diffu输
sion of responsibility": the larg校 the group , the lower thεpsychological cost of
not helping. Second , "audíence í:nhibitìon": thεlarger the group , the greater the
embarrassment suffered by a helI咒r ín case the event turns out to be one in whìch
help is ínappropriate (becau挫， for example, ìt is not ín fact an emergenc拧. Third ,
"socìal influence": a person infers the appropríateness of helping from others' behavíor, so that ín a large group everyone else's lack of intervention leads any given
person to thínk ínter飞'ention is less likely to be appropriate.
1n terms of the model in Section 4.8 , thesεthree factors raise the εxpected
cost andf or reduce the expected benefit of a person's intervening. 在ley all seem
plausible. However, they are not needεd to explaìn the phenomenon: our gametheoretic analysis shows that even íf the cost and benefit are independel1 t of group
sìze , a decrease ìn the probabilìty that at least one person intervenes is an ìmplication of equilibrium.η1Ís game-theoretic analysis has an advantage ovεr the socîopsychological one: it derives the conclusion from the s
Chapter 4. Mixed Strategy Equilibrium
134
must be ìndifferent between intervening and not intεrvening， and as we have seen
thìs condition leads inexorably to the condusion that an increasεin group size
reduces the probability that at least one person intervenes.
4.9τhe formation of players' beliefs
In 在 Nash equìlibrium, each player chooses a strategy that maximizes her刷 expected
payoff, knowing the other players' strategies. 50 far we have not considered how
players may acquire such information. Informally, the idea underlying the previous analysis is that the players have learned each other's strategies from their
experienc2 playìng the game. In the idealized situation to which the aùalysis corresponds , for each player in the game there is a large population of individuals
who may take the role of that player; in any pl町 of the gam廷， one participant is
drawn randomly from each population. In this situation, a new individual who
joìns a population that is in a steady state (i. e. is using a Nash equi 1ibrium strategy
pro fiI e) can learn the other players' strategies by observing their actions over many
plays of the game. As long as the turnover in players is small enough , existing
players' encounters with neophytes (who may use nonequilibrium strategies) will
be sufficiently rare that their be1iefs about the steady state will not be disturbed , so
that a new player's problem is simply to learn the other players' actions.
τhis analysis leaves open the question of what might happen if new players
simultaneously join more than one population in sufficient numbers that they ha飞'e
a significant chanεe of facing opponents who are themselves new. In particulaζ
can 飞Ne expect a steady state to be reached when no onE has experience playing the
game?
4. 绞 1
Efiminating dominated actions
In some games the players may reasonably be expected to choose their Nash equi
librium actions frorn an introspective analysis of the game. At an extreme, each
play衍 's best action may be independent of the other players' actions , as in the
Prisoner's Dilemma (Example 14.1). In such a game no player needs to worry about
the other pla严凹 'actions. In a less extreme case, some player's best action may
depend on the other players' actions , but the actions the other players will choose
may be clear beωuse each of these players has an actìon that strictly dominates
all others. For example, in the game in Figure 135.1 , player 2's action R strictly
dominates L, so that no matter what play臼 2 thinks player 1 v气1m do, she should
choose R. Conse气飞lently， player 1, who can deduce by this argument fhat player 2
will choose R, may reaso口出at she should choose B. τha挝t 岭
i s. even 妇
in
时
仪
e
xp
严
衍ri怡
e
en
时
Cεd
d
抖la玛巧
p
y沱刷
时rsmayb
e
检
e le
时
dt怡
ot白
he unique N ash eguilibrium (B , R) in this game.
η飞is line of argument may be extended. For example, in the g法me 如 Fig­
ure 135.2, player l's action T is strictly dominated , so player 1 may reason that
4.9 The formation of players' belief革
135
-v
咱I
吁BA
inU
唱
吁，
"nu
Figur告 135.1
-AHV
irrFJF
Tinnu
A game in which player 2 has a strictly dominant action and player 1 do巳 s nol
L
R
T 0, 2 0, 0
九1
B
2, 1
1, 1
t2
2, 2
Fìgure 135.2 人 game in which player 1 may reason that she should choose B b芒cause player 2 will
reason that player 1 will not choose 了'， 50 that player 2 飞'/ìll choose R
player 2 will deduce that player 1 will not choose T. 仁onseque时ly player 1 may
deduce that player 2 will choose R, making B a better action for her than M.
The set of action profiles that 陀main at the end of such a reasoning process
contains all Nash equilibria; for many games (unli检 these examples) the set contains many other action profiles as 认rel l. Ir飞 fact， in many games no action profiles are eliminated, because no player has a strictly dominated action. Nevertheless , in some classes of games 由e process is powerful; its logical consequences are
explored in Chapter 12.
4.9.2 Learning
Another approach to the question of how a steady state might be reached assumes
that each player starts with aηunexplainedμprior" bdief about the other players'
actions, and changes these beliefs-//learns"… in response to information she receives. S队em盯1 learn, for exampk from observing the fortunes of other players
like herself, from discussing the game with such players, or from her O\O\'n experience playing the game. Here 1 briefly discuss two theories in which the same set of
participants repeatedly play a game, each partícipar飞t changing her beliefs about
the others' strategies in response to her observations of their actions.
Best 陀
r es
珩
por狞ISε dy
沪nt仰
1η11巾l1C臼s
Apa
盯
戎ticu
r
吼la
创盯r町忖 s
创nTIηpl沁
et白
heωO
创rya
削ssume
白s 往
t ha
拭t 白
i n each period
after the 如
fi让rs纣t， each player believes that the other players will choose the actions
they chose in the previous period. ln the first period, each player chooses a best
response to an arbitrary deterministic 民lief about the other players' ac tÎ ons. In
every subsequent period, each player chooses a best response to the other players'
actions in the previous period. This process is known as best response dynamics. An
actìon profile 位at remains the same from period to períod is a pure Nash equílibrium of the game. Further, a pure Nash equilibrium in which each player's
action is her only best respoηse to the other players' actions is an action profile
that remains the same from period to period.
136
Chapter4. 草草 ixed Strategy 在quilibrium
ln some games the sequence of action profiles generated best response dynamics converges to a pu陀 Nash 叫 u i1ibrium， regardless of the players' initial
be1iefs. 丁he ex阳市le of Cournot' s duopoly game studied in Section 3. 1.3 is such
a game绸 Looking at thεbest responsεfunctíons in Fígure 59 .1, you can convínce
yourself that from arbitrary initial actions , the players' actions approach the Nash
equilibrium (町，毛 L
① EXERCIS主 136.1 (Best responsεdynamìcs in Courno t' s duopoly game) Find the
se气uence of pairs of outputs chosen by the firms in Courno t' s duopoly game under
the assumptions of Section 3. 1. 3 if both fìrms ìnìtiall y choose O. (If you know how
to solve a first-order difference 叫uation，拉ìd a formula for the 0过tputs in each
period; íf not, find the outputs in the first few periods.)
Z EXERCISE 136.2 (Best response dynamics in Bertrand's duopoly game) Consider
Bertrand's duopoly game in which the set of possible prices is discrete , und仔 the
assumptions of Exercise 67.2. Does the 优守口ences of prices under best response
dynamics cor飞verge to a Nash equilibrium when both prices initially exceed c 十 1?
What happens when both prices are initi在 lly equal to c?
For other games there are initial belìefs for which thεsequence of action profiles
generated by the process does not converge. In BoS (Example 18.2), for example , if
player 1 initially believes that pl玛rer 2 wilì choose StravÎnsky and player 2 initially
believes that player 1 will choose Bach , then the players' choices will subsequently
alternate ìndefinitely between the action pairs (Bach , StravÎnsky) and (Stravinsky ,
Bach). This example highl唱hts the lìmited exte时 to which a player is assumed to
reason in the model , which does not consider the possibility that she cottons on to
the fact that her opponen t' s action is always a best response to h衍 own previous
action.
Fictitious play Under best response dynamics, the players' beliefs are contìnually
revealed to be incorrect unless the starting point is a Nash equilibrium: the players'
actions change from period to period. Furtheζeach player believes that every
other pl的'er is us如g a pure strategy: a player's belief doεs not admìt thεpossibilìty
that her opponents' actions are realizations of mixed strategies.
Another theorτknown as 卢ctitious play, assumes that players consider actions
in all the previous periods when forming a be !ief about their opponents' strategies.
They trεat these actions as reaìizations of mixed strategies. Consider a two-player
game. Each player begins with an arbitrary probabilistic belief about the other
player's action. In thεfirst play of the game she chooses a best response to this
belief and observes the other player's action, s冯r A. She then changes her belief
to one that assigns probabilìty 1 to A; in the second period , she choos础 a best
response to this beHef and observes the other play时 's action, say B. She then
changes her be!ief to one th拭部鸣ns p时abìlity to both A ç叫 B， and 巾。ses
a best responsεto this be !i ef. She continues to change her belief each period; in
any p时iod she adopts t始 be !ief that her opponent is using a mixed strategy in
!
4.10 在xt窜犯sio拟在我di法gallmixed 主trategy
Head
'a il
Nash equilibria
Head
1, 1
-1 , 1
…
137
Taíl
1, 1
1, … 1
Fìgure 137.1 Malcltillg Pl'llllies.
whích the probability of each action is proportional to the frequency with which
her opponent chose that action in the previous periods. (If, for example, in the first
six periods player 2 chooses A twice , B three times , and C once, player 1's belief in
period 7 assigns probability ~ to 人 probabìlity ~ to B, and probability ~ to c.)
In the game Matching Pennies (Example 19 .1), reproduced in Figure 137.1 , this
process works as follows. Suppose that player 1 begins with the belief that player
2' s action w i1l be Tail , and player 2 begins wìth the be !ìef that player 1's action w iU
be Head. τhe双 in perìod 1 both players choose Tail. 百lUS ín períod 2 both play翩
ers belìeve that the泣。pponent will choose Ta 迁， so that player 1 chooses Tail and
player 2 chooses Head. Consequently í口 period 3, player 1's bεlief is that player 2
will choose Head with p时ability ~ and Tail with probability , and player 2's belief is that player 1 will definitely choose Taíl. Th us i只 period 3, both Head and
Tail are best respons巳s of pl巧er 1 to her belief, so that she may take either action;
the unique best response of player 2 is Head. 百le process continues similarly in
subsequent perìods.
In two-player games lìke Matching Pennies , in which the players' interests are
directly opposed , and in any two-player game in which each player has 队ro ac翩
tions , this process converges to a mixed strategy Nash equilibrìum from any initial
beliefs. That is , after a sufficiently large number of periods , the frequencìes with
which each player chooses her actions are close to the frequencies induced by her
mixed strategy in the N础h equilibrium. For other games there ãre initial beliefs
for which the process does not converge. (The simplest example is too complìcated
to prese时 compactly.)
τhe people involved ìn an interaction that we model as a game may form beHefs
about their opponents' strategies frorτ1 an analysis of the structure of the players'
payoffs, from their obs校vatior飞S of their opponents' actions, and from information
they obtaìn from other people involved in similar interactíons. The models 1 have
outlh飞ed in thís section explore thεlogical ímplications of two ways in which p]a沪
ers may draw inferences from theír opponen毡'actions. Models that assume the
players to be more sophisticated may give more insights into the circumstances
in which a Nash equilibrium is likely to be attained; this topic is an active arεa of
current research.
1
4.10
Extension: 白 nding all mixed strategy Nash equilibria
We can find a11 the mixed strategy Nash equilibria of a 帆10胸player game in which
each player has two actions by constructing the players' best response functions ,
as we have seen. In more complicated games , tl让s method is usually not practica l.
Chapter 4. Mixed Strategy Equilibrium
138
τhe fo11owing syst.εmatic method of finding a11 mixεd strategy Nash equilibria of a game is suggested by the characterization of an eq飞lilibrium in Propo到哪
tion 116.2.
• For each player i, choose a subset 5j ofher set Ai of actions.
• Check whεther there exists a mixed strategy profileαsuch that 仲 the set of
actions to which each strategy lX i assigns positive probability is Sj and (i i) IX
satisfies the condìtions ín Proposition 116.2.
• Repeat the analysis for every co11ectíon of subsets of the players' sets of
actìons.
丁he following example illustrates this method for a two-player game in which
each player has two actions.
EXAM 扫卫 138.1 (Finding all mixed strategyεquilibria of a two-player game in
which each player has two actíons) Consider a two-player game in which each
player has two actions. Denote the actions and payoffs as in Figure 139. 1. Each
player's set of actions has thrεe nonempty subsets: two each consisting of a sinω
gle action , and one consisting of both actions. Th us there are nine (3 x 3) pairs of
subsets of the players' action sets. For each pair (5 1, 52 ) , we check if there is a pair
(11: 1 ，的) of mixed strategies such that each strategy D: j assigns positive probability
only to actions in 5j and the conditions in Proposìtion 116.2 are satìsfied.
• Checking the four pairs of subsets in which each player's subset cor飞sists of
a single action amounts to checking whether any of the four pairs of actions
is a pure strategy equilibrium. (For each playe毛 the first condìtion in Proposition 116.2 ìs automatically satisfied because there is only one actìon in each
subset.)
• Cons战r the pair of subsets {1二 B} for player 1 and {L} for player 2 The
second condìtion in Proposition 116.2 is automatically satisfied for player 1,
who has no actions to which she assigns probability 0, and the first condition
is autor孜atically satisfied for player 2, because she assigns positive probability
to only one action. Thus for there to be a mixed strategy equilibrium in which
play佼 1's probabìlìty of using T is p we need 1111 = U21 (player 1's payoffs to
her two actions must be equal) and
‘
P V ll 十 (1
p) 归去 P V 12 十(l
p)V22
(L must be at least as good as K given player 1's mixed strategy). If un f.
1121 , or if there is no probabi1i ty p satisfying the inequality, then thεre is no
吨ui1ibrium of this type. A sìmil盯 argument applies to the three other pairs
of subsets in which one player's subset consists of both her actions and the
other player's subset consists of a single action.
• To check whether there is a mixed strategy 叫uilibrium in which the subsets a 陀 {T， B} for player 1 and {L ， 时
4.10 Extension: finding all m议时 strat程gy Nash 坦 quilibria
139
L
R
T
Ull , V11
U12 , V12
B
U21 , V21
μ 22 ， V22
Figur暗 139.1
A two-player slrategic gam亿
mixed strategies that satis豆es the first condition in Propositio扫 116.2 (thεsεc­
ond condition is automatically satisfied because both players assign positive
probability to both their actions).τhat is, we need to find probabilities p and
q (if any such exist) for which
qUn 十 (1 - q)U12
= q U 21 个 (1 … q)U22
P V ll 十 (1 - P)V21 = p V 12 个 (1 … P)V22
For example , in BoS we fir吐 the two pure equilibria when we check pairs of
subsets in whïch each subset consists of a sin。'飞
2:1e action , we find 口 o equilibria when
we check pairs in which one subsεt consists of a single action and the other consists
of both actions, and we find the 1孜以θd strategy equilibrium whεηwe check the
pair ( { B, S} , {翠， S} ).
( EXERCISE 139.1 (Finding a11 mixed strategy equi1ibria of two-player games)
the method described in Example 138.1 to find a11 of the mixed strategy 叫出libria
of the games in Figure 114 .1.
ln a game in which each player has two action白， for any subset of any player's
set of actions at most one of the two conditions in Proposition 116.2 is relevant
(the first if the subset contains both actions and the second if it contains only one
action). \N'hen a player has three or more actions and we consider a subset of her
set of actions that εontains two actions, both conditions are relevan t, as the next
example ill议strates.
EXAMPLE 139 .2 (Finding all mixed strategy eq出libria of a variant of BoS) Co部idel
the variant of Bo5 given in Figure 139.2.
by inspection we see that the gam伫
has two pure strategy Nash equilibria , namely (B , B) and (5 ，的
Now consider the possibility of an equilibrium in which player l's strategy is
pure whereas player 2's st1'at台gy assigns posìtìve probability to two 0 1' mo 1'e ac
tions. If playe 1' l's strategy is B, then player 2's payoffs to her three actions (2 , 0,
and 1) are a11 differe跤， so the 豆rst condition in Proposition 116.2 is not satis加d.
τhus there is no 吨uilibrium of this type. Similar rεasoning ruìes out an 叫lll­
líbrium in which player l's strategy is 5 and playεr 2's strategy assi萨lS positive
B
5
B
5
4， 2
旺， 0
0, 0
2, 4
X
0, 1
1, 3
Figurfi! 139.2 A variant of the game BoS.
Chapter 4. Mixed Strategy Equilibriul冒2
140
probabìlity to more than one action, and also an equilibrium in which player 2's
strategy is purεand player 1's strategy assigns positive probability to both of her
aε t1ons.
:\;但t consider the pos5ib i1i ty of an 吨uilibrium iηwhich player γ5 5trategy
a5signs positi飞'e probability to both her action5 and player 2'5 5trategy a55ign5 P05iti飞'e probability to two of her three action5. Denote by p the probabìlity player 1'5
始就εgy a55ìgn5 t。 在 Therεare three p05sibilities for the pair of player 2'5 action5
that have posìtive probabìlity.
B and 5: For the conditions in Proposition 116.2 to bεsatisfied we need player 2's
expected payoff to B to be equal to her expected payoff to 5 and at least her
expected payoff to X. That is, we need
2p
4(1-p) 主 p
+ 3(1
…
p)
The equation imp 1ies that p = ~，州ich does not satisfy the inequality. (τhat
i5, if P i5 5uch that B and 5 yield the 5ame expectεd payofC then X yields a
higher expected payoff.) Thu5 ther巴 is no equilibrium of this type.
B and X: For the condition5 in Prop05ition 116.2 to be 5ati5fied 认re need player 2'5
expected payoff to B to be egual to her expected payoff to X and at lea5t her
εxpected payoff to 5. That i5 , we need
2p = p 个 3(1 - p) 三点 1 ~ p)
Thεeq1.1ation implle5 that p 口 iFWi灿创i5f阴阳 111εquali牛 For the first
condition in Proposition 116.2 to be satì5fied for player 1 we need player 1's
expected payoff5 to B and 5 to be equal: 4q 乙二 1 - q, where q Îs the probability player 2 as吨ns to B, or q
~. Th1.15 the pair of mixed 5t附gie5
((~，~)， 佳 , 乱 ; )) is a m i以xed 纣
s tra
州t怆e鸣g
白
啪ye叫
罕伊仰
q
臼
u
úl丑ib
忱
bri
挝1U
5 al只飞dX
尘 For 队盯'e
盯ry 洪
s trab饺
egy of play町 2t出
ha
时ta
阳ssi培
gn冶s pos引itive pr噜obabi出
1丑It守
yon
挝
1让i守
yt怕
05
and X, player 1's expectεd payoff to S eXCeed5 her expected payoff to B. τhus
there Îs no equilibrium of this sort.
τhe fìnal possibility is that there is an eq 1.1 iJibrium iηwhich player 1's sh'at栩
egy assigns posítive probability to both her actions and play咽. 2's strategy assigns
positive probability to all three of her actions. Let p be the probability player 1 、
strategy assigns to 旦 τhen for player 2's expected payoffs to her three actions to
be equal we need
2p
是 (1 - p) 口 p+3 口一 p).
Fo
臼E呻出胁εef巾
i汀咐
rs
盯st 叩al且it咛
川
y
附ε n
w
附
e优ed p = i F吟
violati
挝础
t往i归
n
吨
gtl川e配
e∞
co
on飞d 叫叩ua
泊硝
a
no 飞v功咄
川i沁
a
ue of p for which player 2's expected payoffs to her threε actions are equa l,
and thus no equilibrium in which she chooses each action with positivεprobability.
认le concludεthat the game has three mixed strategy 吨uilibria: ((1 ，的，在，乱的)
(i.e. the pure strategy equilibrium (B ， 日片， ((0， 1) ， (0， 1 ， 0)) (i.e. the pure strategy
equilibrium (5 , 5)) , and ((~，书，(~， O，~))
4.10 Exte时 ion: finding all mixed strategy r苦ash equilibria
L
M
141
R
T 2, 2 0, 3 1, 3
B 3, 2
Figur告 141.1
1, 1
0, 2
The strategic game with vNM prefere只 ces I 只 Exercise 141.1
? Ex 在 RC1SE 141.1 (Finding all mixed strategy 叫uilibria of a two-player game) U使
the method described in Example 139.2 to find a11 the mixed strategy Nash eqllilibria of the strategic game in Figure 141.1.
As yOll can see from the examples , this method has the disadvantage that for
games in whichεach player has several strategies, or in which there are several
pl巧ers， the numhεr of possibilities to examine is huge. Even in a two-player
gameiηwhich each player has thr，εe actions, each player's set of actions has seven
nonempty sub优 ts (three each consisting of a single action, thr台e consisting of two
actions, and the entire set of actions) , so that there are 49 (7 x 7) possible co11ections of subsets to check. ln a symmetric game, like the one in the next exercise,
many cases involve the same argument, reducing the nllmb衍 of distinct cas白 to
bechecked理
? EXERCI记 141. 2 (Rock , Paper, Scíssors) Each of two players simultaneously announces either Rock, or Paper, or Scissors. Paper beats (wraps) Rock, Rock beats
(blllnt吟 Scissors， and Scíssors beats (cuts) Paper‘ τhe playεr who names the winn站在 object r'εcei飞'es $1 from her opp∞ent; if both players make the samεchoice，
then no payment is made. Each player's preferences are represented by the ex刷
pected amount of mor飞ey she receives. (Aηexample of the variant of Hotelli口g's
model of electoral competition considered iηExercise 75.3 has the same payoff
structure. Suppose th在e are three possible positions, A , B, and C, and three citizens , one of whom prefers A to B to C, one of whom prefers B to C to A , and one of
whom prefers C to A to B. Two candidates simultaneously choose positions. If the
candidates choose different positions , each citìzen votes for the candidate"年hose
position she prefers; if both candidates choose the same position, they tie for first
place.)
a. Formlllate this situation as a strategic game and find a11 its mixed strategy
e电uiJibria (give both the e气llilibrium strategies and the equilìbrium payoffs).
b. Find a11 the mixed strategy equilibria of the modìfied game in which player 1
is prohibitεd from announcir飞g Scíssors.
EXERCISE 14 1.3 (Election campaigns) A new polìtical part予 A ， is chaIlenging a口
establìshed party, B.τhe race involves three Iocalities of different sizes. Party A
can wage a strong campai♂1 in only one locality; B must commit resources to defend its position ir飞 one of the localities, without knowing which locality A has
targeted. If A targets dìstrict i and B devotεs its reso旧ces to some other dìstrict,
then A ga ir1s ai votes at the expense of B; let a1 > 句 > a3 > O. If B devotes
Chapter 4. Mixed Strategy Equilibrium
142
A
A
B
B
0, 0, 0
0, 0, 0
1, 1, 1
0, 0, 0
A
A
B
A 0, 0, 0 0, 0, 0
B
0, 0, 0
4, 4, 4
B
Figure 142.1 The three-player game in Exercise 142. 1.
resources to the district that A targets , then A gains no votes. Each party's preferences are represented by the expected number of votes it gains. (Perhaps seats in a
legislature are allocated proportionally to vote shares.) Formulate this situation as
a strategic game and find its mixed strategy equilibria
Although games with ma町 players cannot in general be co日ve口ie口tly represented in tables like those we use for two-player games , three-player games can
be accommodated. We construct one table for each of player 3's actions; player 1
chooses a row, player 2 chooses a colurnn, and player 3 chooses a table. The next
exercise is an example of such a game.
? EXERCISE 142.1 (A three-player game) Find the mixed strategy Nash equilibria of
the three-player game in Figure 142.1 , in which each player has two actions.
4.11
Extension: games in which eacn player has a continuum of actions
In all the games studied so far in this chapter each player has Íinitely many actions. In Chapter 3 we saw that ma口y situations may conveniently be modeled as
games in which each player has a continuum of actions. (For example , in Cournot' s
model the set of possible outputs for a firm is the set of nonnegative 口umbers， and
in HotelIing's model the set of possible positions for a candidate is the set of nonnegative numbers.) The principles involved in finding mixed strategy equilibria of
such games are the same as those involved in finding mixed strategy equilibria of
games in which each player has finitely many actions , though the techniques are
different
Propositio日 116.2 says that a strategy profile in a game in which each player has
finitely many actions is a mixed strategy Nash equilibrium if and only if, for each
player, (a) every action to which her strategy assigns positive probability yields the
same expected payoff, and (b) no action yields a higher expected payoff. Now, a
mixed strategy of a player who has a continuum of actions is determined by the
probabiIi ties it assigns to sets of actions , not by the probabilities it assigns to single
actions (all of which may be zero , for example). Thus (a) does not 自t such a game
However, the following restatement of the result, equivalent to Proposition 116.2
for a game in which each player has finitely ma口y actions, does fit.
11
PROPOSITION 142.2 (Characterization of mixed strategy Nash equilibrium) A
mixed strategy proftle 旷 in a strategic gaJ时 ωith vNM prl价rences is a mixed strategy
Nash equilibrium 扩 and only if, for each player i,
4.11 Ext牵 nsion: ga lT在牵s in whkh each play'哇 r has a continuum of actions
143
·α 1 assigns probability zero to the set of actiOlls aí for which the actíon pro.卢 Ie 叭 I a i
yields player i an expected payoff less tltan her expected payoff to 旷
• for 110 action aí does the octíO l1 pro.卢 le 仙 ，1\:*
greater than he川
yield player i 011 expected payoff
A significant cJ ass of garnεs in which each player has a continuurn of actions
consists of garnes Ín which each player's sεt of actions is a one-dirnensional interval of nurnbers. Consideτsuch a garne with two players; let player i's set of actions
be the interval frorn 经ì to G;, for i 1, 2. Identífy each player's rnixed strategy wíth
a cumu[ative probability distribution on this inter飞la J. (See Section 17.6.4 i只 the appendix on rnathernatics if you are not farnilìar with this notion.) 丁hat is, the mixed
strategy of each player i is a no日decreasing fllnction 乓 for which 0 三日创)三 l
for every action Oí; the number Fi 但/) is thεpr巾ability that player i's action is at
注10st
aj.
The form of a rnixed strategy Nash 叫uilibrium in such a game rnay bεvery
complex. Some such games, however, have 叫uilibria of a particularly simple
form , ín which each player's e气uilibrium mixed strategy assigns probability zero
except in an interva l. Specificall乒， consider a pair
乌) of rnixed strategies that
satisfies the following conditior飞 s for i = 1, 2
• There are numbers Xj and Yi such that player i's rnìxed strategy Fi assigns
pro七 ability zero except in the ínterval from Xj [0 Yi: Fi(z) = 0 for z <工 i， and
Fí(z) 口 1 for z 三 Yí'
• Player i's expected payoff 飞γhen her action ís ai and the other player uses her
mixed strategy 乌 takes the form
for Xi ai <二 yi
for a; < Xi and ai > Yi ,
where Ci is a constant.
(丁he second condition is illustrated in Figure 144 .1.) By Proposition142.2 , such a
pair of mixed strategies, if it exists, is a rnixed stratεgy Nash equilibrium of the
game, in which playεr ì's expected payoff is Ci , for i = 1, 2.
The next example illustrates how a rnixed strategy equilibrium of such a game
may be found 编百le example is designed to be very simple; be 叭'arned that in
most such games an analysis of the equilibria is, at a mÍl飞imur凹， somewhat more
compl凹. Furth盯， my analysis is not complete: 1 merely find an equilibrium , rath衍
than studying all equili己ria. (In fact , the game hasηo oth号 r equilibria.)
EXAMPLE 143.1 (All-pay auction) 丁wo people submit sealed bids for an object
worth $K to each of them. Each person' s bid may be any nonn鸣在 tìve number
up to $1ζ 百lew让mer is the person whose bid is higher; 如 the evεnt of a tie each
person receives half of the object, whlch she values at $K /2. (Note that this tie唰
breaking rule differs frorn the one considered in Section 3.5.) Each persoηpays her
Chapter 4. Mixed Strat牵gy Equilibrium
144
C2
ε1
主i
Yl
X1
111
旦2
工2
y2
a2
a2
al
Player 2'5 已 xpected payoff given F]
Player 1'5 expected payoff given F2
Figure 144.1 日的 F1 assigns positive probability only to actions ìn the ìr飞 ten'al from x] to 叭， (i i) F2
assìgns positive probability 0川 Y to actions ìn the interv在! from X2 to 112 , (iii) gíven pJayer 2'5 mixed
strategy F2' player l's expected payoíf takes the form shown in the left p吕 nel ， and (ív) given player 1 、
们飞 ixed strategy Fl' player 2's expected payoff takes the form shown in the right pane l. then (Fj ， 1引 15 a
l1ì ixed strategv 问 uilibrium
t予 id , 陀gnrd)css 句
of d
江， {υd
圳
山
川山
1)h川
r
an飞ou
山
1川l川
1让tof mτ
叭飞one
盯Y shεreceí飞res.
This sìtuation may be IτlOdeled by the following strategic game , know口 as an
all-pay auction. (飞/盯iants of this game are considered ín Exercìse 89.1.)
PIαyers
The two bidders.
Actions Each player's set of actions is the set of possible bids (nonnegatíve
numbers up to K)
Payo疗 fυηζtioηs
Each player i's preferences are represented by t七号 εxpected
飞'alue of the payoff function give口 by
-VAVA
…
gtt飞
，，飞
川内
μ
n
/IIE
r
叫，
"
nH
一一
nu
…
d
if ni < aì
if l1 i 之二I1 J
if 11[ >町，
飞气.hεre i is th伫 otl飞er player
One situation tl飞的 may be modeled as such an auction ís a lobbying pr。但5S in
which each of two ìnterest groups spends resources to pθrsuaçle a government to
carry out the policy it prefers , ，u飞d the group that spends thεmost wíns. Another
situation 日ìat may b伫 modeled as such an auctíon is thεcompetition between two
firms to develop a new product by some deadline , 飞气'here thε finn that spends the
most de\吧 !ops a bctter product, which capturε5 the entire marke t.
An all-pay auction has 口。 pure strategy Nash equilibrium, by the following
argur到 en t.
• No pair of actìons (x , x) with x < K is a Nash equilibrium because eith们
player can increase her payoff by slightly increasi口g her bid
• (K , K) is 叭。 t a Nash 叫uilibrium because either player can increase her payoff
from -K /2 to 0 by reducing her bid to O.
• No pair of actions (吨， 112) with /.1 1 并 112 is a Nash equilibrium becau使 the
player whose bid is higher can increase her payoff by reduci口g her bid (ar飞d
4.11 Extension: games in which each player has a continuum of actions
145
the player whose bid is lower can , if her bid is positive , increase her payoffby
reducing her bid to 0).
Consider the possibilìty that the game has a mixed stratεgy Nash equilibrium
Denote by Fí the Ill ixed strategy (i .ε. cumulative prc必过，ility distributìon ovεr the
interval of possible bids) of player i. 1 look for an equilibrí Ulηiηwhich neitl飞er
mixed strategy assign5 positive probabìlity to any single bid. (Remember that tl 飞ere
a陀 infinitely many possible bids.) In this case Fí((/i) is both the probability that
player i bids at most úi and the probabi]ity that she bids less than a 俨 1 further啕
restríct attε挝ion to strategy pairs (Fl' 巳) for which , for i = 1, 2, there are numbers
.t j and !J i such that 1-; assig口s positivεprobability only to the inlerval from xi to 1J i'
To investigatεthe possibility of such an equilibriulll , consider player l's ex
pected payoff when she uses thεaction 町， given player 2's mixed strategy F2 刷
‘
• If aj < .t2, then Új is less than player 2'.s bid with probabílity one , so that
player l's payoff is … Ú ,.
• If (/1 > 1J 2 , then Új exceeds player 2's bid 认'ith probabilíty one, 50 that
player l's payoff is 1王一。1 制
• If .\2
日 1 <二1f2 ， then player 1' s expected payoff is ca 1c ulated as follows. With
probability F2 (Úl) player 2's bid is less than nl , in vvhich case player l's payoff
isK 一 Ú1; with probability 1 一巳 (n ，) pla J 衍了 2' s bid exceeds nj , in which case
pl咛er l's payo Íf is … Úl; and , by assumption , the probability that player 2's
bid is exactly equal to Ú1 is zero. Thus player l's expected payoff is
(K … ndF2(ú1 ) 卡(一 σ1)(1 … F到 Ú1))
KF2 ((/j) …们
Lγe need to find values of .t 1 and 1J 1 and a strategy
such that player l's exω
pected payoff sa 挝sfies the condition illustrated in the left panel of Figure 144.1: it
is cor飞stant on the interval from .t1 to 叭， and less than this constant for (/1 < .\1
and (/1 > !J l' The constancy of the payoff on the inter飞划 frorn X1 to 1J 1 requires that
k 石问)一。 1 = C1 for X1 三。 1 主扫， for some consté\l1 t [1. We also ne时乌拉βo
and F2 (Y2) 口 1 (beca use 1 am 陀stricting attention to equìlibria in \vhich neither
player's s钉在
assigns positive probability to any single actio对， and 巳 rnust
be nondecreasing (so tÌ1at it is a cumulative probability distribution). Ar飞alogous
conditions rnust be satisfied by 巧，如， and F1
We see that if 川江与二 O，1J l = y2 K, and 乓巳(=) zjKforallzwith
O 三 Z 豆豆， then a11 these conditions are satisfied. Each player's expected payoff is
constant, equal to 0 for all her actions.
Thus the gamεhas a mixed strategy Nash 吨曰libriurn in which each player
randomizes "uniforrnly" over a11 her actions. 1n this equilibriurn each player's
expectεd payoff is 0: 。在 average， the amount a player spends is exactly 叫ual to
the 飞ralue of the objec t. (A more involved argurner飞t 驰。飞vs that this 吨uilibriurn is
the only mixed strategy Nash 叫出1ibrium of the game.)
( EXERCISE 145.1 (All-pay auction with many bidders) Consider the generaliza位。n
of the game considered in the previous exarnple in which there are 11 ~二 2 bidders.
Chapter 4. Mixed Strategy Equilibrium
146
Find a mixed strategy Nash equilibrium in which each player uses the same mixed
strategy. (Jf you know how，在1飞d each player's mean bid in the 叫uilibrium.)
百 EXERCI叩 146.1 (Bertrar飞d's duopoly gam苟 Consider Bertrand's oligopoly game
(5ection 3.2) when there are two firms. Assume that each firm's preferences are
陀presented l巧r its expected profiι5how that if the function (p - c) 0 削 is inc叫"
íng in p, and 拙
i Iη飞 cr陀
ea
部
Sε
白s without bour只lda
挝s p incr陀
ea
臼se
臼S 飞rv
吟￥
p > ει， the game has a m
古双li以×附
ωd strategy Na
仪
e
部sh 叫
e qu
川ilibrium 垃
in 飞W
Nhic巾
叮
h each 缸
fi让主m
τηlU
础Sε
臼S
t址he 曰
s ame
盯刊
m
I红吉η
挝lÍ 肥
xed 纣
s tra
川t怡
egy
with F(p) 0 and F(p) > 0 for p > P
In the games in the example andεxercises， each pla)咽 's payoff depends only
on her action and whεther this action is greater than, equ在 1 to, or less than the other
play时S' actions. The limìtεd dependence cf each player's payoff on the other pl巧"
ers' actions makes the calculation of a play臼 's expected payoff straightforward.
In many games， εach pl叫er's payoff is affected morεsubstantially by the other
players' actions, m点ing the calculation of expected payoff rnore complex; more
sophisticated mathematical tools are required [0 analyze such games.
4.12
Appendix: representing preferences by expected payo仔s
4.12.1
Expected payo陆
5uppose that a decision-maker has preferences over a set of deterministic out…
comes, and that each of her actions results in a lottery (probability distribution) over
these outcomes. To determine the action she chooses, we need to kno飞气r her prε
ences over these lotteries. 人s argued in SectiOI飞 4.1.3， we cannot deriuc these pr己 f­
erences from her preferences over deterministic outcomes; rather, we must specify
them as part of thεmodel.
50 assume tha t w心 are given the decisiOI子maker's prefercnces over lottεn岱­
As in thεcase of preferences over deterministic outcomes, under som已 fairly weak
assumphOI飞s we can represent these prefcrences by a payo Íf function. (Refer to
kεtìon 1.2.2.) That is, when there are K detenninistic outcomes we can find a
function , say U , over lotteries such that
U(Pl ," ., PK) > U(p; ，...， p~)
if and only if the decision-maker prefers the lottery (Pl"'" pd to the lottery
(p; ，...， p~) (where (Pl ,..., PK) is the lot松ry in which outcome 1 occurs with
probability ].1 1, ou. tcorne 2 occurs 飞/11吐飞
For mη1a衍ny purposes仇， however.乙J 飞w
气ve need more st衍ru
泣C
吐tu
旧re
巳 we ca
拍
rηìI只飞ot get ve
盯ry
i
fa主 飞wi坟thOl时
1此t 陀
res纣t甘t
臼
出ìc
往tì损只g OL叽lrselves to p陀f仨
er陀匈它
ences
for which there îs a 11飞o
仪}疗re specific representation. The standard approach, developed by von Neumanr飞 and
Morger飞stern (1944) , is to impose an additional assumption-the "ind叩endence
axíom"-that allows us to concludεthat the decisior刊naker's preferences can be
represeηted by an e.工pected payoff卢mctíon. More preciselYI the independence ax附
iom (which I do not describe) allows us to conclude that there is a payoff function
4.12 Appendix: representing preferences by expected payoffs
147
U over deterministic outcomes such that the dεcision-rτìaker's preference relation
over lotteries is r叩resented by the function U(Pl ," ., PK)
ak is the kth outcome of the lottery:
L Pku(ad > L p~U(ílk)
Lf=l Pku(aÙ where
(147.1)
if and 0均 ìf the decisior刊naker prefers the lottery (Pl , … , PK) to the lottery
(p; ，...， p~). That 也 the decision-makεr eval础 tes a lottery by ìts e工pected pa沪
off accordìng to the functìon u, which is known as the decisìon-maker's Bemoulli
payojj卢mctio刀，
Suppose, for example, that therεare three possible determinìstìc outcomes: the
decisìon-mak盯 may 陀ceìve $0, $1 ，但如， and naturally she prefers $5 to $1 to $0.
Suppose that she p陀fers the lottery
(i , 0, ~) to the lottery 忡， ~， !) (where the first
l飞 umber in each lìst is the probability of $0 , the second number is the probab i1i ty of
$1 , and the third number is the probability of $5).τhìs prefer巳nce is consistent wÌth
prεferenc巳s represented by the expected valuεof a payoff function u for which
叫。) = 0, u(1)
1, and 叫= 4 becaωe
i ， 0+i.4>~.1 十 !.4
(i i)
(Many other payoff functions are co
仪∞阴
陌
n
1冶S臼 t怆G仅扭棍
挝1t with a preference for
n
, O,
over
(O，~，
Among those ìn which 叫 0) == 0 and u
= 4, for example, a町 function
forv巾ch u (1) < j does the job.) Suppose , 0到 the 时her hand , that the decisionis pr陀efe悄
f陀
ren
枉
阳nce们1阳O 川 5
maker prefers the lottery (O，~， !) to the lottery ( 抖, i且) τh旧
t怜佼衍nt wíth pr
陀
efe
盯rence
础s repr拮
esent怡
ed by the expect怡
ed value of a payof
旺ffurηìct挝ìon μ
t1. for
which μ叫(忡0) = 0 ，叫 1) 3, and 叫 5) = 4 because
!).
‘
~， 0+i.4<~.3+1.4
CD EXERCISE 147.2 (Preferεnces over lotteries) There are threεpossible outcomes; in
the outcorτ泛泛 ì a decision-mé飞ker gains $aj , where al < a2 < a3. The decisìonmaker prefers G3 to 向 to al and she prefers the lottery (0 .3, 0, 0.7) to (0. 1， 0礼 0 .5)
to (0.3 , 0.2, 0.5) to (0 .4 5, 0, 0.5 5). Is thìs i时ormation consistent with the decisionmaker's preferences' being represented by the expected value of a payo任 function?
If so, fÌ1ìd a payoff function consistent with thεinforrr飞atìon. If not, show why not.
Answ盯 the sarne quεstions when, alternatìvely, the decìsiorνrnaker prefers the
lottery (0.4 , 0, 0.6) to (0 , 0.5, 0.5) to (0 .3, 0.2, 0.5) to (0.45 , 0, 0.55).
P陀ferences represented by 出e expected value of a (Bernoulli) payoff functìon
have the great advantagεthat they are completely specified by that payoff functIo孔Once we know u(ak) for each possìble outcorne ak we know the decisionrnaker's preferences among all lotterìes. This significant advantage does , howeve乙 carry wi出 it a srnall price: it is very easy to confuse a Bemoulli payoff functÌon with a payoff furïction that represents the decísion-rnaker's preferences over
deterministic outcon飞es.
Chapter 4. Mixed Strate苦苦盖quilibrium
148
To describe the relatìon behveen thεtwo， suppose 住在 t a decisiorνmaker's preferences over lott<εries arεrepresented by the expεcted value of the BernouIIi payoff function u. η1en certainly u is a payoff function that represents the dedsionmak盯 's preferences over deterministic outcomes (which are spedal cases of lotter
ies, in which a single outcomεis assigned probability 1). However, the converse is
not true: if the decision-maker' s pr!εferences over det，εrministic outcomes are represer飞ted by thεpayofffunctionμ(i.e. the decision-mak时 prefers a to a' if and only if
u 归) > u(a')) , then u is 110t necessariIy a BernouIIi payoff function whose expected
valu号 represents the decision-maker's preferencεs over lotteries. For instance , suppose that the deci仰卧makerp陀fers $5 to $1 to $0 and p陀公rs the lottery , 0, i)
to the lottery (O，~，
Then her pre公rences over deterministic outcomes are con
4
sisìent with the payoff fu削ion u for which μ(0) = 0, μ(1) = 3, and u(5)
However, her preferences ovεr lotteries are 110t consistent with thεexpected value
of this function (since i 自 +i 4<i 3 十! .4). The moral is that you ShOl必 be
careful to determine t检 type of payoff function with whìch you are dealing.
(i
!).
4.12.2 Equivalent 8ernoulli payoff functions
If a decision-maker's preferences in a deterministic environment a .re represented
by the payoff function u, thεnthey 及re represented also by any payoff function that
is an iηcreasing function of u (see Sec悦。n 1.2.2). The analogous property is not
satisfied by Bernou \l i payoff functions. Consider thεexample discussed above.
A Be主noul\i payoff function u for which 叫的
0， u(l) 口 1 ， and u(号= 4 is
consiste的认怕 a pr!陀e时
时f仨e盯
的renc纪刊for the lottery , O,!) over (0 , i , !), but t1时unction
i言， for which 叫。) = 0, u(l) = 1, and 叫5) 口 2， is not consistent with such a
p时ere附(~ .0 什 2<i l 十 j 斗， though the 叩are root f肌tion is increa5ing
(larger r飞机rnbers have larger square roots).
Under what circum5tances do the expected 飞ralues of two 日ernoulli payoff fu挝南
tions rεpresent the same preferences? 丁he next result ShOW5 that thεY do 50 if and
or飞Jy if one payoff function is an i口cr'εasing IÍ17car function of the other.
(i
翻 LEMMA 148.1 (Equivalence of BernouIIi payo日 functions) Suppose thcre are at least
three possible outcom邸. The expected values of the Bemollllì payoff乒mctions μ a l1 d v
repre5ent the 5ame p功rel1 ces oZJer lotterie5 if aηdo咐 if there exi5t l1 um 仰'叫 Gl1 d ewith
e > 051川
If the expected value of u rεpresents a decision-maker's preferences over lottεries， then so , for example, do the expected values of 2u , 1 + u, and 1 + 4u; but
…
the expected values of u 2 and of YU do no t.
Part of the lemma is easy to establish. Let u be a Bemoulli payoff fur飞ction
whose expected val时 represents a decisíon-maker's prεferences， and let v(x)
夺十。叫 x) for all x , whereηand e are constants with e > O. 1 argt始 that the
expected values of 11 and of v represer飞t the same preferences. Suppose that the
decision-maker prefers the lottery (Pl" . ., PK) to the lottery 时，... ， l如. Th enher
expected payoff to (Pl ,"', PK) εxceeds her expected payoff to (，牛 ..， l白， or， as
4.12 Appendix: r告presenting preferences bγexpected payoffs
149
given in (147.1) ,
L Pku(ak) > L p~u(ak)
(149.1)
Nov气l，
K
K
K
K
V (a句
:汇二 p内k 叫
ω
左二) 土 汇 附主刊厅
Pk 十 汇 Pk 仇e 归叫
ut叫(阳
G向kρ)
叫
厅十 e~汇二 阳p马驯
kμiμ巾
i
usin
口19 毡
th飞e fact 出
th
挝
说t the sum of 也
a
th飞e probabilities 问
pk 巳
1 s 1. Similarly,
[p~叫ak)
可十 e
K
L p~u(ak)
Substituting for u ift (149 .1) we obtain
τ
/ K
\
哺
/
\lit-
K
盯叶，
~ ~lf; Pkv(ad 才) > ; ~k~ p~v(ad
,/
which , given e > 0, is equi飞ralεntto
K
K
LPkv(ak) > 汇 p~v(ak) :
according to v, the expected payoff of (Pl ,"" PK) exceeds the expected payoff of
(p~ ，...， p~). wεconclude that if u 叩 resents the decision-maker's preferences ,
then so does the function v defined by v 忡口 11 个 eu(x).
1 omit the more difficult argument that if the expected values of the Bemoulli
payoff functions u and v represe时 the same preferences over lott肝iεs， then v(x) =
1]十 eu (x) for some consta赂 1J and e > o.
台 EXERCISE 149 .2 (Normalized Bemoulli payoff functions) S江ppose tha t a decision
maker's prefεrences can be represented by the expected value of the Bemoulli pa沪
off function u. Fi nd a Bemoulli payoff fu只ction whose expected value represe且也
the decision悦laker' s preferencεs and assigns a payoff of 1 to the best outcome and
a payoff of 0 to thεworst outcome.
4.12.3
Equìvlαlent strategic games with vNM pref专rences
Turning to games , coτlsider thεthree payoff tables in Figure 150. 1. All three tables
represent the same strategic game with deterministiζpreferεnces: in each case,
player 1 prefers (B , B) to 币，创始 (B， 句，￥γhich she regards as indifferent to (S , B) ,
and player 2 prefe黯币，的 to 凹，的 to 恨，的， which she reg盯ds as indiffere时如
何， B). However, only the le丘 and middle tables represent the same strategic game
with vNM preferenc巳. The reason is that the payoff fWlctions in the middle table are linear functions of the payoff functions in the left table , whereas the pay伴
。ff functions in the right table are not. Specifically, denote thεBernoulli payoff
functions of player ì in the three games by U j, (}i, and 叫. Then
'01 但) = 2ul(a) and V2(α) =-3 十 3吨位)，
Chap生er 4. Mixed Strategy 主quilibrium
150
户hjd'''
IFF
AV7
』
AHV 节i
吨，
4nv
户3
τiAU
nD
B
S
3, 2
0, 1
0, 1
1, 4
5
B
B
4,
S
0, -3
0
B
S
0,… 3
2, 3
Figure 150.1 人 11 three tables represent tl飞e same strategic game with ordi扒在 1 preferences, but only the
left and middle games , not the right one, rcpresent th日 same strategic game with vNM preferences
so that the left and middle tables represe时 the sam己 strategiεgame with vNM
preferences. However, Wl is not a linear function of u}. lf it were, there would
以ist constants 17 and e > 0 such that 叫(的=与十 eu} (a) for each action pair a, or
equivalently
0=17 十 e.o
1f +e.1
1
3 =号十 e.2.
Ho飞J飞'ever， these three equations have no solution.
Thus the left and right tables
r巳prεsent different strategic games with 飞'::-JMprεferen优s. (As you can check, W2
is not a linear function of U2 either; but for the gamesτlOt to be equivalent it is sufficient that ol1 e player's preferences be differen t.) Another \^.'ay to s茫茫 that player l's
vNM preferences in the left and right games are different is to note that in the left
table player 1 is indifferent between the certaÎn outcome (5, S) and the lottery ìn
which 律， B) occurs with P时ability and 伍， B) occurs wìth p时ability (each
yields an expected payoff of 1), whereas in the right table she prefers the latter
(since it yields an expected payoff of 1.5)
i
i
? EXE 我 CISE 150.1 (Games equivalent to the I勺'Îsoner's Dilemmα) Which of the tables
in Figure 150.2 represents the samεstrategic game with vNM prefe段时es as the
l习'ÍSOI对r 、 Dile/lllna as specifìed in the left panel of Figur号 107 .1，飞"1hεn the numbeτs
are ìnterpreted as Bernoullì payoffs?
哇
吱句/』
叫
Ji
jnHV 句'h
飞
毛FA
1
M 门υ
aij 鸣、
:''
CD
C
o
C
。
6, 0
9, -4
0, 2
3, -2
Figure 150.2 The p在 yoff tables for Exercísε150 .1
Notes
The ideas behind mixed strategies and p 肥ferences represer飞ted by expected payoffs date back in Western thought at least to the eíghteenth century (see Guilbaud (1961) and Kuhn (196时， and Bernoulli (1738)，四日pectively).τhe modern
formulation of a míxed strategy is due to Borel (1921; 1924, 204-221; 1927); the
model of the representation of preferences by a口 expected payoff function is due
to von Neumann and Morgenstern (1944 , 15-31; 1947, 617-632). The model of
a mixed strategy Nash equilibrium and Proposition 119.1 on the existence of a
Note军
151
mixed strategy Nash eq江ilibrium in a finite game are due to Nash (1950a , 1951).
Proposition 122.1 is an implìcation of the existence of a "trembling hand perfect
equilibrium" , due to Selten (1975 , Theorem 5).
The example in the box on page 104 ís taken from Allais (1 953) Conlisk (1989)
discusses some of the evidence on the theory of expected payoffs; Machina (1 987)
and Hey (199ηsllrvey the subjec t. (The purchasing power of the largest prize
in Allais' example was roughly U.s .$6.6 million in 1989 (the date of Conlisk's
pape乙 in which the prize is U.s. $5 million) and roughly U.S.$8 million în 1999.)
The model in Section 4.6 is due to Pitchik and Schotter (1987). The model în Section 岳 .8 1S a special case of the one in Palfrey and Rosenthal (1984); the interp陀w
tation and analysis that 1 describe is taken from aηunpublisheà 1984 paper of
认Tilliam F. Samuelso孔 τhe box on page 133 draws upon Rosenthal (1964) , Latan是
and Nida (1981) , Brown (1986L and Aronson (1995). Best response dynamics were
first studied by Cournot (1838, Chaptεr VI巧， in the context of his duo户)ly game.
Fictitious play was suggested by Brown (1951). Robinson (1951) shov气rs that the
process converges to a mixed strategy Nash equilibrium in any two-player game
iηwhich the players' interests are opposed; Shapley (1964, Section 5) exhibits a
game outside this class in whìch the process does not converge. Recent work on
learning in games is sur飞'eyed by Fudenberg and Levine (1 998).
在le game in Exeróse 118.1 is due to David L. Silverman (see Silvennan 198182 and Heuer 1995). Exercise 118.2 is based 011 Palfrey and Rosenthal (1983). Exercise 118.3 is t在 ken from Shubik (1982 , 226) (who 白nds 0口lv or飞e of the continuUlη
of equilibria of the game).
百le model in Exercise 128.1 is taken from Peters (1984). Exercise 130.2 is a 叫fl­
antofan 以ercise of Moulin (1986b , 167, 185). Exercise 132.3 is based on Palfr句rand
去。senthal (1984). 丁he game Rock, Paper, Scissors (Exercisε14 1.2) was first studied
by Borel (1924) and von Neumann (1928). Exercise 141.3 is based on Karlin (1959a ,
92-94) , who attrìbutes the game to an unpublìshed paper by Dresher.
Exercise 145.1 is based on a resu 1t in Baye, Kovenock , and de Vries (1996). The
mixed strategy Nash equilibria of Bertrand' s model of duopoly (Exercise 146.1) are
studied in detail by Baye and Morgan (1999).
The method of finding a11 mìxed strategy equilibrium descrìbed in Section 4.10
is computati∞aIly 飞rery ìntense in a11 but the sìmple
‘
Extensive Games with Perfect Information:
Theory
5.1
5.2
5.3
5.4
5.5
Extensive games with perfect information 153
Strategies and outcomes 159
Nash equilibrium 161
Subgame perfeζt equilibrium 164
Finding subgame perfect equilibria of finite horizon games:
backward induction 169
Prerequisite: Chapters 1 and 2
HE MODEL of a strategic game suppresses the sequential structure of decision-
making. When applying the model to situatíons in which decisio仔makers
move sequentially, we assume that each decìsion-maker chooses h前 plan of actior飞
once and for a11; she is committed to this plan , which she cannot modify as events
unfold. The model of an extensive g在m台， by contrast, descr也部 the sequential
structure of decìsion-making explicitly, allowìng us to study sìtuations in which
each decision-maker is fr!时 to change her mind as events unfold.
In this chapter and the next two we study a model in which each d配isio仔
maker is always fu11y informed about a11 pr!εvious actions. In Chapter 10 we study
a more gen盯在1 model, which a l10ws each decisìon-maker, when taking an actìon,
to be imperfectlv informed about previous actions.
5.1
Extensive games with perfect information
5.1.1
Definition
τo describe an extensi飞re game wìth perf，εct ìnformatìon, we need to specify the
set of players and their preferences, as for a strategic game (De自IÙtíon 13 .1). In
additìon , we need to specify the order of the players' moves and the actions each
pl的'er may take at each poin t. We do so by spεcifying the set of a11 sequences of
actions that can possibly occur, together with the player who n飞。ves at each point
in each sequence. We refer to eac且 possible sequence of actions as a 缸'minallústory
and to the functìon 在at gives 白e player who moves at each point in each terminal
the
叩
乒la
p
呼
ye
衍r巾
'fi
μ
川
u
11昭
1
history as 由
• playe盯rs
• terminal histories
• player function
• preferences for the players.
'在 3
154
Chapter 5. Extensive Games with Perfect Information: Theory
在efore giving precise definitions of these components , 1 give an example that
i1lustrates them informally.
EXAMPLE 154 .1 (在批ry game) An incumbent faces the possibility of entry by a
challenger. (The challenger may, for example , be a firm ∞nsidering er飞try into
an industry currently occupied by a monopolist, a politician competing for the
leadership of a party, or an animal considering competing for the right to mate
with a congenεr of the opposite sex.)τhe challenger may enter or no t. If it enters,
the incumbent may 臼ther acquiesce or figh t.
We 血aymodε1 this situation as an extensive game with perfect iruormation in
which the terminal histories are (In ， Ac伊拉sc叶 ， (I n, Fight) , and 0说 and the player
function assigns the challenger to the start of the game and the incumbent to the
history In.
At the start of an extensivεgame， and after any sequence of events , a player
chooses an action. The sets of actions available to the players are not, howεver，
given explicitly in the description of the game. Instead , the description of the game
specifies the set of terminal histories and the player function , from which we can
deduce the av在ilable sets of actions.
In the entry game , for example, the actions availablεto the challenger at thε
start of the game are 111 and Out , because these actions (and no others) begin ter唰
minal histories, and thεactions available to the incumbent are Ac伊 iesce and Fight ,
because thεse actions (and no others) follow In in terminal histories. More generally, s叩pose that (C , D) and (C , E) a肥 terminal histories and the pl町erfunction
assigns player 1 to the start of the game and player 2 to the history C. τhen two
of the actions available to player 2 after player 1 chooses C at the start of the game
are D and E
The terminal histories of a game are specified as a set of sequences. But not
every s仗。f sequences is a legitimate set of terminal histories. If (C , D) is a terminal
histo巧1， for example, there is no sense in specifying C as a terminal history: the fact
that (C , D) is terminal implies th挝 after C is chosen at the start of the game, some
play衍 may choose D , so that the action C does not end the game. More generally,
a sequence tha t is a proper subhist01γof a terminal history cannot itself be a terminal
history.τhis rεstriction is the only one we need to ímpose on a set of sequences in
order that the set be interprεtable as a set of terminal histories.
To st在tεthe restriction precisely, define the subhistories
5.1 Extensive games with perfect informatio!l
155
In the entry game in Example 154.1 , the subhistories of ([11 , A碎的esce) are the
empty history 0 and the sequences 1n and (111 , Acquiescc); th巳 proper Sl剖1istories
are the empty history and the sequence 111.
时冈州'τION
155.1 (Extensive game with pelfect infom川仰) An extensÌve game
with perfect information consists of
• a set of players
• a set of sequences (tenninal histories) with the property that no sequence Ìs
a proper subhistory of any other sequenεe
• a function (the player function) that assigns a player to every sequence that
is a proper subhistory of some terminal history
• forεach play毗 preferences over the set of terminal histories.
丁he set of terminal histories is the set of aU sequences of actions that may occur;
the player assigned by the player function to any hìstory h is the player who takes
an action after h.
As for a strategìc game, we may specify a player's preferences by giving a payoff问 ctiOI1 that represents them (see Section 1. 2.2). In some situations an outcome
is associated wìth each terminal hìstory, and the players' preferences are naturally
defìned over these outcomes , rather than dìrectly over the terminal histories. For
example, if we are mode1ing firms choosing prices , then we rnay thìnk in terrns of
each firrn's carìng about its profìt… the outcorne of a profile of prìces-rather than
directly about the profìle of prìces. However, any preferences over outcornes (e.g.
profits) rnay be translated into preferences over tεrminal historìes 价 .g. sequences
of prìces). In the general definition, outcornes are convenientìy identified with terrnìnal histories and preferences are defined d让ectly over these histori白， avoìding
the need for an additional elernent in the specification of the garne.
EXAMPLE 155.2 (Entry garne) In the situation described in Example 154.1 , suppose
that the best outcome for the challenger is that it enters and the iHcumbent acqui翩
esces , and the worst outcorne is that it enters and the incurnbent fights , whereas
the best outcome for the incurnbent is that the challenger stays out, and the worst
outcorne is that it en伦rs and there is a fìgh t. Then the situation rnay be rnodeled as
the following extensive game with perfect inforrnation.
Players The challenger and the incurnbe时，
Terminal historíes (111, Acqμ Íesc<叶 ， (In，F垃h斗， and Out.
Playerfunction P(的二 Challen俨r and P (In) 工 1ncumbel1t.
Preferences The challeng时 's preferences are represented by the payoff function
Ul forwhich 叫1n， Acquiesce) = 2, Ul(Out) = l , and ul (In， F结ht) 0, and the
Ìtî curnbent's preferences are represented by the payoff function U2 for which
U2( Out) 口 2， u2 (I11, Acquiesce) = 1, and U2 (111 , Fight) = O.
156
Chapter 5. Extensive Games with Perfed Inform就ion:Th哇。ry
Challenger
I-
VPEEA
lncumbent
A吁 wesce
Fight
2, 1
0, 0
1, 2
Figure 156.1 The entry game of Example 155、 2 ， The challeng肝、 payoff is the fírst number in each paí r.
Th is game is readily illustrated in a di吨ram ， The small circle at the top
of Figure 156 1 represents the empty history (the start of thεgameì 丁he la栅
bel above this circle indicates that the challenger chooses an action at the start
of the game (P( θ
Clwllenger). The two branches labeled ln and Out represent the challenger's choices. The braI飞ch labeled 111 leads to a small black
dísk, the label beside \vhich indìcat部 that thεincumbent takes an action after the history 111 (that is, P(I n)
Incumbent). The two branches emanating
from the dísk represent the inεumbent's choices , Acquiesce and Figh t. The pair
of numbers beneath each terminal history gi飞'es the players' payoffs to that history, with the challenger's payoff listed firs t. (τhεplayers' payoffs may be given
ín any ordeL For games 1i ke this 0时 in which the players move in a welldefined ord仔， 1 generally list the payoffs in that order. For games in which
the pJayers' names are 1, 2, 3, ar飞d so on , 1 list the payoffs in the order of th的r
names.)
,
D时inítion 155.1 does not directly specify the sets of actions available to the
players at their various moves As 1 discussed briefly before the definitior飞， vve
can deduce these sets from the set of terminal historiεs and the play臼妇nction.
1f, for some no时erminal hístory 11 , the sequence 怡，叫 is a history, then a is one of
the actions a飞!ailable to the player who mo\'es after h. Thus the set of all actions
available to the player 认'ho moves after h is
,
A (f z)
{a: (11 , a) is a history}
,
(156.1)
For example, for the game i旦 Fígure 156.1 , the histories are
111 , Out , (I11 ,
Thus the set of actions available to the player who
moves at the start of the game , namely the challe吨er， is A( 圳工 {II1， O 叫， and
the set of actions available to the player wl飞冶o moves 祉
a t仕te
凹r th冶
e history 扫
1 3η1乙， na
旧
amηlel马yF
the ír口飞￡飞泣
umη
旧巾
b咒
创1时， 抬
e
i sA (J叫 口 {Ac守仰归
i仪n归扣ε5皮ceι，告
'g11
挝叫
f叶}.
Acquiesc吟， and (I1l,F igftt).
? EXERCI 刊 156.2 (Examples of extensive games with perfect information)
a. R叩rε问nt in a diagram líke Figure 156.1 the two盼player extensive game with
perfect information in which the tεrminal historìes are 伍，日 ， (C , F) , (D , G) ,
and (D , H) , the player function is given by P(ø)
1 and P(C) = P(D) =
乙 player 1 prefers (C , F) to (D , G) to (C , E) to (D , H) , and player 2 p陀fers
(D , G) to (C ,F) to (D , H) to (C , E).
5.1 Extensive games with perfect information
157
b. Write down the set of players, set of terminal histories, player function , and
players' preferences for the game in Fígure 160. 1.
c.
ηle political 均ures Rosa and Emesto have to choose either Berlìn 扭) or Ha-
vana (H) as the location for a party congress帽 They choose sequentially. A
thlrd person, Karl , determines who chooses firs t. Both Rosa and Ernesto care
only about the actions they choose , not about who chooses first. Rosa prefers
t始 outcome in which both she and Emesto choose B to that in which they
both choose H , and prefers this outcome to either of the ones in which she
and Emesto choose different actions; she is 如different between these last two
outcomes. Ernesto's preferences differ from Rosa's in that the roles of B and
H are reversed. Karl's preferences are the same as Emesto' s. Model this situation as an exèensÌve game with perfect information. (Specify the components
of the game and 陀present the game in a diagram.)
Definition 155.1 allows terminal histories to be infinitely long. Thus we can usε
the model of an extensi飞re 伊me to study situations in which the participants do
not consider any particular fixed horizon when making decisions. lf the length
of the longest terminal history is in fact fìni栓， we say that the game has a fin Ït e
horÌzon.
E飞咄ì a game \^l ith a finite horizon may have infinitely many terminal histories ,
because some player has infinitely many actions after somεhistory. If a game has
a finite horizon and finitely many terminal histories wεsay it is finite Note t虹的 a
game that is not 自nite cannot bεrepresented in a diagram like Figure 156 .1, because
such a figure allows for only finitely many branches.
An extensive game with perfect information models a situation in which each
player, whεn choosing an action, knows a11 actions chosen previously (has perfect information) and always moves alone (rather than simultaneously with othεr
players). Somξ 仓conomic and political situa tÍons that the model encompasses are
discussεd in the next chapter. The competition between int时已st groups courting
legislators is one example. 丁his situa tÍon may be modeled as an extensive game
in which the groups sequentially offer payments to induce the legislators to vote
for their favorite version of a bill (Section 6.3). A race (between firms developing
a new technology, or between directors making competing movies , for instanc时，
is anotheτexample. This situation is
‘
Chapter 5. Ext盼到￥'e Games with Perfect Information:τheory
158
5.1 .2 Sofutions
1n the entry game in Figure 156. 1, it seems cJ ear that the challenger will enter and
the inεumbent will subsequently acquiesce. 丁he challengεr can reason that if it
enters then the incumbent will acquiesce, becausεdoing so is better for the incumbent than fighting. Given that the íncumbent will 陀spond to entry in this way,. the
challenger is better off entering.
Th is líne of argument is called backward induction. Whenever a player has to
move, she deduces, for each of her possible actions, the actions that the players
(including herself) 飞vill subsequently rationally take , and chooses the action that
yields the terminal history she most prefers.
While backward induction may be applied to the game 如 Figure 156.1, itεan­
not be applied to every extensive game with perfect information. Consìder, for
example, the variant of this game shown in Figure 158.1 , in which the incumbent' s
payoff to the terminai history (In , Fight) is 1 rather than O. If, in the modified game ,
the challenger enters, the incumbent is indifferent between acquìεscing and fight伽
ing. Backward induction does not tell the challenger what the incumber飞t will do in
thìs case, and thus leaves open the question of whìch action the challenger should
choose. Games with infinitely long histories present another diffic时 ty for backward induction: they have no end from which to start the induction.τhe generalizê飞tion of an extensi陀 game with perfect information that allows for simultan夺
ous moves (studied in Chapter 7) poses yet another problem: when players move
simultaneously'时 cannot in general straightforwardly deduce each player's optimal actior飞. (As in a strategic game, each player's best action depends on the other
players' actions.)
Another approach to defining equilibrium takes off from the notion of Nash
equilibrium. It seeks to model patterns of behavìor that can persìst in a steady
st挝已 τhe resulting notion of equilibrium applíes to a11 extensive games with perfect information. Because the idea of backward induction is more limited, and
the principles behind the notion of Nash equilibrium have been established in
prevìous chapters , 1 begin by discussing the 5teady 5tate approach. In game5
ìn which back飞A'ard inductior飞 is we11-definεd ， this approach turn5 out to lead to
the back飞'\1ard ÌI叶飞.iCtion outcome , 50 that there is no conflict betw时n the two
ideas.
Challenger
。ut
1, 2
2, 1
0, 1
Figure 158.1 A varial飞 t of the entry game of Figure 156.1. The challenger' s payoff is the fírst number ir飞
pair.
e在ch
5.2 5trategies and outcomes
5.2
159
Strategies and outωmes
5.2.1 5trategies
A key concept in th巳 study of extensíve games is that of a strategy. A player's
strategy specifíes the actio扒出εplayer chooses foτ every history after which it is
her turn to moVe.
DEFINITION 159.1 (Strategy) A strategy of player i ìn an extensive game with
perfect information is a function that assigns to each history h after which it is
player i's turn to move (ì.e. P(h) = Î, where P is the player function) an actìon ín
A(的 (the set of actions available after h)
Consider the gamεir飞 the top pal飞el of Figure 159.1.
• Player 1 moves only at the start of the game (i. e. after the empty history) ,
when the actior飞s avaìlable to her are C aηd D. Thus she has two strategi岱:
one that assigns C to the empty history, and one that 附signs D to thεempty
histoτF
• Player 2 moves after both the history C and the hìstory D. After the hìstory C
the actìons a飞划lable to hεr are E and
and after the hìstory D the actìons
available to her are G and H. 丁hus a strategy of player 2 is a function that
assigns either E or F to the history C, and either G or H to the 1飞istory D.
That is, player 2 has four strat笃ies， which are shown in the bottom panel of
Figure 159.1.
1 refer to the strategìes of player 1 ìn thìs game simply as C and D , al飞d to the
strategies of player 2 simply as EG , EH , FG , and FH. For many other fìnite gamεs 1
use a simìlar shorthal飞d: 1 wrìte a player's strategy as a list of actions , one 如r each
E;<
1
f
D
2
H
G
\
2, 1
0, 2
3, 0
Actior飞 assigned
tohistory C
Strategy 1
Strategy 2
Strategy 3
Strategy4
E
E
F
F
1, 3
Action assigned
tohistory D
G
H
G
日
Figure 159.1 An extensive game with pe出ct informatio口 (topp甜甜甜d thefour 舵在tegies of player 2
in thisgame 作ottom panel).
Chapter 5. Extensive Games with Perfect Information: Th eory
160
1
1
C
1, 2
0, 0
Figure 160.1 人n extensÎve game in which player 1 moves both before and after pl击 yer 2.
history after which it is the player's turn to move. 1n general 1 write the actions in
the ordεr in which they occur in the game, and , if they are available at the same
"stage" , from left to right as th町 appear in the diagram of the game. When the
meaning of a list of actions is undear, 1 explici t1 y give the history after which each
action is taken.
Each of player 2's strategìes in the game in the top panel of Figure 159.1 may 汩
汩terpreted as a plan of action or contingency plan: it specìfies what player 2 do部
扩 player 1 chooses C, and what she does 扩 player 1 chooses D. In every game ,
a player's strategy provides sufficìent ínformation to determine her pla l1 of actioll:
the actions she intends to take , whatever the other players do. In particular, if a
player appoints an agent to play the game fo 1' he 1', and tells the agent her strategy,
then the agent has enough information to carry out hεr wishes , whatever actìons
the other players take ‘
In some games some players' strategies are more than plans of action. Consider
the 伊me in Fi gure 160. 1. Player 1 movεs both at the start of the game and after
the history (C , E). In each case she has two actions, so she has four strategies: CG
(i.e. choose C at the start of thεgame and G after the history (C , E)) , CH, DG , and
DH. 1n particula乙 each strategy specifies an action after the history (C , E) εven 扩
11 sF忧功ies the action D at the beginnÎng 吃f the 伊?时， in which ca使 the histo巧 (C， E)
does not occur! 丁he poínt is that Definition 159.1 requires tl飞at a strategy of any
player i specify an action for eve i' y history after which it is pJayer i's turn to move ,
eve l1 for histories th风 if the strategy is followed , do 110t occur
In view of this point and the fact that "strategy" is a synonym for "plan of ac
tion" inεveryday language, you may 陀gard the word "strategy" as inappropriate
for the concept in Definition 159. 1. You a陀 right. You may also wondεr why we
cannot restrict attention to plans of action.
For the purposes of the notion of Nash equilibrium (discussed in the nεxt sectio吟 we could in fact work with plans of action rath盯 than strategies. But, as we
sha11 see, the notion of Nash 叫出librium for anεxtensive game is not satísfactory;
the concept we adopt depends on the players' full strategies. When discussing
this concept (in S时tion 5.4.4) 1 elaborate on the interpr性ation of a strateg予 Atthe
morτl.ent， you may think of a player's strategy as a plan of what to do , whatever
5.3 I咱击sh 垣quilibrium
161
the other players do, both if the player carríes out her intended actions and also if
she rnakes rnistakes. For exarnple, we can iηterpret the strategy DG of player 1 in
the garne in Fígure 160.1 to rnean "1 intend to choose D , but if 1 rnake a rnistake
and choose C instead, then 1 will subsequently choose G". (Because the notion of
Nash equilibriurn depends only on plans of action, 1could delay the definitìon of a
strategy to the start of Section 5.4. 1do not do so because the notion of a strategy is
central to the study of extensive garnεs， and its precise definition is rnuch sirnpler
than that of a plan of actíon.)
? EXERCISE 16 1. 1 (Strategies in extensive garnes) What are the stratεgies of the players in the entry garne 阴阳rnple 155.2)? What are Rosa's strategies in the garne in
Exercise 156.2c?
5.2.2 Outcomes
A strategy profile deterrnines the terrninal history that occurs. Deno也 the strategy
profile by 5 and the player function by P. At the start of the game player P( ø)
rnoves. Her strategy is 5p( 的， and she chooses the actio川 P问 (ø). Denote this a心
tion by al . If the history a1 is not terrninat player P(a 1 ) rnoves next Her strategy
is 5P(a1ì' and she chooses the action SP(a l )(的. Denote this action by a2• If the history (川 ，( 2 ) is not terrninal , then again the player function s伊cifies whose turn it
is to move , and that player's strategy specifies the action she chooses.ηle process
continues until a terminal history is constructed. We refer to this termìnal history
asthe 。前com就 of 5, and denote it O( 5)
ln the garne in Fi gure 160 .1, for example, the outcome of the strategy pair
(DG , E) ís the terminal h就ory D , and the outcome of (CH , E) is the terrninaJ
history (C E, H).
Note that the outcome 0(5) of the strategy profile 5 depends only on the p1ay呻
ers' plans of action, not their full strategi出 τhat is, to determine 0(5) we do not
need to refer to any component of any player's strategy that specifies her actions
after hlstories precluded by that 处rategy.
5.3
Nash equilibrium
As for strategíc games, we are interested in notions of equiJibrium that model the
players' beha可 ior in a steady state. That is, we look for patterns of behavior with
the property that if every player knows every otl飞er player's behavior, she has no
reason to changεher own behavior. 1start by defini到g a r、如 sh equilibriurn: a strat响
心gy profile from which no player 切lÎshes to deviate, given the other players' strategies. The definition is an adaptation of that of a Nash equi1ibrium in a strategic
game (23 .1)‘
DEFINITION 161.2 (Nash equilibrium of extensive game wìth perfect informatio的 The
strategy profile s年 in an extensive game with perfect inforrnation is a Nash equilibrium if, for every player i and every strategy rì of player Î，位le terminal history
ChapterS. 主xtensiv告 Games
162
with Perfect Information:τheory
。 (s 本) generated by ♂ is at least as good according to player i' s preferences as
the t柑柏ml
player í 巾。ses tj while every other player j 巾。ses sj
player i,
问lÍvale均， for each
Ui( O(s*)) 兰州。 (rρ 巳)) for every strategy rj of player i,
where Ui ìs a payoff function that represents player i's preferences and 0 is the
outcome function of the game.
One way to fìnd the Nash equilibria of an extensìve game in which eaεh player
has finìtely many strategies ìs to list each player's strategíes, find the outcome of
each strategy profìle, and analyze this inforrnation as for a strategic garne. That
ìs, we construct the following strategic game, known as thεstrategic fonn of the
extensíve game.
Players The set of players ìn the extensive garne.
Actions
主ach player's set of actions is her set of strategìes in the extensive
garne.
尸references
Each player's payoff to each action profile is her payoff to the
terminal hìstory generated by that 浅ctior飞 profile ín thεextensive garne.
From Definitìon 161.2 we see that
the set of Nash equilìbrìa of any extensì飞re garne with perfect inforrnatìon ìs the set or r、~ash equilìbria of its strategìc forrn.
EXAMPLE 162.1 (Nash equilibria of the entry garne) In the entry garne in Fi段
ure 156.1, the challenger 趴as t飞"l0 strategìes, 111 and Out , and the incumbent has
two strategìes , Acquiesce and Figh t. ηle strategic forrn of the garne is shown in Figure 162.1. We see that it has two r也sh equilibria: (I11, Acquiesce) and (0吼 F告ht).
The first 叫uilìbriurn ís the pattern of behavîor isolated by bàckward ind飞lction，
discussεd at the start of Section 5.1 .2.
ln the second eguìlibrium the challenger always chooses Ou f. This strategy
ís optimal given the incumben t' s strategy to fight in the event of entry. Further,
the incurnben t's strategy F拉加 is optirnal given the challenger's strategy: the εhal­
lenger chooses Out , so whether the incurnber飞t plans to choose Acquiesce or Fig l1 t
rnakes no difference to its payoff. Thus neither player can ìncrease its payoff by
chÇlosing a different strateg予 givcn the other player's strategy.
吁/
』存
figur暗 162.1 丁he
o
EFF
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ohv vi
刀，仗
c'haii en e
L凹ηH 门υ
FL
ρ‘
v
A mT21
Jrr
?gaafhM
JU 们飞 01
KM 乱
hk12
mm b Ij
UE
strategic form of the e只 try game in Figure 1 气 6.1
5.3 挝ash
equilibrium
163
丁hinking about the extensìve game ìn this example raises a question about the
Nash equilibrium (Out , Fight) that does not arìse ìn the strategic form: How does
the challenger know that the ìncumbent will choose Fight if it enters? We interp陀t
the strategic game to model a situation in which, whenever the challenger plays
the game, it observes the incumbent's action, even if ìt chooses Ou t. By contrast ,
we interpret the extεnsive game to model a situation in which a challengεr that
always chooses Out never observes the incumbent's action, because the incumbent
never moves. In a strategic game, the rationale for the Nash equilibrium condition
that each player's strategy be optimal given the other players' strategies is that in
a steady state, each player's experience playíng the 伊 me !eads her belìef about
the other players' actions to be correct. τhis rationale doεs not apply to the Nash
equ i1ibrium (Out , Fïght) of the (extensive) entry game, because a challenger who
always chooses Out never observes the incumbent's action after the history 111.
We can escape from this difficulty in interpreting a Nash equilibrium of an
extensive game by cons:dering a slightly perturbed steady state in which, on rare
occasiöns, nonequílibrium actions are taken (perhaps players make mistakes, or
deliberately experiment) , and the perturbations allow each player eventually to
ob优队'e every other player's action after evenJ history. Given such perturbations,
each player eventually learns the other players' entire strategies.
Interpreting the Nash equilibrium (Out , Fight) as such a perturbed steady state,
however, wεrun into another problern. Or飞 those (rare) occasions when the challenger enters, the subsequent behavior of the incumbent to fight is not a steady
state in the remainder of the game: if the challenger enters , the incumbent is better
off acquiescing than fighting. That is, the Nash equilibrium (Out , Fíght) does not
correspond to a robust steady state of the extensive game.
Note that the extensive gar附 embodies the assumption that the incumbent cannot commit, at the beginning of the game, to fight ìf the challenger enters; it is free
to choose either Acquiesce or Fight in this e飞咄飞t. If the incumbent could commit
to fight in the event of entry, then the analysis would be different. Such a commitment would induce the challenger to stay out, an outcome that the incumbeI飞t
prefers. In t
Chapter5. 主xtensive
164
Games with Perfect Information:τh晤。 ry
S.4 Subgame perfect equilibrium
5.4. 1 Definition
The notion of Nash equílibrium ignores the sequentia1 structure of an extensìve
game; it treats strategies as choices made once and for all before p1ay begins. Con栅
sequently, as we saw in the pre抖。us section, the steady state to which a Nash
吨uilibrium corresponds may not be robus t.
1 now define a notion of equilibrium that mode1s a robust steady state. This
notion requires each player's strategy to be optima l, given the other players'
strategies, not on1y at the start of the gamεbut after 已very possib1e history
To define this concept, 1 first defir飞毛 the notion of a subgame. For any nontermÏnal history 11 , the subgmne following h is tl冶 part of the game tha t rema如s after 11
has occurred. For example , th仓 subgame following the his如ry In in the entry game
(Example 154.1) is the game in which the incumbent is the only playε乙 and there
are two terminal histories, ACQl山SC川ndF也ht
DE Fl NlTl ON 1 创 .1 (Subgame of extensíve game with pe价ct information) Let r be an
以tensive game with perfect information, with player function P. For any nonter-
minaJ history 岛 of
r, the s巾 game r (11) following the history 11 is the 扣followin
似
e
xt沁
ensive game.
Players τhe play巳rs in
r
Terminal histories The set of all sequences h' of actions such that (11 , h') is a
terminal history of r.
Player function The player P(h , 11') is assigned to each proper subhistory h' of a
terminal histo巧，
Pre必rences Each player prefers 11' to h" if and on1y if she prefers (扎的 to (h , h")
in 仁
Note that the subgame following the 臼npty history ø is the entire game. Every
other subgame is called a pl叩ersu告'game. Because there is a subgame for every nontermìnal history, the number of subgames is equal to the number of nonterminal
histories.
As an example, the game in the top panel of Figure 159.1 has 吕立ee nonterminal histories (the empty hístor予C， and D) , and hence three subgames: the whole
game (the part of the game following the empty history) , the game following the
history C, and the game following thεhistory D. 古飞e two proper subgames are
shown in Fígu陀 165. 1.
The game in Figure 160.1 also has three nonterminal histories, and hence three
subgames: the whole game , the game following the hístory C, and the 伊me
following the history (C , E). The two proper subgames are shown in Figure 165 .2‘
⑦ Ex在 RCISE 164.2 (Subgames) Fìnd all the subgamεs of the 伊meiηExercìse 156.2c
1
川
2川川
5.4 Subgame perfect equilibrium
;气
3, 0
2, 1
165
1, 3
0, 2
Figure 165.1 The two propeτsubgamesοf the extens认'e game in the top panelof Figure 159.1
2
G/2
0, 0
1, 2
0, 0
Figure 165.2 The two proper subgames of the extensive game in Figu陀 160.1.
In an equilibriurn that corresponds to a perturbed steady state in which eveηf
history sometìmes occurs, the players' behavior IτlU st correspond to a steady state
in every subgmne , not only in the whole garne. Interpreting the actions specified
by a player's strategy in a subgame to give the player's behavior if, possibly after
a series of rnistakes , that subgarne is reached , this condition is ernbodîεd in the
following informal definition.
A subgame pe币'ct equilíbrium is a strategy profile 扩 with the property
that în no subgan飞e can any player i do better by choosing a strategy
different frorn , given that every other player j adheres to 57
5i
(Cornpare this definition with that of a Nash 叫 uilibrium of a strategic game given
onpage 22.)
For example, the Nash 叫uilíbrium (0吨 Fight) of the entry game (Ex在mple
154 .1) is not a subgame perfect 叫uilibrium because in the sub伊me following the
history 111 , the strategy Fight is 我ot optimal for the incumben t: Íl1 this 5U告'game，
the incumbent is better off choosing Acquiesce than it is choosing Fight. The Nash
equilibrî泣m (111 , Acquiesce) is a subgame perfect equilibriun飞: each player's strategy
is optimal, given the 位her player's strategy, both in the whole garne and in the
S飞lbgame followiI飞g the history 111.
To define the notion of subgame perfect equilibrium precisel予 we need a new
piece of notation. Let h be a history and 5 a strategy profile. Suppose that h occurs (even though it is not necessarily consistent with 5), and afteræard the players
adhere to the strategy profile s. The resulting terminal history (consisting of h foIlowed by the outcome generated in the subgame followìng h by the strategy profile
induced by 5 in the subgame) is denoted Oh (斗. Note that for any strategy profile
5, we have Oø( s) = O( s) (where ø , as alwa)电 denotes the empty history).
As an example, consider again the entry garne. Let 5 be the strate部T profile
(Ou t, Fight) and let h be the histo巧r ln. If h occurs, and afterward the players adhere
怡 S， t挝 resultìng terminal histo可 is Oh(5) 口 (In， F守ht).
Chapter 5. Extensiv带 Gamωwith Perfect Information: Theory
1 晕晕
DEFINITION 166.1 (Subgame pelfect equílibrium of extensive game with 阿拉ct i1份 r­
mation) The strategy profile ♂ maηextensi飞le game with perfect informati。在总 a
subgame perfect equilibrium 迁， for eve巧r player i, every history 11 after which it is
player i's turn to move (i.e. P(h) i) , and every strategy l' i of player Î, tl飞e terminal
history Oh( 叫 genera ted by ♂ after the history h is at least as good according to
player i' s pref，时ences as the terminal history OII(r川 ~í) generated by the strategy
profile (r川 ~i) in which player i chooses rí while every other player j chooses sj
Equi飞lalently， for every player i and every history 11 after which it is player í's turr飞
to move,
μ i(Oh(S*)) 主 Ui(Oh(ri'S~í 日 fo1' every strategy ri of player Î,
where Ui is a payoff function that rep泛使llts player i' s preferences and 0 11 书 ) is thε
terminal history consisting of 11 followed by the sequence of actions gεnerated by
s after h.
The ÌI丑portant point in this defìnitíon is that each player's 没rategy is requir役d
to be optimal for every history after which ìt is the player's turn to 11飞ove， not only
at the start of the game as in the definition of a Nash 叫uilibrium (16 1.2).
5.4.2
Subgame perfect equilibrium and Nash εqui1ibríum
In a subgame perfect equilibrium every player's strategy is optimal, in particul矶
时ter the empty history (put 11 =θin the definition, and remember that 0 0 (5)
O(s)) τhus
every subgame pe飞fect equilíbriu111 is a Nash equilíbriu111
In fact , a subgame perfect equilìbrium generat岱 a 民ash equilibrium ir飞机~
ery subgame: if 5 那 is a subgame perfect equilibrium, then, for any history h and
player i, thεstrategy inducεd by in the St均ame following h is optimal given
the strategies Ìl飞duced by s"'-i in the subgam已 Further， any strategy profile that
generates a Nash equilibrium i只 every subgame is a subgame perfect equilibrium,
so that w心 can give the fo l1owing alternative definition.
si
A subgamc 严r如t eq 1l ilibriu 11I is a strategy pro去le that induces a Nash
吨uilibrium in evεry subgame.
ln a Nash egui1ibrium every player's strategy is optimal, given the other pla沪
ers' strategies, in thεwhole game. As we have seen, it may lWt be optimal in some
subgames. 1 claim, however, that it is optimal in any subgame that is reached
whεn the players follow th的r strategi性. Gi飞!en this claim, the significance of the
requirement in the definition of a subgame perfect 吨uilibrium that each player's
strategy be optìmal after every hístory, relative to the reguirement in thεdefi时，
tion of a 出ash equilibrium, is that ead飞 player's strategy be optimal after histories
that do not occur if the players follow t挝ír strategies (like the hístory 111 when the
challenge内 actíon is Out at the beginning of the entry game).
5.4 Subgame perfect equilibrium
167
10 show my claim, supp05e that 扩 is a Nash equilibrium of a game in which
you are player i. Th阳en
川O
∞飞
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时
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咱 S
When the other players follow their strategies , there comes a poí挝 (possibly
the start of the game) when you have to move for the first tíme. Suppose that at
this poí对 yO Ll follow youτstrategy sj; denote the action you choose by C. r、Jow，
after choosing C, should you change your strategy in the rest of the game, given
that the other player5 will cor飞tinue to adhere to their strategies? 齿。! If you could
do better by changing your strategy after choosing C-say by switching to the
strategy s; in the subgam then you could have 白
d Oη
肥
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t挝
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am
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怆
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if句
you
plan is optima l, given the other players' strategies , at the start of the game, and
you stick to it, then yO Ll nevεr want to changεyour mind after play begi邸， as long
as the other players stick to thεir strategies. (The general principle is known as the
p灯 nciple of opti111alíty in dynamic programming.)
•
5.4.3 Exampies
EXAMPLE 1 盯.1 (伍
E挝f巧
y game) Consider ag
伊
ai如
lη1 白
t he el时
1让t巧 game of Ex
川
an
口
mple 154
令.1 ,，
W
亏认
吨
、V
r
F咆
$
t扎h况
e N;:肘王污时
sh飞 equ
山il且ibrium (01时
It， 引
F i除
旷ht片) 沁
g
i s not a subgarn
只 e perfect eql山
ülibr约ium
引1 follo飞w
咄
from the formal definìtìon as fo11ows. For s*
(Out , Fígh片， í
lncumbent,
rl = Acquies伴， and h
阳， we have O/t( ♂ = (1 n, Fíght) and Oh(rμ51i)
(I11 , Acquiesc时 I so tha t the ineq旧lity in the definition is violated: 叫 0i1(日本))
0
and Ui(Oh(ri ， S 二 1))tl
τhe 到 ashe年tilibrium (1 11, Ac守 uiesce) ís a subgame perfect equilibrium because
伊) it is a Na5h equilibrium , 50 that at the 5tart of the game the cha11enger's strategy
1n is optimal, giveηthe incumben t' s strategy Ac守uzes时， and (的 after the history 111 ,
the incumbent's strategy Acquiesce in the subgame is optímal. In the language of
the formal definitio丑， let s浅口 (111， Acquíesc咛.
• The cha11enger moves after one history, nι:\mely fl 口豆豆 We have Oh
(II1, Acquiesce) , and hencεfor í
challenger we have llí(O/1 (S 咐
2，
whereas for the only other strategy of the cha11enger, fi 工 Out， we have
叫 OJt(rirs=-J)
1.
• The in艺臼
um
incumbent we have 以 011 伊拉升 = 1,
whereωfor the only other strategy of the incumbent , rl 口 Fight， we have
ui(OI1 (ri , s=-i)) = O.
(I扣
7η1 ， Ac叫号 Ules町Cα
叫
E叶)， and hence for i
Evëry subgame perfect equilibrium is a Nash equilibrium, so we conclude that
the game has a unique subgame perfect equilibrium, (1J1, Ac伊iesce ).
EXAMPLE 167 .2 (Variant of entry game) Consider the variant of the entry game in
which the incumbent is indifferent between fighting and acquiescing if t抬 al­
lenger enters (s时 Fi凯ue 158.1). This game, like the original game , has two Nash
168
Chapter 5. Extensive Games with Perfect Information: Theory
equilibri矶 (In， Acquiesce) and (0吼 F'ight). B时∞w both of these equilibria are sub-
game perfect equilibria , because after the history 111 , both Fight and Acquiesce are
optimal for the incumbent.
In particu1a毛 the game has a steady state in which every challenger a1ways
chooses In and every incumbent always chooses Acquiesce. lf you, as the cha1lenge乙 were playing thεgame for the first time, you would probab1y regard the
action 111 as "risky" because after the history 111 the incumbent is indifferent bet毛<veenA句uiesce and Fight, and you prefer the terminal history Out to the termina1 history (扣， Fight). Indeed , as discussed in Section 5. 1.~立俨
does 贝
r1旧
∞
时t yield a c1e
O
俯
ar solution of 白
t hi总s 苔伊
ame
已. But the subgame p εrfect equilibri um
(扣 ， Acquiesce) corresponds to a per坠ctly reasonable steady sta但If you had p1ayed
the game hundreds of times against opponents drawn from the same population ,
and on every occasion your opponεnt had chosen Acquiesce: you cou1d reasonably
expect your next opponent to choose Acquiesce, and thus optimally choose 111.
⑦ EXE 反CISE 168.1 (Checking for sub伊 me perfect equilibria) Which of the Nash
叫uilibria of the game in Figure 160.1 are subgam泛 perfect?
5.4.4 Interpretation
A Nash equilibrium of a strategic game corresponds to a steady state in an idealized setting in which the participants in each play of the game are drawn ran
dom1y from a collection of populations (see Section 2.6). The idea is that each
player's long experience playing the game 1eads her to correct beHefs about the
other players' actions; given these beliefs , her equili如 ium action is optima l.
A subgame perfect equilibrium of an extensive game corresponds to a slightly
perturbed steady state in which all players , on rare occasions, take no口号quilìb­
rium actions , so th挝 after long exp臼ience each player forms correct beliefs about
the other players' entÌrεstrategies and thus knows how the other p1ayers will behave in every subgame. Given these beliefs , no player wishes to deviate from her
strategy either at the start of the game or after any history.
This interpretatíor飞 of a subgame peτfect 吨uilibriur口， like the interpretation
of a Nash equilibrium as a steady state, does not re气uire a player to l<now the
other players' pref役rences， or to think about the oth巳r pJayers' rationality. It en..
tails interpreting a strategy as a plan specifying a play肝 's actions not only after
histories consistent with the strategy, but a1so after histories that result when the
player chooses arbitrary alternative actíons, perhaps because she makes mistakes
or delibεratelyexperiments.
τhe subgame perfect equilìbria of some extensive gam々 can be gi飞ren other interpretations. In some cases , one alternative interpretation is particularly attractîve. Consider an extensive game with perfect information in which each player
hasa 田ùque best action at every history after which it ìs her turn to move, and the
horizon is fir飞it巳 In such a game , a player who knows the other players' preferences and knows that the other players are rational may use backward induction
5.5 Finding subgame perfect equilibria of finite horizon games: backward inductíon
岳♀
to deduce her optimal strategy, as discussed in Section 5. 1. 2. Thus we can ìnterpret
a subgame perfect equilibrium as the outcome of the player毡， rational calcul挝ions
about each oth旧 's strat巳gies
This ìnterpretation of a subgame perfect equilibrium entails an interpretation of
a strategy different from the one that fits the steady state interpretation. Consider,
for example, the game in Figure 160.1. When analyzing this game , playe川 must
consider the consequences of choosing C. Thus she must think about player 2's
action after the histo巧 and her飞cem飞1St form a belief about what player 2 thinks
she (player 1) will do after the history (c, E). τhe component of h缸 strategy that
specifi岱 h号r action after this history reflects this belie f. For instance , the strategy
DG means that play附 1 chooses D at the start of the game and believes that were
she to choose C, player 2 wo议ld believe that after the history (c, E) she would
choo能 G. In an arbitrary game , the interpretation of a subgame perfect equilibrium as the outcome of the players' ratìonal calculations about each oth衍 's strategies entails interpreting theεomponents of a player's strategy that assign actions
to histories inconsistent with other parts of the strategy as specifying the play时 's
be !ief about the other players' beliefs about what the player will do if one of these
histories occurs.
在lis interpretation of a subgamεperfect eq山librium is not free of dif在cultíes，
which are discussed in Section 7.7. Further, the interpretation is not tenable in
games in which some player has more than one optimal action after some histor丑
。r in the more general extensive games considered in Section 7.1 and Chapter 10.
Nevertheless, in some of the 伊mes studied in thìs chapter and the next it is an
appealing alternative to the steady state interpretation. Furthε乙 an extensior飞 ofthe
procedure ofbackward induction can be used to find a11 su1:予gamep时fect equilibria
of finíte horizon games, as we sha l1 see in the next 忧ction. (This extension cannot
be given an appealing 挝havioral interpretation in games 出 which some play泛r
has more than one optimal actíon after some history.)
5.5
Finding subgame perfect equilibria of finite horizon games:
backward induction
We four飞d the subgame perfect equilìbria of the games in Examples 167.1 and 167.2
by fi口ding the Nash equilibrìa of the games and checki口g whether each of these
e电uilibria is subgame perfec t. 1n a game with a finite horizon the set of subgame perfect 叫uilibria may be found more directly by using an extension of the
procedure of backward ir飞duction discussed briefly in Section 5. 1.2.
Define the length of a subgame to be the length of the longest history in the subgame. (在le lengths of the subgames in Figure 165.2, for example, are 2 and 1.)
The procedure of backward induction works as fo11ows. We start by finding the
optimal actions of the players who move in the subgames of length 1 (the "last"
subgames). Then, taking these actions as given, we 位ld the op泣mal actions of the
players who move first in the subgames of length 2. We continue working back to
Chapter5. E茸tensiveGam费s with Perfect Information: 有始。ry
170
C
D
F
G
2
E
2, 1
3，。
0, 2
飞
2
1
1, 3
Figure 17队 1 A game illustrati只 g the procedure of backward induction.τhe actions selected 扣y back阳
ward induction are indicated in black.
the begim1ing of the game , at each stage k finding the optimal actions of the play棚
ers who move at the start of the subgames of length k, given the optimal actions
wεhave found in all shorter subgames.
At each stage k of this procedure , the optimal actions of the pl可erswho move
at the start of the subgames of length k are easy to determine: they are simply the
actions that yield the players the highest payoffs, given the optimal actions in all
shorter subgames.
Consider, for example, the game in Figure 170. 1.
• First consider subgames of length 1. 在le game has two such subgames ,
in both of which player 2 moves. In the subgame follmνing the histo巧r C,
player 2's optimal action is E, and in the subgamεfollowing the history D ,
her optimal actìon is H.
• Now consider subgames of length 2. The game has one such subgame ,
namely the entire game, at the start of which player 1 movεs. Given the optimal actions in the subgames of length 1, player l's choosing C at the start of
the game yields her a payoff of 2, whereas her choosing D yields her a payoff
of 1. Thus player l's optimal action at the start of the game is
The game has no subgame of length greater than 2, so the procedure of backward
inductio问明lds the strategy pair (c, EH).
As another example, consider again the game in Figure 160..1. We first deduce
that in the subgame of length 1 following the history (C , E) , player 1 chooses G;
then that at thξstart of the subgame of length 2 following the history C, player 2
chooses E; then that at the start of the whole game , player 1 chooses D. Thus the
procedure of backward induction in this game yields the strategy pair (DG, E).
In any game in which this procedure selects a single action for the player who
moves at the start of each subgame, the strategy pr。在le thus selected is the unique
subgame perfect equilibrium of the game. (You should find this result very plausible , though a complete proof is not trîvial.)
What happens in a game in which at the start of some subgames more than
one action is optima l? In such a game an 仪tension of the procedure of backward
induction locates all subgame perfect equilibria. This extension traces back separately t挝 implications for behavior in the longer subgames of every combination of
optimal actions in the shorter subgames.
5.5 事沁dingsubgam暗 perfect
equilibria of ñnit费 horìzor在 games: back四ard inductìon
171
Consider, for example, the game in Figure 172. 1.
·τhe game has thrεe subgames of length 1, in each of which player 2 moves. 1n
the subgames fo11owing the histories C and 0 , player 2 is indifferent betweer飞
her two actions. ln the subgame fo11owing the history E, player 2's unique
optimal action is K. Thus there are 卢 1I r combinations of player 2's optimal
actions in the subgames of length 1: FH K, FI K, (二日瓦， and GIK (where the first
component in each case is player 2's actior飞 aft已r the histo巧1 C, the second
component is her action after the history 0 , and the third component is her
action after the history E).
·刊le game has a single subgame of length 2, namely the whole game, in which
player 1 moves firs t. We now consider player l's optimal actioηin this game
for every combination of the optimal actions of player 2 in the subgames of
length 1川
o For the combinations FHK and FIK of optimal actions of player 2,
playerγs optimal action at the start of thεgame is C
o For the combination GHK of optimal actions of player 2, the actions C,
0 , and E are a11 optimal for player 1.
o Forthe ωmbinationζIK of optim<:.l actions of play附 2， player l's optimal action at the start of the game is O.
τh时 the strategy pairs isolated by the procedure are (c, FHK) , (C, FIKL (C， ι付K) ，
(0 , GHI<) , (E , GHK) , and (O , GIK).
Thεprocedure， which for simplicity 1 refer to simply as backward induction,
may be described compactly for an arbitrary game as fo11ows.
• Find, for each subgame of length 1, the set of optimal actions of the player
who moves fi以 Index the s巾games by j, and denotεby 5j (1) the set of
optimal actions in subgamε j. (If the player who moves first in subgame j
has a unique optimal action，出en 5j (1) co始 ins a single action.)
• For each combination of actions consisting of one from each set 5气巧， find ,
for each subgame of length 2, the set of optimal actions of the player who
moves firs t. The result is a set of strategy profiles for each subgame of
length 2. Denote by 但) the s的。f strategy profiles 阶 subgameι
• Cor飞tinue by examining successively long时 subgames until you reach the
start of the game. At each stage k, for εach combination of strategy profiles
consisti吨 of one from each set 5; 作气 1) const古ucted in the previous stage,
find, for each subgame of length k, the set of op到mal actions of the player
who moves first, and hence a set of strategy profiles for each subgame of
length k.
The set of stra tegy pro副部位at this procedure yields for the whole game is the
set of subgame perfect equi主bría of the game.
Chapt岳rS. 在xtensive Games with Perfect Information:τheory
172
K
3, 0
1, 0
1, 1
2W\
2/L/2
1, 1
2, 1
2,2
1, 3
2W\
3, 0
\d
1, 0
1, 1
2, 1
2, 2
ZW\
2 ___共-----'2
U\
\JV
3, 0
1, 0
1, 1
2, 1
2, 2
γ----Z
3, 0
\f
H/'
1, 0
1, 1
1, 3 3, 0 1 , 0
2
飞
2, 1
2,2
1, 3 3, 0
1, 1
M\
1, 0
2, 1
2, 2
2, 1
1, 3
\J
\d
1,1
1, 3
2uy
1
升V
1, 0
1, 3
2/L/-2
y
γV
3, 0
2, 2
2, 1
2, 2
1, 3
Figur告 172 ， 1 人 game in which the first-mover in some subgames has multiple optimal açtions. The
top diagram sho飞刊 the full game. 了he six small diagrams iIl ustrate the six su七game perfect equilibria;
in each case , the actions speci 白 ed by the equilibrium strategies 在 re indicated by blaξk lines, and the
rema江ling actìons are indìcatcd by gray Ii nes.
• PROPOSIτrON 172.1 (Subgame perfect equilibrium of finite horizon games and
backward induction) The set 01 subga 附 pe~如t e号 uíl加ia 01 a 卢nite horizon extensive game wit l1 pe拉ctm其mnatiol1 is equal to the set 01 s幻挝egy pro.卢 les isolated by the
procedure 01 backωard inductiol1 啕
You should find this result, like my claim for games in which th~ player who moves
at the start of every subgame has a single optimal action, very plausible，校lOugh
again a complete proof is not trivial费
1n the tεrminology of my d巳scription of the general procedure , the analy也s for
the game in Figure 172 .1 is as follows. Number the subgames of length 1 from left
to right. Then we have 5}(1) =枉， G} , 52 (1) = {H , I} and S3( 斗士{互}. The陀
are four lists of actions consisting of one action from each se t: FH K, FIK, GH K,
and GIK. For FHK and FIK , the action C of player 1 is optimal at the start of the
game; for GHK the actions C, D , and E arεall optimal; and forιIK the action D
必 optimal. Thus the set 5 节) of strategy profilεs consists of (C, FHK) , (C , FIK) ,
(C , GHK) , (D ， GH岛 ， (E , GHK) , and (D， ιIK) ， There are no 10吨er subgarnes , so
this set of strategy pr。我les is the set of subgarn巳 perfect equilibria of thεgame.
Each example 1 have prεsented so far in this section is a finite game-that is ,
a game that has n时 only a finite horizon but also a 豆nite number of terminal hisF
5.5 Finding subgame perfect equilibria of fmite horizon games: backward induction
173
tories. ln such a game, the player who moves first i口 any subgame has finitely
many actions; at least one action is optìma l. Thus in such a game the procedure of
backward induction isolates at least one strategy profile. Using Proposition 172.1 ,
we conclude that every fiηite game has a subgame perfect equilibrium.
戴 PROPOSITION 173.1 括对stence of subgame perfect equilìbriurr飞)卢 nite αten­
sivc g'β me with perfect ín声rmation has a subgame pCl女ct equilíbrium.
Note that this result does not claim that a 且nite extensive 伊me has a single
subgame perfect equilibrium (As we have seen, the game in Fìgure 172.1 , for
example, has more than one subgame perfect equilibrium.)
A finite horizon game in which some player does n0t have 自nitely many actions after some history may or may not possess a subgame perfect 叫uilibrium.
A simple example of a game that does not have a subgame perfect 叫uilibrium is
the trivial game in which a single player chooses a number less tl1 atl 1 and receives
a payoff equal to the 1飞umber she chooses.τhεre is no greatest number less than
on飞 sothεsingle player has no optimal actiOI飞， and thus the game has no subgame
perfect equilibrium.
‘
? EXERCISE 173.2 (Finding subgame perfect e弓uilib公司Find the subgame perfect
equilibria of the games in parts a and C of Exercisε156.2，夜nd in Figure 173.1.
1
λ
D
C
2
2
E
2, 1
H
F
t1
2, 0
1, 0
Figure 173.1 One of the games for Ex告rcise 173.2
? EXERCISE 173 .3 (Voting by altemating veto) Find the subgame perfect e气也libria of
the game in Exercìse 163.2. Does the game have any Nash equìlibrium that is not
a subgame perfect 吨uilibrium? Is any outcome generated by a Nash equilibrium
not generated by any subgame perfect equilibri飞lITI? Consider variants of the game
in which player 2's preferences may differ from those specified in Exercise 163.2.
Are there any preferences for which the outcome in a subgame perfect equi1ibrium
of the game in which player 1 moves first differs from the outcome in a subgame
p在fecte电时librium of the game in which player 2 moves first?
Army 1, of country L must decide whether to
attack army 2, of country 2, which is occupying an island between the two coun相
tries. In the event of an attack, army 2 may figl飞t， or retreat over a bridge to its
mainland. Each army prefers to occupy the island than not to occupy it; a fight is
the worst outcome for both armies Model this situation as an extensive game with
这 EXERCISE 173 .4 (Burning a bridge)
‘
174
tensive Game军 with Perfect Informati怠n: Theory
Chap在er 5. Ex
perfect ìnfor丁nation and show that army 2 can ìncrease its subgame perfect equilibrium payoff (and reduce army l's payoff) by burning the bridge to its mainland
(assume this act enta i1 s no cost) , eliminating its optio口 to 陀treat ìf attacked.
( EXERCISE 174.1 (Sharir飞g heterogeneous objects) A gro叩 of n people have to share
k objects that they value differently. Each person assigns values to the objects; no
one assigns the same value to two different objects. Each perso口 evaluatεs a set
of objects according to the sum of the valu部 she assìgns to the objects in the set.
The following procedure is used to share the objεcts. 卫le players are ordered 1
through n. Person 1 chooses an object，在en person 2 does so, and so on; ìf k > 刀，
then after pεrso贝 n chooses a双 object， person 1 chooses a second object, then person 2 chooses a second 0问ect， and so on. Objects are chosen l川i1 none remain
(Canadian and U.S. professional sports teams use a similar procedure to choose
I飞ew players.) Denote by G 识，均 the extensi飞!e gam心 that models this procedure.
If k 三 n， then obvìously G 归， k) has a subgame perfect e电ui1ibrium in whichεach
player's strategy is to choose her favorite object among those remaining when her
turn comes. Show that if k > n, then G(n , k) may have 矶。 subgame perfect equi呻
librium in which person 1 chooses her favorìte 0问 ect on the first round. (You can
give an example in whïch 11
2 and k = 3.) Now fìx n = 2. De如le xk to be thε
object least prefεrred by the person who does not choose at stage k (i. e. who does
not choose the last object); define xk-l to be the object, among all those except xk'
least preferred by the person who does not choose at stage k - 1. Sìmilarl如 for any
j with 2 ~二 j 三 k， given Xj" . ., xk , define Xj一 1 to bεthe object, among alì those
exdudì吨 {Xj ， . . ., Xk} , least preferred by the person who does not choose a t stage
j …1. Show that the 伊me G(2 ， 到 has a subgame per检t equilibrium in which for
every j
1, . . . , k the object X i is chosen 拭 stage j. (Th is resuÏì ìs true for G(2 , k)
for a11 values of k.) If 11 ;:::: 3, then interestingly a person may be better off in a11
subgame perfect equilibria of G 忡 ， k) when she comes later in the orderir飞gofplay­
ers. (An example , howeve乙 is difficult to construct; one is given in 日rams and
Straffin 1979.)
The next εxercìse shows how backward inductìon can causë a relatively minor
change in the way in which a game ends to rever州检rate to the start of the game ,
leadiηg to a very different action for the first-mover二
言 EXERCISE 174.2 (An entry game with 浅 financially constrained fìrm) Aηincumbent
in an industry faces thεpossibility of entry by a challenger. First the challenger
chooses whether to enter. If it does not enter, neither firm has any further action;
the incumbent' sμyoff is T M (it obtains the profit M inεach of the following T ~三 l
periods) and the challenger's payoff is O. If the challe口gerεnters， it pays the entry
cost f > 0, and in each of T periods the incumbent fìrst commits t。在ght or cooper唰
ate wìth the challeng衍 in that períod , then the challenger chooses whether to stay
ín the índustry or to exí t. (Note that the ord衍 of the firms' moves within a period
differs from that in the game in Example 154.l.) If, in any period, the challenger
stays 埠， each firm obtains i贝 that period the profit - F < 0 íf the incumbent fights
5.5 Finding subgame perfect equilibri矗 of finite horizon game嚣: backward induction
175
and C > max{ F, f} if ìt coope川es. If, in a町 period， the challenger exíts, both
firms obtain the profit zero in that period (陀gardless of the incumbent's action);
the incumbent obtains thεpro甜 M > 2C and the challenger the profit 0 in ev叫f
sllbsequent period. Once the challenger exits, it cannot subse弓uently reenter二 Each
firm cares about the sum of its profits.
。.
Find the subgame perfect equilíbria of the extensÌve game that models this
sltuatlOn.
b. Considel' a variant of the sitllation in which the challenger is constrained by
巾 financial war ches t, which allows it to survive at most T - 2 fights. Specifω
ically, consider the game that differs from the one in part a in one respec t: the
histories in which (i) at the start of the game the challenger enters and (i ì) the
incumbent fight3 in 了一 1 periods are terminal histories (the challenger has to
exit). For the terminal history 如 whlch the incumbent fights in the first T … l
periods, the incumbe时， s payoff is M … (T - 2)F and the challe吨时 's payoff
is-f-( 了一 2)F (in períod T … 1 the incumben t' s pay。在 is 0, and in the last
period its payoff is M). For the terminal history in which the incumbent co叩
operates in one of the first T … 1 periods and 主ghts in the remainder of these
pe∞ds and in the last period , the incumben t' s payoff is C 一 (T 斗 )F and the
challenger's payoff is
十 C …(了一 2)F. Find the subgame perfect 吨位 librìa
of this game.
EXAMPLE 175 .1 (Dollar auction) Consider an auction in which a到 object is sold to
the highest bidder, but both the highest bidder and the second-highest bidder pay
their bids to the auctioneer. When such an auction is conducted and the object is
a dollar, the outcome is sometimes that the object is sold at a price greater than a
dolla卫 (Shubik writes: A total of payments between three and five dollars is not
uncommon" (197 1, 110).) Obviously 5uch an outcome is inconsistent with a subgame perfect 吨uilibrium of an extensive game that models the auction: every par翩
ticipant has the option of not bidding, so that in no subgame perfe时叫uilibrium
can aηyone's payoff be negative.
Why, then , do such outcomεs occur? Suppose that ther泛 are two participants,
and both start bidding. If the player making the lower bid thinks th拭 making a bid
abovε 岳飞e other player's bid will induce the other player to 午1泣， she may be better
off doing 50 than stopp如g bidding. For example , if the bid5 a肥 currently $0 .50
and $0 .5 1, the player bidd出g $0 .50 is better off bidding $0.52 扩 doing 50 induces
the othe
ff
176
Chapter5. E茧tensiveGar辑部 with Perfect Information: lbeory
that "In playing this game, a large crowd is desirable ... the best time is during a
party when spirìts are high and the propensity to calculate does not settle in until
at Jeast two bids have been made" (1971 , 109).)
1n the next exεrcise you are asked to find the subgame perfect 叫uilibria of an
extensive game that models a simple example of such an auction.
:1 EXERCISE 176.1 (Dollar auction) An object that two people each value at v , a positive integer, is sold ìn an auction. In the auction, the people takεtums bidd Lrlg; a
bid must be a positive integer greater than thεprevious bid. (In the situation that
gives the game its name, V is 100 cents.) On her turn, a player may pass rather
than bid , in whích case the game ends and the other player receives the object; both
players pay theìr last bids (if any). (If player 1 passes initially, for example , player 2
receives 出e objεct and makes no payment; if player 1 bìds 1, player 2 bids 3, and
then player 1 passes, player 2 obtains the object and pays 3, and player 1 pays 1.)
Each person's wealth is w , which exceeds v; neither player may bid morεthan her
wealth. For v 2 and w = 3, model the auction as an exte找到ve game and find its
subgame perfect equilibria. (A much more ambitious project is to find a11 subgame
perfect e弓uilibria for arbitrary valuεs of v and w.)
ln a11 the extensive games studied so far 如 this chapte乙 each player has available finitely many actions whenever she moves.τhe next examples show how the
procedu陀 of backward induction may be used to find the subgame perfect equilibria of games in which a continuum of actions is available after some hístories.
EXAMPLE 176.2 (A synergistic relationship) Consider a variant of the situation in
Example 39. 1, in which two individuals are involved in a synergistic relationship.
Suppose that the players choose their effort levels sequentially, rather than simultaneously. First individua11 chooses her effort level al , thεn individual 2 chooses
her 社fort 1εvel a2. An effort level is a nonnegative numb肘; and individual i's pr性
erences (for i = 1, 2) are represented by the payoff function ai(c 十 ai - Gi) , where j
is the other individual and c > 0 is 在 constar飞t
To find the subgame perfect equilibria , we first consider the subgames of length
1, in which indÍ\划 ua12 chooses a value of a2. Individual2's optimal action after
the hístory Gl is her best response to al which we found to be i(c 十 a1) in Exa附
ple 39. 1. Thus indi飞ridual 2's stratεgy in any subgame perfect equilibrium is the
function that associates with each history al the action (c 忖d
峙。飞tV consider individual 1's action at the start of the game. Given individual 2's st时egy， individual l' s payoff if she chooses al is al (c 十 !(c 十 al) … a升，
or !al (3c - a1). This f飞mction is a quad川ìc that is zero when al
0 and when
al 3c, and reaches a maximum in between. Thus individua11's optimal action
atthe 伽~t of the game is 句
~c
We conc1 ude that the game has a unique subgame perfect 吨uilibrium， in w hich
individua11's 阳tegy is a1
~c and individua12's 阳t吻 is the function that
associates with each histoηr al the action ~ (c 忖 IJ. 丁heo忧。me of the equilibrium
is that indivíduall chooses al
~c and ir咄vidua12 巾。使sa223c
l
i
5.5 Finding subgame perfect 岳quilibri晶。f finite horizon game革: backward induction
177
字 EXERCl SE 177.1 (Fi rm-union bargaining)
A firm's output is L(100 - L) wheηit
uses L 三 50 units of labor, and 2500 when it uses L > 50 units of labor. The
price of output is 1. A union that represents workers presents a wage demand (a
nonnegative number w) , which the firm either accepts or rejects. If the 主rm accepts
the demand , it chooses the numb前 L of workers to employ (which you should take
to be a continuous variable, not an integer); if it rejects the demand, no production
takes place 杠，口。).如飞e 位m's preferences are represented by its profit; the union' s
preferences are representεd by the value of w L.
成
Formulate this situation as an extensÎve game with perfect information.
b. Fi nd the subgame perfect 问uilibrium (equilibria?) of the game.
C.
Is there an outcome of the game that both parties prefer to any subgame
perfect 叫uilibrium outcome?
d. Find a Nash 吨uìlibrium for which the outcome differs from any subgame
perfect equilibrium outcome.
? EXERC1SE 177.2 (The "rotten kid theorem") A child's action a (a number) affects
both her own private income c(a) and her parents' income p(a); for all values of
a we have c 权) < p( a). The child is selfish: she cares only about the amount of
money she has. Her loving parents care both about how much money they have
and how much their chìld has. Specifically, model the parents as a single individual
whose preferences are represented by a payoff 叫ual to the smaller of the amount
of money they have and the amount of money the child has. The parents may
transfer money to the child. First th较 child takes an action, then the parents decide
how much money to transfer. Model this situation as an extensive game and show
that in a subgame perfect equilibrium the child takes an action that maximizes
the sum of hεr private income and her paren怜'income. (Jn particul矶 the child's
action does not maximize her own private income. The result is not limited to the
specific form of the parents' preferences, but holds for any preferences wÎth t阳
property that the amount of money the child receives in the optimal allocation of
a fixed amount x on money between the parents and the child increasεs with x.)
ø EXERCISE 177.3 (Comparing simultaneous and sequential games) The set of ac 町
tions avaílable to player 1 is A 1; the set available to player 2 is A 2 舍 Player l's pref慨
erences over pairs (al , a2) are represented by the payoff Ul (al , a2) , and player 2's
preferences are repr台sented by the payoff U2 (町，向). Compare the Nash equìlibria
(in pure strategies) of the strategic game in which thεplayers choose actions simultaneously with the subgame perfect 叫uilibria of the extensi飞re game in which
player 1 chooses an action, then player 2 does so. (For each history al in the
extensive game, the set of actions available to player 2 is A 2 .)
a.
Show that if, for evεry value of al , a unique member of A2 maximizes
召2(肉， αz)， then in every subgame pεrfect equilibrium of the extensive 伊me，
player l's payoff is at least equal to her highest payoff in any Nash equi1ibrium of the strategic gam已
Chapter 5. Extensive G毒 mes with Perfect Information: Th唱。 ry
178
b. Show that player 2/s payoff in every subgame perfect equ挂ibrium of the extεnsive game may be higher than her highest payoff in any Nash equîlîbrium
of the strategic game.
c. Show that if for some values of 41 more than one memb时 of A2 maxìmizes
U2( 肉 ， 42)1 then the extensive game may ha飞呛 a subgame perfect equilibrium
than her payoff in a11 Nash equilibria of the
in which player l's payoff is
strateg 1C game.
(For parts b and c you can give exa皿ples in which both .1 1 and A 2 εontain two
actior毡. Se e Example 320.2 for further discussion of the implicati∞ of the order of
play.)
TICKTACKTOE , CHESS AND RELATED GAMES
1
TicktacktoE' chess , and 陀latεd games may be modeled as extensive games wìth
perfect information. (A history is a se气uence of moves and each player prefers
to win than to tie than to lose.) Both ticktacktoe and chess may be modeled as
finite games, so by Proposition 173.1 each gamεhas a subgame perfectequilibrium
(The official rules of chess a110w indefìnitely long sequencεs of moves, but the
game seems to be well modeled by an extensive game in which a draw is declarεd
automatically if a position is repeated three times , ra在 t血
hε
衍r than a player ha
飞冶浅肌盯飞
vin飞gt白
he
optìon ofde配c1a盯r拉g a drmN in 出
t hiβsc
怆
as优
已， 滔
e
a s in the of仔ficí臼
aln吐
1丘le
臼叫圄.)
s. The 川
s ubg
伊
amεpe
盯rf伦
肘ctt
e
equílibria of tìcktacktoe are of course kno兀wγTf飞飞， whereas those of chess are not (yet).
Ticktacktoe and chess are strictly competitive games (Definition 365.2): iη
every outcome , either one player loses and the other w Ì!飞鸟 or the players draw.
A 陀sult in a later chapter implies that for such a gamεa11 Nash equilibri在 yield
the same outcome (Corollary 369. 斗. Further a player's Nash equilibI如m strategy
yields at least her 叫uilibrium payoft regardless of the oth臼 players' strategies
(Proposition 368.1a). (The same is definitelyηot true fo 1' an arbitrary game that is
not strictly competitive: look , for example, at the game in Figure 31.1.) Because any
subgame perfect 吨uilibrium is a Nash equilibrium the same is true for subgame
perfect equilibrium strategies.
W与 conc1ude that in ticktacktoe and chess , either (a) one of the players has a
strategy that guarantees shεwins， or (11) each player has a strategy that guarantees
at worst a draw.
ln ticktacktoe, of course we know that (b) is true. Chess is more subtle. 111
particula巳 it is not known whether White has a strategy that guarantees it wins
or Bl ack has a strategy that guarantees it wins, or each player has a strategy that
guarantees at worst a draw. 丁he empirical evidence suggests that Bl ack does not
have a winning strate白r， but this result has not been proved. When will a subgame perfect equilíbrium of chess be found? (The ar飞 swer "never underestimates
human ingenuity!)
1
ll
II
l
l
l
l
ll
Notes
179
? EXERCISE 179.1 (Subgame perfect equilibrìa of ticktackto的 Ticktacktoe has subgame perfect equilibrìa in which the first player puts her first X in a com但. The
second player's mo飞re is the same in all these equilibria. What is it?
? Ex 巨 RCIS 充 179.2 (Toetacktick) Toetacktìck ìs a variant of ticktacktoe ìn which a
player who puts three marks in a line loses (rather than wins). Find a strategy
of the first翻mover that guarantees that she does not Jose. (Jf fact, in a11 subgame
perfect equìlibria the game is a draw.)
①在 XE 民CISE 179.3 (Three Men's Morris , or Mill) The ancient game of Threε 过en's
Morris is played on a ticktacktoe board. Each pl町er has three counters. The players move alternatel y. On each of her first three turns , a player places a counter
0日 an unocεupied square. On each subsequent move , a player may move one of
her counters to an adjacent square (vertica11y or horizontally, but not diagonally).
The first player whose counters are in a row (vertica11y, horizonta11)斗。r diagona11y)
wÌns. Fi nd a subgame perfect equilibrium strategy of player 1, and the equîlibrium
outcome
Notes
The notion of an extensi飞产e game Ìs due to von Neumann and Morgenstern (1944).
Kuhn (1950b , 1953) suggested the formulation described i到 this chapter.τhe description of an extεnsive game in terms of historiεs was suggested by Ariel Rubinstein. The notion of subgame pεrfect equilibrium is due to Selten (1965). Proposition 173.1 is due to Kuhr飞 (1953). The interpretation of a stratεgy when a subgame perfect equilibrium is interpreted as the outcome of the players' reasoning about each other's rational actions ìs due to Rubinstein (1991). T1飞e principle of optimality in d严lamic programming is discussed 问1 Be11man (1957, 83) , for
example.
The procedure ìn Exercises 163.2 and 173 .3 was first studied by Mueller (1978)
and Moulin (1981). The idea in Exercise 173 .4 goes back at least to Sun-tzu, who ,
in
art of Iva小 re (probably written between 500 and 300 B. C.), advises "in surrounding the enem予 leav心 him a way out; do not press an enemy that isξor­
nered" (end of Chapt自 7; see, for example, Sun仅u (1993 , 132月‘ (That is, if no
bridge exists in the situation describ时 in the exercise , army 1 should build one.)
Schelling (1966, 45) quotes Sun也u and gives exampl销。f the strategy's being
used in ar飞tiquity. My formulation of the exerCÌse comes from Tirole (1988 , 316).
The model in ExerCÌse 174.1 is studied by Kohler and Chandrasekaran (197 1) and
Brams and Straffin (1979).η\e game in Exercise 174.2 is based on Benoît (1984 ,
Section 日 .The dollar auction(Exercise176.1)was iIItroduced into the literatureby
Shubik (1971). Some of its subgame perfect equilibria , for arbitrary values of v and
w, are studied by O'Neill (1986) and Leininger (1989); see a1so 丁'aylor (1995 , Chap-
180
Chapter 5. Extensive Games with Perfect Information: Theory
ters 1 and 6). Poundstone (1992 , 257-272) writes informally about the game and its
possib1e app1ications. The resu1t in Exercise 177.2 is due to Becker (1974); see a1so
Bergstrom (1989). The first forma1 study of chess is Zerme10 (1913); see Schwa1be
and Wa1ker (2000) for a discussion of this paper and re1ated wor k. Exercises 179.1 ,
179.2, and 179.3 are taken from Gardner (1959, Chapter 4), which includes severa1
other intriguing examp1es
主xtensive Games with Perfect Information:
lII ustrations
6.1
丁he ultimatum game, the holdup 9在 me， and agend在
6.2
6.3
6.4
control 181
Stackelberg's model of duopoly
Buying votes 192
A race 197
Prerequisite: Chapter 5
187
HE first three 照εtions of this chapter illustratεthe notion of subgame p臼fect
equilibrium in
in which the longεst history 仨 as length two or tl飞reεThe last section studies a game with an arbitrary finite horizo孔 Some games with
infinite horizons are studied in Chapters 14, 15, and 16.
6.1
The ultimatum game, the holdup game, and agenda control
6.1.1
The ultimatum game
Bargaining over the division of a pie may naturally be modeled as an extensive
game. Herε1 analyze a very simple gamεthat is the basi5 of a richer model studied
game is 50 simple, in fact , that you may 口 ot inìtially think of it
in Chapt创始‘
as a model of "bargaining气
Two people use the following procedure to split 和. Person 1 offεrs pεr50日 2
an amount of money up to $c. If 2 accepts this off时~ then 1 rec台ives the r!εmainder
of the $c. If 2 rejects the offer, thm neither person recei飞!es any payoff. Each pε卜
son cares only about the amount of money she receives , and (naturally!) prefers to
receive as much as possible.
Assume that the amount person 1 offers can be any number, not necessaríly
an integral number of cents. Then the following extensive game , known as the
ultimatum game , models the procedure.
Players The two people.
7旨rminal histories
The set of 叫uences (x ,
where x is a nurr快r with 0 三
x <二 c (thεamount of money that person 1 offer5 to persoη2) and Z ìs eìther
Y ("yes, 1 accεpt") or N ("no, 1 reject")
Player 此Jnct
1Q 守
Chapter 6. Ext牵 nsive Games with Perf最ct Information: lII ustrations
182
N
C …工，工
0， 0
Figure 182.1 An illustration of the ultimatum game. The gray triangle represents the contir飞口泣 m of
possible offers of player 1; the black línes indicate the terminal histories that 5tart with the offer x.
Pref亏rεnces
Each pe1'so丑 's p 1'efe1'ences a 1'e 1'ep 1'esented by payoffs equal to the
amounts of money she receives. Fo 1' the tel 口lÍnal histo 1'Y (x , Y) pe 1'son 1 1'e∞口 2 l'陀
ec
岱e
创
创i认:陀
仍
e s 川x; fo 1' 血
t h飞e te
肝
err
门mi
ceives c - x and pe1'陀Sο
pe
时1'son 1'ecei飞vesO 署
This game Îs illustrated in Fi gure 182 .1, in which the contìnuum of offers of
play机 1 is l'叩1'esented by the gray t1'íangle , and the b 1a ck lines índicate the terminal histori硝 that sta 1' t with the offe 1' x. The game has a finite horizon, so we can
use backwa 1'd induction to find its subgame perfect 吨u i1ib1'ia. First consider the
subgames of length 1r 比 which person 2 either accepts 0 1' rejects an offer of per翩
son 1. For every possíble offe1' of person 1, tl飞ere is such a subgame. 1n the subgame
that fo11ows an offe1' x of pe 1'son 1 for whlch x > 0, pe1'son 2's optimal aεtìon is to
accept (if she rejects ，说1号 gεts nothing) In the subgame that fo11ows the offer x 0 ,
pe1'son 2 is indifferent bet认reer飞 accepting and rej仅 ting. Thus in a subgame pe 1' fect
吨uilibrium pεrson 2's strategy ei出台r accepts all offers (including 的， or accepts all
offe1's 工
o and rejects the offe 1' x
O.
Now conside1' the whole game. For each possible subgame perfect equilibrium
st1'ategy of person 2, we need to find the optimal strategy of pe1'so日 L
‘
• If person 2 accepts all offe 1's (including 的 1 then pe 1'son l's optimal offer is 0
(which yields her the payoff $c).
• lf person 2 accepts all offers except zero , tf飞en 110 offe 1' of pe1'son 1 is optimal!
No offe1' x > 0 is optimat because the offe 1' x 12 (for exampl的 is better, given
that person 2 accept both offers. And an offer of 0 is not optimal bεcause
pe1'soη2 1'ejects it , leading to a payoff of 0 fo 1' pe1'son 1 ，飞《叮!点
T
offerir叭飞gaηy positive ar盯no飞un飞t Jε
岱ss than $c.
We conclude that the onJy subgame perfect equilib 1'ium of the game is the
strategy pair in which 严rson 1 offers 0 and pe1'son 2 accepts all offe凹. 1口 this
equilib 1'ium , person l's payoff is $c and pe1'son 2's payoff is zero.
This one.心ided outcome is a consequer飞ce of the one-sided structure of the
game. If we allow person 2 to make a counteroffer aftεr rejecting person l' s opening offer (and possibly allow further responses by both player剖， so that the model
corresponds more closely to a 飞argaining situatio饥， then unde1' some drcumll
6.1 The ultimatum game, the holdup game, and agenda control
183
stances the outcome is less one-sided. (An extension of this type is explored in
Chapter 16.)
( EXERC15E 183 .1 (Nash equilibria of the ultimatum game) Find the values of X
for which there is a Nash equilibrium of the ultimatum game in which person 1
offers X
( EXERCI5E 183.2 (Subgame perfect equilibria of the ultimatum game with indivisible units) Find the subgame perfect equilibria of the variant of the ultimatum
game in which the amount of money is available only in multiples of a cent.
? EXERCI5E 183.3 (Dictator game and impunity game) The "dictator game" differs
from the ultim è. turn game only in that person 2 does not have the option to reject
person l's offer (and thus has no strategic role in the game). The "impunity game"
differs from the ultimatum game only in that person l's payoff when person 2
叫 ects any offer X is c - x , rather than O. (The game is named for the fact that
person 2 is unable to "punish" person 1 foτmaking a low offer.) Fi nd the subgame
perfect equilibria of each game.
?l EXERCI5E 183.4 (Variants of ultimatum game and impunity game with equity-
conscious players) Consider variants of the ultimatum game and impunity game
in which each person cares not only about the amount of money she τeceives， but
also about the equity of the allocation. Specifically, suppose that person i's preferences are represented by the payoff function given by Ui(Xl , X2) = Xi 卢 i[X1 工2[ ，
where Xi is the amount of money person i receives，卢 i > 0, and , for any number z,
[z[ denotes the absolute value of z (i.e. [zl 二 z if z > 0 and [z[ = -z if z < 0). Assume c = 1. Find the set of subgame perfect equilibria of each game and compare
them. Are there any values of ßl and ß2 for which an offer is r叫 ected in equilibrium? (An interesting further variant of the ultimatum game in which person 1 is
uncertain about the value of 卢2 is considered in Exercise 227. 1.)
EXPERIMENT5 ON THE ULTIMATUM GAME
The sharp prediction of the notion of subgame perfect equilibrium in the ultimätum game lends itself to experimental testing. The first test was conducted in the
late 1970s among graduate students of economics in a class at the University of
Cologne (in what was then West Germany). The amount c available varied among
the games played; it ranged from 4 DM to 100M (around U .s .$2 to U. S.$5 at the
time). A group of 42 students was split into two groups and seated on different
sides of a room. Each member of one subgroup played the role of player 1 in an
ultimatum game. She wrote down on a form the amount (up to c) that she demanded. Her form was then given to a randomly determined member of the other
group , who, playing the role of player 2, either accepted what remained of the
amount c or rejected it (in which case neither player received any payoff). Each
184
Chapter 6. Extensive Games with Perfect Information: lIIustrations
player had 10 minutes to make her decision. The entire experiment was repeated
a week later. (Güth, Schmittberger二 and Schwarze 1982.)
ln the 在rst 假periment the avεrage demand by people playing the role of
player 1 was 0.65c, and in the second experimen专 it was 0.69c , much less than
the amount c or c - 0.01 predicted by the notion of subgame perfect 叫uilìbrium
(0.01 D时 was the smallest monetary unit; see Exercise 183.2). Almost 20% of offers
were rejected over the two experiments, including one of 3 DM (out of a pie of
7 DM) and five of around 1 DM (out of pies of between 4 DM and 6 DM). Many
other experiments, including one in which the amount of money to be divided was
much larger (Hoffman, McCabe, and Smith 1996), have produced similar results.
ln brie f, the results do not accord well with the predktions of subgame pεrfect
叫址librium.
Or do they? Each player in the ultimatum game cares only about the amount of
money she receives. But an experimental subject may care a1so about the amount of
money her opponent receives. Further, a variant of the ultimatum gam仑的 which
the players are equity conscious has subgame perfect 叫uilibria in which offers are
significant (as you will have discovered if you did Exercise 183 .4).
Howeτ尼r， if people are equi咛 conscious in the strategic environrnent of the
ultimatum game，在何 are presumably 叫uity conscious also in related environments; an explanation of the experimental 陀sults in the ultimatum game based on
the players' preferences' exhibiting equity conscience is not convincing if it applies
only to that environment. Several related games have been studied, amOI哩.them
the dictator game and the impunity game (Exeröse 183.3) 竭 ln the su饨amεper­
fect equilibria of these games, player 1 offers 0; in a variar吐出 which the players
are equity conscious, player 1's offers are no higher than they are ir飞 the analogous
飞'ariant of the ultimatum game, and, for moderate degrees of 叫uity conscience, are
low盯住ee Exercise 183 .4). These features of the equilibria a陀 broadly consistent
with the experimental evidence on dictator, impunity, and ultimatum games (see,
for example, Forsythe, Horowitz, Savin, and Sefton 1994, Bolton and zv飞lick 1995,
and Güth and Huck 1997).
One feature of the experimental rεsults is inconsistent wìth subgame p时fect
equilibrium even when players are equity conscious (at least given the form of
the payoff functio
6.1 The ultimatum game, the holdup game, and agenda control
185
that work well in the situations in which we are typically involved; wεdo not
calculate our rational actions in each situation. Further, we are not typically inω
volved in one-shot situations with the structure of the ultimatum game. lnstead ,
we usua11y engage in 陀pωted interactions, where it is advantageous to "punish"
a player who makes a paltry off叹， and to build a reputation for not accepting
such offers. Experimental subjects may apply such rules of thumb rather than
carefully thinking through the logic of the game , and thus rεject low offers in an
u Itimatum game but acc叩t them in an impunity game , where rejection does not
affect the proposer. The expεrirnental evidence so far collected is broadly consistent with both this explanation and the explanation based on the nature of players'
prεferences
? EXERCJSE 185 .1 (Bargaining over two indivisible objects) C仅lsider a variant of the
ultimatum gam役， with indivisible units. Two people use the fo11owing procedure
to a110cate two desirable identical indivisible objects. One person proposes an allocation (both objects go to person 1, both go to person 2, onεgoes to each person) ,
认lhich the other person then either accepts or rejects. In the event of rejection,
neither person receives either object. Each person cares only about the number
of objects she obtains. Construct an extensive game that models 出ìs situatìon and
find its subgame perfect 吨uilíbria. Does the game have any Nash equilibrìum that
is not a subgame perfect equìlibrium? Is there any outcome that is generat时 bya
Nash 叫uilibriurn but not by any subgame perfect 吨uilìbrium?
苦 EXERCISE 185 .2 (Dividing a cake fairly)τ\vo players use 出e following procedure
to divide a cake. Player 1 divides the cake into tvγo pieces, and then player 2
chooses one of the pieces; player 1 obtains the remaining piece. The cake is contiη
uously di飞也ible (no lumps巧， and each player likes a11 parts of it.
。.
Suppose that the cake is perfectly homogeneous, so that each player cares
only about the size of the piece of cake she obtains. Hovγis the cake divided
in a subgame perfect equilibrium?
b. Suppose that the cake is not homogeneous: the players evaluate different
parts of it differently. Represent the cakεby the set C so that a piece of
thεcake is a subset P of C Assume that if P is a subset of P' not equal to
P' (smaller than P') , then each pl可er prefers p' to 旦 Assurne also that the
players' preferences are continuous: if player i prefers P to p' , then there is a
subset of P not equal to P that player i also prefers to P气 Let (P1 ，乌) (where
P1 and P2 together constitute the whole cake C) be the division chosen by
player 1 in a subgame perfect equilibrium of the divide-and-choose game，马
being the piece chosen by player 2. Show that player 2 is indifferent between
P1 and 屿， and pl可衍 1 1ikes P1 at least as much as P2. Give an example in
which player 1 prefers P1 to P2.
Chapt.叙岳. Ex
186
6. 1.2
tensive Games with Perfect Information: liIustrations
The holdup game
Before engaging in an ultimatum game in which she may accept or reject an offer of
pεrson 1, person 2 takes an action that affects the size Cof the pie to be divided. She
m町 exert little effort, resulting in a small pie , of size cL , or g陀at effort , resulting
in a large pie , of size CH. She dislikes exerting effor t. Specifically, assume that her
payoff is x … E ifh佼 share of the pie is x, where E = L if she exerts little effort and
E H > L if she exerts great effort.丁he extensive game that models this situation
is known as the holdup game.
cv EXERcrsE 186.1 (Holdup game) Formulate the 沁ldup game pr仅isel予 (Write
down the set of players , the set of terminal histories , the player function , and the
?layers' preferences.)
轩hat is the subgame perfect 叫uìlibrium of the holdup game? Each subgame
that follows pers∞ 2' s choice of effort is an ultimatum game, and thus has a unique
subgame perfect equilibrium , in which persoη1 offers 0 and person 2 accepts all
offers. Now consider person 2's choice of effort at the start of the game. If she
chooses L, then her payoff, given the outcome in the following subgame, Ïs L
whereas if she chooses H , then her payoff is - H. Consequently she chooses L.
τhus the game has a unique subgame perfect 叫uìlibrium， ìn which person 2 exerts
little effort and person 1 obtains a11 of the resulting small pie.
This equilibrium does not depend on thεvalues of CL , CH , L, and H (giver飞 that
H > L). In particula乙 even íf CH is much larger than CL , but H is only slightly larger
than
严rson 2 exerts little effort in the 吨uilibrium， although both players could
be much better off if persoη2 were to exert great effort (which , in this case, is not
very great) ar飞d persoη2 were to obtain some of the extra pie. No such superior
outcome is sustainable in an equilibrium because person 2, having exerted gr!εat
effort , may be 咋leld uγfor the entire pie by perso日 1.
This rεsult does not depend sensitively on the extreme subgame perfect equi
librium outcome of the ultimatum game. A similar result emerges when the bargaining fo11owìng person 2's choice of effort generates a more equal division of the
pí巳 By exertíng gre挝 effort， player 2 increases the síze of the pie.τhe point is
that if the negotia tion results in some (not necessarily a11) of this extra pie going to
player 1, then for some values of player 2's cost of exerting g陀at effort less than the
value of the extra pi役， player 2 prefers to exert líttle effor t. ln these circumstances ,
player 2's exerìing great effort generates outcomes in which both pl町εrs are better
off than they are when player 2 exerts little effort, but because the bargaining puts
some of the extra pie in the hands of player 1, player 2's incentive is to exert little
effor t.
•
>
6.1 .3
Agenda control
In some legi坦latures， proposals for modifications of the law are formulated by
committees. Under a 气 losed rule" , the legislature may either accept 0主叫 ect a
6.2 Stackelberg's model of duopoly
187
proposed modificatìon, but may not propose an alternative; in the event of rejectìor飞， the existing law is unchangεd.τhat is, the committee controls the 飞 genda"
(1 n Section 10.9 1 consider a reason why 在 legislature might cede such power to a
committee.)
Model an outcome as a number y. Assume that the legislature and commit如
tee have favorite outcomes that may differ, and that the pr，εferences of each bod y
are represented by (j single-peaked payoff function symmetric about its favorite
outcome , like the voters' prefer'εnces in Hotelling's model of elεctoral competition
(see Pigure 71.日. Assign nurnbers to outcomes so that the legislature's favorite
outcome is 0; denote the corr孜孜ittee's favorite outcorne by yc > O. Then the foIIowing v盯iant of the ultirnatum garne models the procedure. 在le players are the
comrnittee and the legislature 丁hεεomrnittee proposes an outcome !J， 叶飞ich the
legislature either accepts or rejects. In the e\唱lt of rejection the outcome is yo , th巳
"status quo飞 Note that the rnain respect in whích thìs game differs from the ultimaturn game Îs that the playe邸， preferences are diametrically opposed only 飞vith
regard to outcomes betwεen 0 and yc; if y' < y" < 0 or yζ < y" < y' , then both
players prefer y" to y'.
‘
? EXERCISE 187.1 (Agenda control) Pind the subgame perfectεquilibrium of this
garne as a functìon of the status quo outcome .11 0. Show, in particular, that for a
range of val飞les of Yo , an increase in the value of yo leads to a decrense in the 飞'aluε
ofthe 叫uilibrìum outcorne.
6.2
Stackelberg's model of duopoly
6.2 .7 General model
In the models of oligopoly in Sections 3.1 and 3.2, each firm chooses its action not
knowing the other firrns' actions. How do the conclusions change 飞!\'hen the firrns
rnovεsequentially? Is a firrn better off rnovìng before or after the other firms?
ln this section 1consider a market in whìch there ar.εtwo firrns, both producing
the sarnεgood. Firm i's cost of producing qi units of the good is Ci(qi); the price at
which output is sold when the total output is Q is Pd(Q). (In Se ction 3.11 denote
this function P; here 1 add a d subscript to avoid a conflict with the player function
of the extensi飞'e game.) Each firm's strategic variable is ou平时， as in Courn时 's
rnodel (Section 3. 巧， but the firms make their decisions sequentially, rather than
simuitaneously: one firrn chooses its outp时， then the other firrn does so, knowing
the output chosen by the first fir rn.
v飞re can model this situation by the following extensÎve garne, known as Stackelberg's duopoly game (Mter an early analyst of duopoly with asynchronous
aεtions).
Players
τhe two firrns.
陀rminal historìes
The set of a11 sequences (ql , qz) of outpu也 for the firrns
(where each qú the output of firm í, is a nonnegative number).
188
Chapter6. 主xt带到 sive Games with Perf，牵ct Information: lII ustratìons
Playerfunction
P( 创口 1 and P(q]) = 2 for all 号 1.
Pr，ε先rences The payoff of firm i to the terminal history
qi Pð(q1 十但)… Ci( 号 i )， fori
们，归) is its profit
1, 2
Firm 1 moves at the start of the game. Thus a strategy of firm 1 is simply an
outpu t. Firm 2 moves after every histo巧如 which firm 1 chooses an output. Thus a
strategy of firm 2 is a户 η CtiOI1 that associates an output for firm 2 with each possible
output offirm 1.
τhe game has a finite horizon , so we may use backward induction to find its
subgam校 perfect equilibria.
• Fi rst, for any output of firm 1, we find the outputs of firm 2 that rr,aximÌze
its profi t. SUFPose that for each output q1 of firm 1 there is one such 0过 tpllt
of firm 2; denote it b2 (ql ). Then in a巧 s山game perfect equilibrium, firm 2's
strategy is b2
• Next, we find the outputs of firm 1 that maximize its profit , givcll th♂ stratcgy
4卢rm 2. When firm 1 chooses the output q1 , firm 2 chooses the outp飞址 b2(qd ，
rεsulti鸣 in a total output of ql 十归 (q叶， and hence a price of Fd ( 们十 b 2 (qd).
Thus firm l's output in a subg在me perfectεquilibrium is a 飞ralue of ql that
maxnnlzes
(1 88.1)
ql P,t( ql 十 b 2 (q1)) - C1(ql)
Suppose that there is one such value of ql; den时e it ql
We conclude that if firm 2 has a unìque best response 172 阳 ) to each output 号 1 of
firm 1, a挝 f如
i订rm
su
川
lb】咆
伊am
g
丑le perfect eqt山Iilibr忖ium of 出
the 伊
g amel巴S (qî ， bβ
fi让nnη 1γ，、s 吨
e qu
低il丑i衫怡rium strategy
2 ): 如
i沁s q;: and 白
f irm 2's 吨
f
e qu
白i过li山
bri
川川
ium 仗
s t川e
咯
gy 总
is 白
t h巳 funct位iorη1 句
b2. The output chosen by
firm 2，飞ren firm l's equilibrium strateg予 Îs 17 2 (q t); de∞te this output qî.
When firm 1 chooses any output ql , the outcome , given that firm 2 uses Íts
equilibrium stra忧g予 is the pair of outputs (们，与何 1) ). That is , as firm 1 varies
its output, the outcome varies along firm 2's best response function b2 . τhus we
ca川 haracterize the Sl问a 血e pe巾ct equilibrium outcome (苟，苟) as the poi挝 on
firm 2's best response function that maximizes firm 1's profit.
6 .2.2
Example: constont unit cost ond lineor inverse demond
Suppose that Ci(qi) ::::: CL] i for i = 1, 2, and
马 (Q)
if Q 空泛
if Q >乱.，
(1 88.2)
where C > 0 and c <α(as in the εxample of Courno t' s duopoly game in Sec
tíon 3. 1. 3). We found that under these assumptions firm 2 has a unique best response to each output ql of fírm 1, given by
f
1(tt-c-ql}
b2 (ql) = ~ ~
lO
ifql :S: αC
if ql
夜一 c
6.2 Stac主elberg's model of duopoly
1 茜9
Thus in a subgame perfect equilibrium of Stackelberg's game firm 2's strategy is
this function 的 and firm l's strategy is the output q1 that maximizes
q1 (iX - C …{们十 H皮 - c
qd)) = !ql ( ι -c …吁1 )
(refer to (188.1)). This function is a qlladratic ìn 1] 1 that ís zero when 们工 o and
when 号1 =α c. ThllS its maximizer is ql 二 !(a c)
专Ve conclude that the game has a unique subgame perfect equilíbrium, in which
firm l's strategy is the output i 住 c) and firm 2' s strategy is b2 . τhe 0忧。me
of the equilibrium is that firm 1 produces the output
二立 ;识 c) and firm 2
prod附s the output q~ 口 b2 (qn
b 2 ( 拎.--c))
!(α … c - !(议 - c))
扫一 c). Firm l's profit is 号1( Pd 时十 q~) - c) = 扫一 C)2， and 主rm 2's profit
is qHPd(qr 个 q~) - c)
1~ 件… C)2. By co此r dSt, in the llnique 芮ashequilibrium
of Cournot's (simultaneous-move) game under the same assumptions, each firm
produces j 但一 c) units of output and obtains the profit ~归. Th us under
our assumptions firm 1 produces more output and obtains more profit in the s飞lb­
game perfect equilìbrium of the sequential game in which it moves first than it
does in the Nash equilibrium of Courno t's game, and firm 2 produces less output
and obtains less profit.
q;
-
CD EXERCISE 189.1 (Stackelberg's duopoly game with quadratic costs) Find the subgame perfect equilibrium of Stackelberg's duopoly game 认
叫
w
F斗i 怡
1 口 1 ， 2， and 马(号}工此… Q for al江i 号三 α(with 乌 (Q
白) = 0 for Q >皮时). Compare
the equilibrium outωme with the Nash equilibrium of Courn时's game under the
same assumptions (Exercise 59.1).
6.2 .3 Propertìes of subgame perfect equilibrium
Fi rst-mover's equilibrium pro.卢 t In the example just studied, the first-mov叹 is bet恤
ter off in the subgame perfect 叫ui1ibrium of Stackelberg' s game than it is in the
Nash equíIi brium of Courn时's game. A weak version of this result holds under
very general condìtions: for any cost and inverse demand functions for which
firm 2 has a unique best respo日se to each Olltput of firm 1 ，如rr飞 1 is at leasì as
well off in any subgame perfect 叫uilibri山n of Stackel快吃's game as it is in any
Nash equilibrium of Courn仗's game. This result follows from the gεneral result
in Exercise 177.3a. τhe argument is simple. One of firm γs options in Stackel萨
berg's game is to choose its output in some Nash equilibrium of Co旧not's game.
If it chooses such an output , then 主rm 2's best action is to choose its output in the
same Nash equilibrium, given the assumption that it has a unique best responsε
to each output of firm 1. Thus by choosing sùch an output, firm 1 obtains its profit
at a Nash 吨曰librìum of Coumot' s game; by choosing a dìfferent output it may
possìbly obtaìn a higher payoff.
Equilibrium outputs 1n the example in the previous section (6.2.2)，在rm 1 produc岱
more output in the subgame perfect equí挂brium of Stackelberg's game than it does
Chapter 6. Extensi￥e Games with Perfect Informa重ion: lII ustrations
190
小 -l
Etiilk?
饨，，也
nη，
\
\
\ \ b1
\
与\\
Grav curves:
COI飞stant唰profit curves
of firm 1
(lower c盯ve =字
hlgher pro豆。
号2
1
1
飞
飞
飞
q2
飞
飞
飞
飞
飞
。
ql
吁;
q1 •
Figure 190.1 τh挝芒 subg伊a n-时丁
咔
e
甲uω
q
州
驯l川librìu
m
泣
衍 (闭可
ëh ， 寻叩
2ρ) of Cωou汀
1m
curve co时rr陀esψpo只ds tωo hí法ghε衍r proοd
峭
f白1坟
缸t 应
t har到1 does 汹
t he upper Ct泣lTve
已. Each curve has a s!ope of zero where ìt
crosses fìr丑1 l's best response function 告1.
in the 1、-Jash equilibríum of Cournot' s gam已， and fírm 2 produces less. A weak form
of this result holds whenever firm 2's best resp∞se function is decreasing where ìt
is positive (i.e. a hígher output for firm 1 implies a lower optimal output for firm 2) 喃
The argument is illustrated in Figure 190. 1. The firms' best response functìons
are the curves labeled b1 (dashed) and b2. Tl曰:、-Jash equilibrium of Cournot' s
game is the intersectìon (ql' 毛) of these curv臼. Along each gray curve, firm l's
profit is constant; the lower curve corresponds to a higher profit. (For any given
valt淀 of firm l's output, a reduction in the output of firm 2 increases the price
and th l1 s increases firm l's profi t.) Each constant阳profit curve o f. firrn 1 is horizontal where it crosses firm l's best response function, because the best response is
p陀cisely the output that maximizes firm l's profit, given firm 2's outpu t. (Cf. Figure 61 袋.1.刊.) τht出
1怒
s the su
飞ul均〉哩
伊
g
盯扭
a
mepe
扭rf始
配
e
ct eq
邬飞
u泣i必
i让ib
址n血
um
双飞 01时
1此tc
∞
omη1ε
←甲………耐仇叩斗.币.
民
b
岱$纣tr陀es咛
e
po
∞
E叭lS
邸使ef归
u
旧配
n1ctωi
始
O川ha
挝ty
抖
f丛i陀
eld
巾st血
h飞ehl均
阱
g
h飞e
岱st profit for finn l-ís the point ( 哨 , q]J
in Fi gure 190. 1. In partícula乙 given 位lat the best response function of 位rn 2 is
downward sloping, fjrm 1 produces at least as much, and firrn 2 produces at most
as much, in the subgame perfect equilibrium of Stackelberg's game as in the Nash
equilibrium of Courno t' s game.
For some cost and demand functions , firm 2's output i丑 a subgame perfect
equilibrium of Stackelberg's game is zero. An example is shown in Figure 191.1.
τhe disco州nuityin firm 2's best response function at qi in this exarnple may arise
because 在nn 2 incurs a "fixed" cost-a cost independent of its output-when it
produces a positive output (see Exercise 惊2). When firm l's output is , firm 2's
qi
6.2 Stackelb曹 rg's model of duopoly
191
?如
飞 b1
飞
Grav curves:
constant-profìt curves
of firm 1
q2
飞
飞
飞
飞
飞
飞
k
。
ql
ql 叫
Figur费 191.1 Th巴 subgame perfect equilibrium output qj of firm 1 ill Stackelberg's sequenti在 I game
when firm 2 incurs a fíxcd cos t. Along each gray curve , firm 1's profit ís constant; the lower cur\' 巳
corresponds to hig头er profit than does the 让ppercurve.
maximal profit is zero, which it obtains both when it produces no output (and does
not pay the fìxed εost) and when it produces the output 夺2. When fìrm 1 produces
less than 哥， fìrm 2's maxìmal pro缸 is positìve, and firm 2 optimally produces a
positive output; when firm 1 produces more than 时， firm 2 optìmally produces no
outp成 Given this 扣rm of firm 2's best response function and the form of firm 1's
constant-profit curves in Fìgure 190.1 , the point on firm 2's best responsεfunction
出at y刨出 firm 1 the highest profit ìs (吭，时，
1 daìm that this example has a UI飞ique subgame perfect 叫uilibrium， ir飞 which
firm 1 produces qi and firm 2's st臼校gy coincid♂s with its best response function
except at qr , where the strategy specifies the output O.τhe output fìrm2's 叫uilib­
rium strategy specifies after εach history must be a best response to firm 1's output,
S即
o the OI趴到飞让i￥
y questìon 陀
r eg
伊
ar址
di凶
ng fìrm 2's strategy 怡
i s 飞w
咄
γh
挝
1隐
et也
h飞较眩r 让
i t specifìes an 0飞utψpu
时t
ofO 时
0 r q2刊￥认叶
L
fec
时t equi让li岛
br丘i飞口1m 妇
ì n which firm 2's strategy 叩
s pe
配
ci能
f在i硝 the out甲
pu
时t 夺仨2 is sim i1 ar to the
argument that there is no subgame perfect equilibrium in thεultimatum game in
which person 2 rejects the offer O. If主rm 2 produces the output q2 in rεsp∞se to
firm 1's output 苟， then firm 1 has no optimal outpu t: it would like to produce a
little more tha只有， inducing firm 2 to produce zero, but is better off the closer 出
output is to qi. Because there ìs no smallest output gre挝er than q~ ， no output is
optimal for fìrm 1 in thìs ca 优 Thus the game has no subgame perfect equilìbrium
in which fìrm 2's strategy sp时ifìes the output q2 in response to fìrm 1 's output qi.
Note that if firm 2 were entirely absent from the market，在rm 1 would pro响
duce 争
than q Thus firm 2's presence affects the outcom巳， even though ìt
produces no output.
r.
ff飞 EXERCISE 191.1 (Stackelberg's duopoly game with 岳阳d costs)
Suppose that the
invers役 demand functìon ìs given by (188.2) and the cost function of each firm i is
Chapt岳r 6.
192
Extensive Games with Perfect Informatio以lIIustrations
given by
f0
ìf qì 0
l f + cqi if qi > 0,
Ci(qi) 工{
where ( ~三 0, f > 0, and c < α ， as in Exercìse 59.2. 51飞机\' that if c 之二 。， IX.
12, and f = 4, 5tackelberg's game has a uni气ue subgame perfect equilibrium, in
which firm 1'5 output is 8 ar飞d firm 2's output is zero. (Use your results from
Exercise 59.2).
The value of com 斤litment Firm l' s output in a subgame p衍fect equilibriurn of
5tackelberg's game is J1 0t in general a best response to firm 2's outp时 if finn 1
could adjust its output after firm 2 has chosen its output, then it would do so! (In
the case shovvn in Figure 190 .1, it would reduce its outpu t.) However, if firm 1 had
this opportur飞ity， and firm 2 knew that it had the opportunity, then firm 2 would
choose a different outpu t. lndeed , if we simply add a third stage to the game , in
飞I\1hich firm 1 chooses an output, then the first stage is irrelevant, and卢rm 2 is effecti\'ely the first mover; in the subgame perfect equilibrium firrn 1 is worse off than
it is in thε 忖 ash equilibrium of the sirnultaneous同move garne. (1 n the exarnple in
5ection 6.2.2, the unique subgame perfect 叫uilibrium has firrn 2 choose the output
{泛… c) /2 and firrn 1 choose the output 仪一 c)/4.) In surnrnary, even tho狂喜h firm 1
can increase its profit by changing its output after f打rn 2 has chosen its ouf:J当时， ln
the game ìn which it has this opportunity it is worse off than it is in the garne in
which it rnust choose its output before firrn 2 and cannot subsequently rnodify this
。这tp时. That is, firrn 1 prefers to bε cornmítted not toεhange its mind.
>
?且 XERCISE 192.1 (5equential variant of Bertra丑d's duopoly game)
Consider tl冶
variant of Bertrand's duopoly game (Sectìon 3.2) in which first firm 1 chooses a
price , then firm 2 chooses.a price. 人ssurne that each firrn is restricted to choose a
price that is an integral number of ce时s (邵阳 Exercise 67 .2), that each 如m's unit
cost is constant and equal to c (an integral nurnber of cents) , and that the monopoly
profit is positive化
a.
Specify an extεnsive game with perfect inforrnation that models this situation.
b. Gi飞'e an example of a strategy of 如m 1 and an example of a strategy of firm 2.
c. Find th号 subgame perfect 吨以 ilibria of the game.
6.3
Buying votes
A legislature has k members, where k is an odd number. Two rìval bills , X and
Y, are bei口g considered. The bill that attracts the votes of a majority of legislators
will pass. Interest group X favors bill X , whereas interest group γfavors bill Y.
Each group wishes to entiζe a majority of legislators to vote for its favorite bill.
First interest group X gives aηamount of money (possibly zero) 怜相ch legislato巳
约1en interest group Y does so. Each interest group wishes to spend as little as
possible. Group X 飞lalues the passing of bill X at $Vx > 0 and the passing of bill Y
6.3 Buying votes
193
at zero, and groupγvalues the passing of bill Y at $Vγ> 0 and the passing of
bill X at zero. (For example, group X is indifferent between an outcome in which
it spends Vx and bill X is passed and one in which it spends nothing and bill Y is
passed.) Each legislator votes for the favored bill of the interest group that offers
her the most money; a legislator to whom both groups offer the same amount of
money votes for bill Y (an arbitrary assumption that simplífies the analysis without
qua 1itatively changing the outcome). For example, if k = 3, the amounts offered to
the legislators by group X are x = (100， 50，的， and the amounts offered by group Y
are y = (100, 0, 50) , then legislators 1 and 3 vote for Y and legislator 2 votes for X,
so that Y passes. (In some actuallegislatur部 the índucements offeredωlegislators
are more subtle than cash transfers.)
目'e can model this situation as the followi吨 extensive game.
Players
τhe two interest groups, X and Y.
Terminal histories The set of all sequences (x , y) , where x is a list of payments to
Iegislators made by interest group X and y is a list of paym仅lts to legislators
made by interest group Y 阳刚 is， both x and y are lists of k r飞onnegative
integers.)
Playerfunction P(ø)
X and P树立 Y for all x
Preferences The preferences of interest group X are represented by the payoff
function
(归一忡十 Xk)
一(工1 个
十工k)
if bill X passes
if bill Y pas优 s，
where bill γpasses after the terminal history 怡 ， y) if and only if the number
of components of y that are at least equal to the corresponding components
of x is at least i(k 十 1) (a ba陀 majority of the k legislators). The p时盼
εnces of ìnterest group Y are rep陀sented by the analogous function (where
Vy replacεs Vx , Y replaces x, and Y replaces X).
Before studying the subgame perfect equi1ibria of this game for arbitrary values
of the parameters, consider two examples. 1二irst suppose that k = 3 and Vx
Vy = 300. Under these assumptions, the most group X is willing to pay to get
bill X P捕捉d is 300. For any payments it makes to the three legislators that sum to
at most 300, two of the payments sum to at most 200 , so that if group Y matches
these payments it spends less than Vγ(= 300) and gets bill Y passed. Thus in
any subgame perfect equilibri飞1m group X makes no payments , group Y makes no
payments, and (given the tìe喘breaking rule) bill γis passed.
泣。'w suppose that k = 3, Vx
300, and Vy = 100. In this case by paying
each legislator more than 50, group X makes matchìng payments by group Y unpr。在table: only by spending more than Vy (= 100) can group Y cause bill Y to
be passed. However, there is no subgame perfect eq双边ibrìum in which group X
pays each legislator more than 50 because it can always pay a lìttle less (as long
Chapter 6. Ext窑泪如e Games with Perfect Information: lII u苦苦rations
194
剖 the payments still exceed 50) and still prevent groupγfrom profitably matchb飞各
1n the only subgame perfect e平lÍlibrium group X pays each legislatoτex­
actly 50 and group γmakes no payments. Given group X's action, group Y is
indifferent between matching X's payments (so that bill Y is passed) and making no payments. However, there is no subgame perfect equilibrium in whìch
gro飞lp Y matches group X's payments bεcause if this were groupγ's response ,
then group X could increase its payments a little , making matching payments by
group Y unprofitable.
For arbitrary values of the parameters, the subg在 me perfect equilibrium outcome takes one of the forms in these two examples: either no payments ar咬 madε
and billγis passed , or group X makes payments that group Y doεs not wish to
match , groupγmakes no payments , and bill X is passed.
τo find the subgame perfect 叫uilibria in generat we may use backward induction. Fi让rst consider group γγ's best response to an arbit饥rary strategy X of group X.
Le仕t 严ft 古 H拉k 十 1盯)， 忡
a ba盯r陀
川ema句JO忖向
t咛
Y of k 怡
1e鸣伊
gl
咱1ωisla挝to盹 a拍
n飞dd
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挝lle
a
础础$纣tfμ1 Cωomponents of x-the total payments γ needs to make to buy off a bare
majority of legislators.
• If 11勺，< F亏(， then group Y can buy off a bare majority of legislators for less
than Vy , so that its best response to 工 is to match group X's payments to the
{I legislators to whom group X's payments are smallest; the outcome ìs that
bill Y ìs passed.
• If mt>V了， then the cost to group Y of buying off aηy majority of 1εgìslators
excεeds Vy , so that group y's best response to x is to makεno payments; the
outcorr飞e is tha t bill X is passed.
• If 111.\.
Vy , then both the actions in the previous two cases are best responses
by groupγto 工.
We conclude that groupγ's strategy in a sub在ame perfect equilibrium has the
following properties.
• After a history x for which l11 x < Vy , group Y matches group X's payments
to the fllegislators to whom X's payments are smalles t.
e Aftεr a history x for v'. l hich 111 x > 屿， group Y makes no payments.
• After a history x for which 111 王，= ì"斗， group Y either makes no payments or
matches group X' s payments to the 11 legislators to whom X' s payments are
smallεst.
Given that group y's subgame perf仅t equilìbrìum strategy has these proper…
ties , what should group X do? If it chooses a list of payments x for which n1 x < Vy ,
then group Y matches Îts payments to a bare majority of legislators, and bill Y
passes. If it reduces a1l Ìts payments , the sarr飞e bill is passed. Thus the onìy list of
payments x vvith mx < Vy that may be optimal is (0 ,..., 0). If it chooses a list of
paym泛nts 工认'ith m r >斗， then group Y makes no pavm号n怡， and bill X passes.
6.3 8uying votes
195
ff it reduces a11 its payments a little (k时ping the payments to every bare majority
greater than V叶， the outcome is the same. Th us r飞。 list of payments x for which
nI x > Vy is optimal.
wεconclude that in any subgamεperfect equilibrium we have either x
仰，… F 的 (group X makesno payments) or nI x = Vy (the smallest sum of group X's
payments to a bare majority of legislators is V讨. Under what conditions does each
case occur? If group X needs to spend more than Vx to deter group Y from matching its payments to a bare majority of legislators, then its best strategy is to make
no payments (x = (0,. ", 0)). How much does it need to spεnd to deter group Y?
It needs to pay more than Vy to every bare majority of legislators, so it needs to
pay each legislator more than Vy I '1 , in which case its total payment is more than
k 吟/
Thus if Vx < kVγ/ 户， group X is better off making no payments than
getting bìll X passed by making payments Jarge enough to deter group Y from
matching its payments to a bare majority of legislators.
If Vx > kVγ/ 严， on the other hand, group X can afford to make payments large
enough to deter group Y from matching. In this case its best strategy is to pay
each legislator Vy / 严， so that its total payment to every bare majority of legislators
is Vy Given this strat守 gy， group Y is indifferent between matching group X's
payments to a ba皮 majority of legislators and making no payments. 1 claim that
the game has no subgame perfεct equilíbrium in which group Y matches. The
argument is similar to the argument that the ultimatum 伊me has no subgamε
perfect equilìbrium in which 严主son 2 陀jects the offer O. S飞lppose that group Y
matches. Then group X can increas伫 its payoff by increasing its payments a little
(1但eping the totalless than Vx) , thereby deterring group Y from matching, and
ensuring that bill X passes.τhus in any subgame perfect equilibrium group γ
makes no payments in response to group X's strategy.
Ir飞 ωnclusion， if 吨乒 kVylfl ， then the game has a unique subgar时 perfect
εquilibrìum， in which group Y's strategy is to
'1
‘
‘
• match group X's payments to the 严 legislators to whom X' s payments are
smallest after a history x for which mx < Vy , and
• make no payments after a history x for which In x 主 Vy
and group X's strategy deper飞ds on the relati飞re sizes of Vx and Vy :
• if Vx < kVγ/ 严， then group X makes no payments;
• if Vx > k Vy / fl , then group X pays each legislator V y I tι
If Vx < kVy / 严， then the outcome is that neither group makes any payment, and
bill Y is passed; if Vx > kVγ/ 严， then the outcome is 在at group X pays each legislator Vy / μgroup Y makes no payments, and bill X is passed. (If Vx = kγγ / ]J.，
then the analysis is more co黯plex.)
Thr树 features of the subgame perfect equilibrium are significan t. First, the outcome favors the second-mover in the game (group Y): only if Vx > kVy / fl , which
is close to 2Vy when k is large , do伫s group X manage to get bill X passed. Second,
Chapt酷 r6. 军xtensive Gam硅s
196
with Perf，程ct Information: lIIustrations
group Y never makes any payr旦ents! According to its e气uilibrium strategy it is
prepared to make payments in response to certain strategies of group X, but given
group X's equíli伊ium strategy, ìt spends not a cent. Third , if group X makes any
payments (as it does in the equilibrium for Vx > kVγ/ 抖， then it makes a payment
to every legislator. If there were no competing interest group but nonetheless each
legislator would vote for bill X only if she werεpaìd at 1εast some amour泣， then
group X would make payments to only a bare majority of legislators; ìf it were to
ad ìn this way in the presence of groupγ， it would supply group Y with almost a
majority of legìslators who could be induced to vote for bill Y at no cos t.
? EXERCISE 196 .1 (τhr优 interεst groups buyir飞g votes) Consíder a variant of the
model in which there are three bills, X， γ， and Z , and thr把 interest groups , X, Y,
and Z , who choose lists of payments sequentíally. Ties 吕re broken in favor of the
group moving later. Assum巳 that if each bill obtains the vote of one legislator, then
bill X passes. Find the bill passed in any subgame perfect equilibrium when k = 3
and (a) Vx = Vy = 巳= 300, (b) Vx = 300, Vy = 哈= 100 , and (c) 内乙二 300，
吟出 202，巧
100. (You may assume that in each case a subgame perfect 叫ui­
librium exists; note that you are not asked to find the subgame perfect equilibria
themselv俗.)
子 EXE RcrSE 196.2 (Interest groups buying votes und在 supermajority rule) Consider
anoth缸 variant of the model in which a supermajority is required to pass a bil l.
There are two bills, X and Y, and a "default outcome气 A bill passes if and only if
it 陀ceives at least k* >
k + 1) votes; if neíther bill pass拙， the default 0氓∞m
occurs. There are two interest groups. Both groups attach value 0 to the default
outcome. Find the bill that is passεd in any subgame perfect equilibrium when
k = 7， γ 口 5, and (吟 Vx
γγ :工 700 and (的 Vx = 750 , Vy 玄之 400. ln each case ,
would the legislators be better off or worse off if a simple m咛ority of votes were
required to pass a bi ll?
H
? EXERCI5巨 196.3 (Sequential positioning by two political candidat臼) Cor飞sider the
variant of Hotellir飞g's model of electoral competition in Section 3.3 in which thε n
candidates choose their positions sequentially飞 rather than simultaneously. Model
this situation as an εxtensive game. Find the subgame pεrfect equilibrium (吨ui棚
libria?) when 11 之二 2.
c1ò EXERCISE 196.4 (Sequential positioning by three political ca叫id在tes) Consid
6.4 Arace
6.4
197
Araξe
6才. 1
Genera/ modei
Firms compete with each other to develop new technologies; authors competεwi 也
each other to write books and film scripts about momentous current εvents; scientists compete with each other to make discover础. In each cas已 the winner 创1jOYS
a significant advantage over the losers, and each competitor can, at a cost, inζrease
挝r pace of activity. How do the presence of competitors and size of the prize affect
the pace of activity? How does the identity of the winner of the race depend ∞
each competitor's initial distancεfrom the finish line?
We can model a race as an extensìve game with perfect information in which
the players alternately choose how many "st叩 s" to take. Here I study a simple
example of such a game, with two players.
Play衍 i is inìtìally k i > 0 steps from the finìsh line , for i
1, 2. On each of her
turns, a player can either not take any steps (at a cost of 0) , or can take one step , at
a cost of c(l) , or two steps, at a cost of c(2). The 在rst player to reach the fìnìsh lìne
wìns a prize , worth Vi > 0 to player í; the losing player's payoff is O.τbmakεthe
game finìte , 1assume that if, on successive turns , neither player takes any step, the
game ends and neither player obtains the prize
I denotεthe game in which player i movεs fì附 by G i (鸟，与). The game
(k 1, k2 ) ìs defìned precìsely as fo11ows.
P/ayers The two parties.
Terminal histories 丁he set of s吨础Icesofthe form (xIFYIrx2， y2r … J勺。r (x 1,
y l, x2, y2 尸..， yT) 扣r some integer 了， where each Xl (the number of steps
taken by play在 1 0到 her tth turn) and each }/ (the number of steps taken
by player 2 on her tth turn) ìs 0, 1, or 2, there are never two successive O's
except possibly at the end of a sequence, and either x 1 + …十 x T k} and
yl 十十 yT <!走2 (player 1 reaches the fìnish line first) , or x 1 十 .. .十立了 < k}
andyî 十. .郁 + yT
k2 (player 2 reaches the finish líne first).
Playerfunction P( 的
1 ， P(x 1 ) = 2 for a11 扎 p(X 1 ， yl) = 1 for a11 (x 1, yl) ,
P(xLyl ，工2) 口 2 for all (x 1 ， yl ， x巧， and 50 on
Pref台rences
For a t盯mìnal history ìn whìch player i loses, her payoff is the neg唰
ative of the sum of the costs of a11 her moves; for a terminal history in which
she wins it is Vi mìnus the sum of these costs.
6.4.2
Subgame perfect equilibria ofan example
A simple example illustrates the features of the subgame perfect equilibria of this
game. Suppose that both vl and V2 are between 6 and 7 (their exact values do not
affect the equilibria) , the cost c (l) of a sìngle st叩 is 1, and the cost c(2) of two steps
Chapter 6. Extensìve Games with Perfect Information: lII ustrations
198
is 4. (Given that c(与 > 2c( 片，但ch pla)吼 in the absence of a corr飞petito乙 would
like to take one step at a tÎm叫
Thεgame has a finite horizon, so we may use backward induction to find its
subgame perfect equilibria. Each of its subgames is either a game Gi (η11 ， m2) with
i = 1 or i 口 2 and 0 < n1 1 .:; k1 and 0 < m2 .:; k2, 0乙 if the last player to move
before the subgame took no steps, a game that differs from Gj (ml' m2) OI飞ly in that
it ends if player i ir飞itially takes no stεps (i.e. the only terminal history starting with
oconsists onlv of 0).
First consider the very simplest game , G1(1 ，片， in which each player is initially
one st咋 from the finisl飞 line. If player 1 takes one step, she wins; íf she does not
m0ve, then player 2 optimally takes one step (ìf she does not, the game ends) and
wins. V飞Te conclude that the game has a unique subgame perfect equilibrium, in
which player 1 initially takes one step and wins.
A sir削
1
t白
h飞e
创I口1p
抖layε
已r嘎 2 ha
拈
s 出
the option of 恼
t ak
挝iI扫19 or口飞e or two st怆
eps. If she takes one step, then
play moves to a subgame ide批ìcal G1(1, 1) , in which we have just concluded that
player 1 wins. Thus player 2 takes two steps , and wins, if player 1 does not move
at the start of Gl 口， 2). We conclude that the game has a u抗ique subgame perfect
equilibrium，如 which player 1 initially takes 。我e step and wins.
Now consider player 1's optio只s in the game G1(2, 1).
• Player 1 takes two steps: she wins , and obtains a payoff of at least 6 - 4 = 2
(her valuation is more than 6, and the cost of two st叩s is 4).
• Pl码'er 1 takes one step: play mo飞'es to a subgame identical to G2(1 , 1); we
know that in the equilibrium of this subgame player 2 initially takes one step
and wins.
• Player 1 does ηot move: play moves to a subgame in which player 2 is the
first们1f只咄嗣 and is one step from the finish line, and , if player 2 does not
mo飞吃 the game ends. In an equilibrium of this subgame , player 2 takes one
stεp and 认rins.
We conclude th在t the game G1(2, 1) has a unique subgame perfect equilibrium, in
which play自 1 initially takes two step骂 and wins.
1 have speJJed out thεdetails of the analysis of these cases to show how we
use the result for the game G1(1 , 1) to find the equilibria of the games G 1 口 ， 2)and
G1(2, 1). ln general , the equilibria of thεgames Gi (k 1, k2 ) for all values of k1 and k2
up to k tell us the COI1S叫uences of player 1's taking one or two steps in the game
G1 (k+1 , k)
@EXERClS五 198.1 (The race Gl (2, 2)) Show that the game G1(2, 2) has a u口iq时 sub伪
game perfect equilibrium outcome , in whìch player 1 坦itially takes two steps , and
wms.
6.4 Arace
199
50 far we have concluded that in any game in which each player is initially at
most two steps from the finish line, the first-mover takes enough steps to reach the
finish li只仑， and wins
Now suppose that player 1 is at most two steps from the finish line, but player 2
is three steps away. 5uppose that player 1 takes only one step (even if she is initially
two steps from the finish line). 11飞已n if player 2 takes either one or two steps , play
moves to a subgame in which player 1 (the first-mover) wins. Thus player 2 is
better off not moving (and not incurring any cost) , in which case p1ayer 1 takes
one step on her next rurn, and wins. (Player 1 prefers to move one st叩 at a time
than to movεtwo steps inìtìally because the former costs her 2 whereas the latter
costs her 4.) We conclude that the outcome of a subgame perfect 叫uilibrium ìn the
game G1 (2 , 3) is thçlÌ player 1 takes one step onher fìrst t盯叽 then player 2 does
not move, and then player 1 takes another step , and wins.
By a simìlar argument, in a subgame perfect 叫uilibrium of any game in which
player 1 is at most two st叩 s from the fÌlùsh line and player 2 is three or more steps
出气.ray， player 1 moves one step at a tìme , and player 2 does not move; pl叮咬r 1 wins.
5ymmetrically, in a subgame p衍fect 叫uìlibrium of any game ìn which player 1 is
three or more steps from the fìnish line and player 2 is at most two steps away,
pl盯在 1 does not move, and player 2 moves one step 挝 a tìme, and wìns.
Our cor飞clusions so far are illustrated i口Figure 200.1 , where player 1 moves to
the left and player 2 moves down. The values of (k] , k2 ) for 时1Ìch the subgame
perfect equilibríum outcome has been determined 50 far are labeled. The label
"1" means that, regardless of who moves first , Ìl飞 a subgame perfect equìlibrium
player 1 moves one step on each rur口， and player 2 does 110t move; player 1 w ÎnS.
Simìlarly, the label "2" means that, regardless of who moves first , player 2 moves
one step on each rurn , and player 1 does not move; player 2 win5. 11飞e label "f"
means that the fìrst player to move takes enough st叩 s to reach the finish line, and
认F江15.
Now consider the game G1 (3, 3). If player 1 takes one step , we reach tl飞egame
G2(2 , 3). From Fìgure 200 .1 we seεthat ìn the subgame perfect equilibrìum of this
game player 1 wins, and does so by taking one step at a time (the point 口， 3) is
labeled ''1''). If player 1 takes two steps, we reach the game G2(l ，剖， in which
player 1 a1so win5. Player 1 prefers not to take two steps unle5s she has to, so
int
Chapter 岳.
200
• 6
1
主2
Extensive Games with Perfect Informa主ion: lII ustrations
1
5
4Fz
,m.-.4
认."2叫
3L
1
1
1
4
2
2
守
2
2
2
2
2
2
5
6
line
2
3
4
k1
Figure 200.1 Thc 以lbgame perfect equilibrium outcomρS of the racc Gi (k j , k2). Playeγ] 1110飞 'es to
the left, anà player 2 n飞机咽 down. The values of (k] , for whîch the subgame perfect eq口 ilibríum
outcome ha5 快en determíned 50 f挝、 are labeled; dots r叩 resent cases that ha\'号烈。t yet been studíed
丁he labels are explaí口号d ín the text
she takes one step at a time, and wins , and player 2 does not mov伫If player 2 takes
either one or two steps in G 2 (3，白， she reaches a game (either G1 (3 ，毛 or G1 (3 , 3))
in which player 1 "气rins.τhus whatev盯 she does, she loses, so that in a subgame
perfect equilíbrium she does not move and player 1 moves one step at a time.
We conclude that iηa subgame perfect eq山librium of both G1 (3 , 5) and G 2 币 f 5) ,
player 1 takes one step on each turn and player 2 does not move; player 1 wins
A similar argument applies to any game in which one player is initíally three
or four steps from the fì时sh line and the other player 站在ve or more steps from the
finish lì时. We have now made arguments to justify the labeling in Figure 20 1.1,
where the labels have the same meaning as in Figure 200. 1, except that "f" means
that the first player to move takes enough steps to reach the fìnish 1i ne 0 1' to reach
the closest point labeled with her name , whíchever is closer.
A feature of the subgame perfect equilibrium of the 伊 me G1 (4，的 is noteworth y. Suppose that, as planned , player 1 takes two steps , but then player 2 de飞riates from her equilibrium strategy and takes two steps (rather than not mov俨
i口g). According to our analysis, player 1 should take two steps, to reach the finish
li口e. If she does so, her payoff is 只egative (less than 7 - 4 … 4=
日. Nevertheless she should definitely take the two steps: if she does not, her payoff is even
smaller (一句， because plaÿer 2 wins. The poi时 is that the cost of her first move
is "sunk"; her decisior飞 after play在 2 deviates must be based on her optìons from
that point on.
丁he analysis of the games in which each player is initially either five or six steps
from the finish line involves arguments simìlar to those used in the previous cases,
with one amendme口1. A player who is inîtially six steps from the finish line is
6.4 A race
201
T 6
I
5
1
4E
3
·认2，叫叫
1
1
2
2
1
1
2
2
1 I
f
k2
2
1
f
1
1
1
1
2
2
2
2
2
2
2
2
4
5
6
k1 •
Fìnìsh
2
3
Figure 201.1 The subgilme perfect eguilibrium outcomes of the race Gi (k j , k2 ). Player 1 moves to
仅 2mo\' 巳sdow 只 τi恕飞ralues of (k j , k2) for which the subgame perfect 叫 uilibrium
the left. and
outc心 me has be巳 n determined 50 far ilr，已 labeled; dots r专 present cases thilt ha 丸'εnot yet been studied
The lilbels <l rt? explained in the tex t.
better off 只 ot movìng at all (and obtairüng the payoff 0) than she is moving two
steps 0ηany turn (and obtaining a negative payoff). An implication is that ìn the
game G 1 怡，白 ， for example , player 1 does not mo飞吃 if she takes only one step , then
player 2 快 comes the fìr叹 -mover and , by taking a single step , moves the play to a
game that she wins. We conclude that the first啕 mover wìns 加 the games Gi {5,
and G i 怡，的 f whereas player 2 wins in G;(6 , 5) a挝 player 1 wins ìn G i 币 ， 6) ， for
1, 2.
A player who is initially more than six steps from the Íinish line obtair飞 sanεg­
ative payoff if she moves , even if she wìns , so in any subgame perfect e 弓 uilibrium
she does not move. Thus our analysis of the game is complete. The subgame
perfect equilibrium outcomes are ìndicated ìn Fìgure 202 .1, which shows also the
steps taken in thεequilibrium of each game w 怆 n player 1 is the first-mover.
? Ex 在反C1 SE 20 1.1 (A race ìn which the players' 飞laluations of the prize differ) Find
the subgame perfect equìlibrium outcome of the game in which player γs valuation of the prìze is between 6 and 7, and player 2's valuation is between 4
and5.
1n both of thεfollowìng exercises , ìnductìve arguments on the length of the
game , like the one for G;(kl' k2 ) , can be used.
:?- Ex 我 RCISE 20 1. 2 (Removìng stones)τ'w o people take turns removìng stones from a
pìle of n stones. Each person ma弘 on each of her turns, remove either one or two
stones.τhe person who takes the last stone is the winner; she gets $1 from her opponent. Fìnd 出e subgame perfect equilibria of the gam仍在atmodel 吐出 situation
for n = 1 and n 口 2. Fìnd the winner ìn each subgame perfect equilibrium for
Chapter 6. Extensive Game革时th Perfect Information: lII ustrations
202
↑与
1•
1•
1 … i •- 1 •- f
6
•
5
中… 1 ←… 1 ←叩 1 ←- 1 •…
「
3. 盟中… 1 ←-l+-f
f
飞-­
.5
I•
f
‘~、-
i
1
生
』 114 ，
.,
句ι11+ 句Liip--ve
泣4
2
匀'711'句411+ 州Lit*7
43 ← 1 ← l 之L f
2l1+211+2ii'2ia*2-4·
f
5
6
k1
Figure 202.1 丁 he subgame perfect equilìbrì 过 m outcomes of the race
2) Player 1 mo飞 'es tù the
left , and pl在 yer 2 moves down. The arrows indícate the steps taken in th芒 subgame perfect equilibrium
outcome of the games ín whích player 1 moves fjrs t. The I 在加 ls are explained i 公 the text.
,1(
,
n = 3, usìng the fact that the subgame 扣 llowing player 1'5 rem O\址。 f one stone is
the game for 污 = 2 in which player 2 is the first吨 n飞机!er， and the subgame following player 1's remo 毛Tal of two stones is the game for n = 1 in which player 2 is the
first-mov 衍 Use the same tεchnique to find the winner ìn each subgame perfect
机Iuilibrium for 11 = 4, and , if you can , for a. n arbitralγ\'alue of n
?? EXERCISE 202.1 (Hungry lions) The members of a hierarchical group of hungry
lions face a piece of prey. If lion 1 does not eat the prey, the prey escapes and the
game ends. If it eats the prey, it becomes fat and slow, and lion 2 can eat it. If lì Ol飞 2
does not eat lion 1 , the game ends; if it eats lion 1, then it may be eaten by lion 3,
and so on. Each lion prefers to eat than to be hungry, but prεfers to be hungry than
to be eaten. Fi nd the subgame perf巳ct equilibrium (equilibria?) of the extensive
game that models this situation for any 1飞 umber 11 of lions.
6.4.3
Cθ nera/ !essons
Each player's equilibrium strategy involves a "threat" to speed up if the other
play轩 de飞Jiates. Consider, for example, the game G1 (3 , 3). Player l's equ过 ibrium
strategy calls for her to take one step at a time , and player 2's 吨 uilibrium strategy
calls for her not to move. Thus in thεequilibrium outcome, player 1's debt cJ imbs
to 3 (the cost of her three single steps) before she reaches thεfinish line.
Now suppose that after player 1 takes her first step, player 2 deviates and takes
a stεp. Then player l's strategy c 在 lls for her to take two steps , raising her debt to
5. If at no stage can her debt 帆 ceed 3 (ìts maximallevel if both players adhere to
their 叫 uilibrium strategies) , then her strategy cannot embody such threats
‘
Notes
203
The general point is that a limìt on the debt a player can accumulate m问1 affect
thεoutcome even if it exceeds the player's d巳bt in th巳 equilibrium outcome in the
abser飞ce of any limits. You are asked to study an example in the next exercise.
( EXERCISE 203.1 (A race with a liguidity constraìnt) Find the subgame perfect 叫山，
librìum of the variant of the game G1(3 , 3) in whìch playεr 1's dεbt may nεver
excεed 3.
In the subgame per伦ct equilìbrium of εvery game G i ( 抖 ， k 2 )， 01让y one player
moves; her opponeηt "gives up". This property of 叫 uilibrium holds in more ge升
衍al games. What added ingredient might lead to an equilibrium in \-v hich both
players are 在ctive? A player's uncertainty about the other's characteristics would
seem to be su r: h an ingredien t: if a player does not kno飞N the cost of its opponen t' s
moves , it may assign a positive probabilìty less than one to its winning, at least
U在 til it has accumulated some evidence of its opponen t' s beha 飞'10 1'， and while it is
optimìstic it may be actiγεeven though its rival is also active. 1ö buíld such consideratìons into the model wεneed to generali泪 the model of an extensive game
to encomp在 ss imperfect information , as w巳 do in Chapter 10.
Another robust feature of the subgamεper伦ct equilibrium of Gi (k 1, k2 ) ìs that
the presence of a competitor has little effect on the speed of the player who moves.
A lone player would move one step at a tim巳 When there arεtwo players , for most
starting poiηts the onεthat moves does so at the same leisurely pace. Only for a
small number of starting points , in a11 of which the players' initial distar飞ces from
the starting line are similar, doεs the preseηce of a competi tor ÌI飞duce the active
player to hasten its progress, and then only in the first perìod.
Notes
The first experiment or飞 the ultimatum game is reported ìn Güth, Schmittberger二
and Schwarze (1982). Grout (1984) ìs anεarly analysis of a holdup game. 丁he
model of agenda control in legislatures is based on Denzau and Mackay (1983);
去omer and Rosenthal (1978) 创rlìer explored a sìmilar idea. The moàel in Section 6.2 derives its name from th己 analysis ìn vor飞 Stackelberg (1 934 , Chapter 4).
The vote-buyiηg gan飞e in Section 6.3 is taken from Groseclose and Snyder (1996).
The model of a race in Section 6.4 is a simplificatìon sugges伦d by Vijay Krislu冶 of
a model of Harris and Vickers (1985).
For more discussion of the experimental evidence on the 飞lltimatum game (discussed in the box on page 183), see Roth (1995). Bo1ton ar飞d Ockenfels (2000) study
thεimplìcations of assuming that players are equity conscious , and relate these
implications' to thεexperimental outcomεs in various games. The explanation of
the experimental results in terrns of rules of thumb is discussed by Aumann (1997,
7-8). 1he problem of fair division, an example of whìch is giveηin Exercíse 185.2,
is studied in detail by Brams and Taylor (1996), who trace the idea of divide-andchoose back to antiquity (p. 10). 1have been unable to find the origin of the idea in
Exercisε202.1; Barton Lipman suggested the formulation in the exεrcíse.
Extensive Games with Perfect Information:
Extensions and Discussion
7.1
7.2
7.3
7.4
7.5
7.6
7.7
Allowing for simultaneous moves 205
IIlustration: entry into amonopolized industry 213
IIlustration: electoral competition with strategic voters 215
IlI ustration: committee decision-making 217
Illustration: exit frùm a declining industry 221
Allowing for exogenous uncertainty 225
Discussion: subgame perfect equilibrium and
backward induction 231
Prerequísíte: Chapter 5
7.1
Allowing for simultaneous moves
λ 7.7
Definítíon
HE model of an extensive game with perfect information (Definition 155. 1) as-
sumes that after every sequence of events, a single decision-maker takes an
action, knowing every decision-maker's previous actions. 1 now describe a more
general model that allows us to study situations in which, after some sequences of
events , the members of a group of decision-makers choose their actions 飞imulta­
neously" , each member knowing every decision-maker's previous actions, but not
the contemporaneous actions of the other members of the group.
In the more general model, a terminal history is a sequence of lists of actions,
each list specifying the actions of a set of players. (A game in which each set contains a single player is an extensive game with perfect information as defined previously.) For example , consider a situation in which player 1 chooses either C or
0 , then players 2 and 3 simultaneously take actions , each choosing either E or F
In the extensive game that models this s阳ation， (C , (E , E) ) is a terminal history in
which first player 1 chooses C, and then players 2 and 3 both choose E. In the general model, the player function assigns a set of players to each nonterminal history.
In the example just described , this set consists of the si吨le player 1 for 由eempty
history, and consists of players 2 and 3 for the history C.
An extensive game with perfect information (Definition 155.1) does not specify
explicitly the sets of actions available to the players. However, we may derive
the set of actions of the player who moves after any nonterminal history 仕om
the set of terminal histories and the player function (see (156.1)). When we allow
simultaneous moves , the players' sets of actions are conveniently speci自ed in the
206
Chapter 7. Extensive Games with Perf，告ct Inforr拢在tion: 在xten纹。ns and Oiscussion
definition of a game. In the example of the previous paragraph, for instance , we
specify the game by giving the eight possiblεtermìnal histories (C or D followed by
one of the fo盯 pairs
E) , (E , F) , (F , E) , and (F , F)) , the player function defined
by P(Ø
别) = 1 ar只ld P(C) 工 P(D) 口 {2， 3} ， theset怡s of actio邵 {C ， D} for pla
盯
ye
仔r 1 at
t出
h飞eω$纣tar吐t of 白
theg
伊
ameand {EJ
月} for bot出
h pla
咛
Y世 2 andp
抖la
叼
yα
旧r 3 af玩t怡e
盯
阳r tl
衍n叫
a
d D， 刽
a1飞叫
d each playe
眩苦，、s preferences over terminal hìstories.
In any game , the set of tεrminal histories, player function , and sets of actìons
for the players must be consistent: the list of aεtìons that follows a subhistory
of any tεrminal history must be a list of actìons of the players as公gned by the
player functìon to that subhisto巧. In the game just describ时， for example, the list
of actions fol1owi吨 the subhisto巧r C of tl飞e terminal history (C , (E , E)) is 怪，日，
which ìs a pair of actions for the players (2 and 3) assìgned by the player function
to the history C.
Precisely, an extensive gamεwith perfect information and simultaneous moves
is definεd as follows.
DEFIN lTI ON 206.1 An extensive game with p盯住ct information and simultaneous moves consists of
• a sεt of players
• a set of sequenc出 (termìnal histories) with the property that no sequence is
a proper subhistory of any other sequence
• a function (the player function) that assigns a set of players to every sequence
that is a prop臼 subhistory of some terminal history
• for each proper subhistory h of each terminal history and each player i that
is a member of the sεt of players assigned to h by the player. function , a set
Ai (J l) (th号 set of actions avail油 le to player i after the histo巧均
• for each player, preferences over the set of terminal historìes
such th挝 the set of terminal histories, player functìon , and sets of actions are conω
sistent in the sense that 11 is a termÌI飞al history if and only if either (i) h takes the
form (a 1, . . . , {/可 for some integer k, the player function is not defined at 11, and for
every f 二 0，...， k … 1, the element af +1 is a list of actions of the players assigned
by the player functior飞 to (a 1 ,..., af ) (the empty history if f 口 0)， or (i i) h takes
thεform (a 1, a2 ,. ..) and for every = 0, 1,..", the element a f 十 1 is a list of 配tions
of the players assigned by the player function to 扫飞 , af ) (the empty history if
e
e 0).
This defìnìtior飞 encompasses both extensive gamεs with perfect ìnformation as
in Definitíon 155.1 and , in a sεnse， strategíc games. Ar飞 extensìve game with per
fect information is an extensive game with perfect infonnation and simultaneous
moves in which the set of players assigr飞ed to each history consists of exactly one
member二 (The definition of an extensive game with perfect information and sirnul翩
taneous moves includes the players' actions , whereas the definition of an extensi飞F仓
7.1 Allowing for simultaneous moves
207
game with perfect infonnation does no t. However, actions may be derived from
the terminal histories and player function of the latter.)
For any strategic game there is an extensìve game with perfect information
and simultaneous moves in which eve巧 te仅归口ôl history has length 1 that models
the same situatìon. In this extensive garr淀j the set of termìnal historìes is the sεt
of action profiles in the strategìc game, the player function assigns thεsεt of a11
players to the empty histor予 and the single set A i ( 的 of actions of each play时 í is
the set of actions of player j in the strategic game.
飞 EXAMPLE 207.1 (Variant of B05)
Fi rst person 1 decides whether to stay home and
read a book or to attend a concer t. If she reads a book , the game ends. If she
decides to attend a concert, then , as in B05 , she anò person 2 independently choose
whether to sample the aural delights of Bach or Stravinsky, not knowing the other
person's choice. Both people prefer to attend the concert of their favorite composer
in the company of the other person to the outcome in whìch person 1 stays home
and reads a book, and p陀fer this outcome to attending the concert of their less
preferred composer in the company of the other pεrso11.; the worst outcome for
both people is that they aUend dìfferent concerts.
The fo11owìng extensìve game with perfect information and sìmultaneous
moves models this situatio11..
j
Players The two people (1 and 2).
7旨rminal hìstories
Book, (Co町E屹 (B ， B)) , (Co町ert， (B ， 5)) ， (Co时ert， (5 ， B)) ， and
(Co町ert， (5 , 5))
Player function
P( 的
1 a11.d P(Co阳的= {1 ， 2} 咽
Actions The set of play时 l' s actíons a t the empty history ø ìs A 1 (叨 = {Concert ,
Book} a 11.d the set of her actìons after the hìstoη1 Concert is A 1 (Co町E的土
{B , 5}; the set of player 2's actions after the hìsto巧， Concεrt is A 2 (Concert) =
{B , 5}.
Preferences Player 1 prefers (Concert , (B , B)) to Book to (Co舵ert， (5 ， 5)) to
(Concert , (B , 5)) , which she regards as ìndifferer飞t to (Concεrt， (5 , B)). Player 2
prefers (Co盯t吃币， 5)) to Book to (Conce吃 (B ， B)) to (Concert , (B ， 5 片， which
she regards as i11.differe时 to (Conce斤， (5 ， B)).
This game ìs ìllustrated in Figure 208.1 , i11. which 1 r叩resent the simulta11.eous
choices betwee们 B and 5 in the way that 1 previously r叩resented a strategic game
(Onlya 伊me in which a11 the sìmulta11.eous moves occur at the end of tenninal
histories may be represe11.ted ì11. a diagram like thJs one. For most other games 11.0
C011.ve11.ie11. t diagrammatic represe 11.tation exists.)
‘
λ 1 .2
5trategies and Nash equ i/ibrium
As in a game without simultaneous moves, a pl町er's strategy speci豆es the action
she chooses for every history after which it is her turn to mov巳 De缸ùtÌon 159.1
208
Chapter 7. Extensi飞fe Games with Perfect Informa主ion: Extensions and Disωssion
万扩产飞吃rt
-MAU
-AAVAHV-1
,,
nu
叫J
nAUCυ
m
气…'''
E
2, 2
Figure 208.1 机飞e variant of 80S described in 在 xample 207.1
requires only minor 主ewording to allow for the possibility that players may move
simultaneously.
D五日 NI了ION 208.1 (5trategy În extc l1E' ive game with pe价ct information and simultane
ous 11l0ve斗 A strategy of player i in an extensìve game with perfect ìnformation
and simultaneous moves is a function that assigns to each history h after which i
is one of the players whose turn it is to move (i 忍 i is a member of P( 的， where P is
the player function of the game) an actìon ìn Aj 何) (the set of actions available to
player i after h).
The definition of a Nash 呵 uilibriu111 of an extensive game wìth 严 rfect information and simultaneous mov硝 is exactly the same as the definition for a game
with no simultaneous moves (Definition 161 勾: a Nash equilibrium is a stratεgy
profile with the prop衍ty that no player can ìnduce a better outcome for herself
by changìng her strategy, given the other playe凹 'strategies. Also as before , the
stratcgic for111 of a game is the strategic game in which the players' actions are their
strategies in the extensìve game (see Section 5 .3), and a strategy profile is a Nash
叫uilibrium of the extensive game if and only if it is a Nash 叫ui1ibrium of the
strategic form of the game.
EXAMPLE 208.2 (时ash equilibria of a variant of 805) ln the game in Example 207.1 ,
a strategy of player 1 specifies her actions at the start of the game and after the
history Concert; a strategy of player 2 specifies her action after the history Conccrt.
Thus player 1 has four strategies, (Conce吃 β) ， (CO/1CC吃 5) ， (Book , 8), and (8ook , 5) ,
and player 2 has two strategies , B and 5. (Rememb盯 that a player's strategy is
more than a plan of action; it specifies an action for evcry history after which the
player mo飞咽， even histories that it precludes嗜 For example, player l's stratεgy
specifies her actiOll after the history COl1ccrt even if it spedfies that she choose Book
at the begìnnìr飞g of the game.)
The strategic form of the game ìs gi飞收1 in Figure 209.1. 认le see that the game
has three pure 队Jash equilibria: (( Concert， 酌，韵， ((Book， 的， 5) ， and ((Book, 5) , 5).
Every extensive game has a unique strategic form. However, some sh'ategic
games are the strategic forms of more than oneεxtensivεgame. Consider, for
example, the strategic game in Figure 209.2. Thls game is the strategic form of the
exter飞sive game with perfect information and simultaneous moves in which the
恼。 players choose their actìons simultaneously; it is also the stra
7.1 Allowing for simultaneous moves
209
(Con臼rt ， B)
(Co配ert， 5)
(Book , B)
(Bo忱 5)
B
5
3, 1
0, 0
2, 2
2, 2
0, 0
1, 3
2, 2
2, 2
Figure 209.1 The strategíc form of the game in 巳xample 207. 1.
L R
T 1, 2 1, 2
B
0, 0
2, 0
Figure 209.2 A slrategic game that is the strategic form of more than one extensive g在 m巳
7.1 .3 5ubgame perfect equilibrium
As for a game in which one player moves aftεr each history, the S1巾 game following the history 11 of an extensi飞re game with p巳rfeεt information and simultaneous
moves is thεextensi飞re game "starting at h". (丁怡扣rmal definitíon is a variar飞t of
Definitioη164. 1.)
For instance , the game ín Example 207.1 has two subgames: the whole game,
and the game in which the players engage after player 1 chooses Concert. In the second s巾 game， the terminal hístories are (B , 町，凹，时， (5 , B) , and (5 ，町， the player
functíon assigns the set {口1γrJ2斗
户 } cωO只
ns剑íst
挝tir吨 ofbo
时th players to the empty history (the
only no时时minal history) , the set of actions of each player at the empty history is
{B， 斗， and the playεrs' prefere挝es are represented by the payoffs in the table in
208. 1. (This subgam己 models the same situatíon as B05.)
A subgame perfect equilíbrium is defined as before: a subgame pe~fect 咛双ilíb­
rium of an extensive game with perfect informatíon and simultaneous mo飞毛s is a
strategy profile with the property that in no subgame can any player increase her
payoff by choosing a different strategy, given the other players' strategies. The
formal definition differs from that of a subgame perfect equilibrium of a game
without simultaneous moves (Definition 166.1) onlv in that the meaniηg of "it is
player i's turn to move" is that i is a member of P(h) , rather than P(h) i.
To find the set of subgame perfect 吨uilibria of an extensive 军ame withp校fect
information and simultaneous moves that has a finite horizon, we can, as before,
use backward in小lctÏon. The only wrinkle is that some (perl飞aps a11) of the situas 由
the
吗叮
y are in
ations we need to analyze are not single-person decision problems , 础
s Ím1咄aneous moves
轧， bu
川tpn伪lem
盯
m冶川
s创1I川叮N"hich severa
叫!p
抖layers choose
the abse缸eof 玩
actions simultaneously. We cannot simply find an optímal action for the player
who使挝rn it is to move at the start of each subgame , given the players' behavior
in the rεmaìnder of the game. 飞Ale need t。如ld a list of actions for the players who
move at the start of each subgame, with the property that each player's action is
optìmal given the other players' simu1taneous actions and the players' behavìor
in the remainder of the game. Th at is, the argument we need to make is the same
Chapter7. Extensive G矗m然 with Perfect Infor苦苦苦辑tion: Ext带 nsions and Discu军事ion
210
as the one we make when 自nding a T、也sh equilibrium of a strategic gam巳 This
argument may use any of the techniques discussed in Chapter 2: it may check
each action profile in turr飞， it may construct and study the players' best response
functions , or it may show directly that an action pr。在le we have obtained by a
combination of intuition and trial and error is an equilibrium.
EXAMPL 旦 210.1 (Subgame perfect equilibria of a variant of B05) Consider the 俨 me
in Figure 208.1. Backward induction proceeds as follows.
• ln the subgame that follows the history Concerf , there are two Nash e电uilibria
(in pure strategi耐， namely (5 ，的 and (B , B) , as we found in Section 2.7.2.
• If the outcome in the subgame that follows Conc衍't is (5 , 5) , then the optimal
choice of play时 1 at the start of the game is Book.
• If the outcome in the subgame that follows Concert is (B , 的， then the optimal
choice of player 1 at the sta 1' t of the game is Concer t.
We ω时lude that the game has two s山game perfect equilibria: 在 Book， 5) , 5) and
((Concert , B) , B).
Every fìnìte extensìve game with perfect informahon has a (pure) subgame per往往 equìlibrium (Proposìtion 173.1). The same is not true of a fìnite extensìve game
with perfect information and simultaneous moves because , as we know, a finite
strategic game (which corresponds to an extensive game with perfect ìnformation
and simultaneous moves of length 1) may not possess a pure strategy Nash equilibrium. (Consider Matclúng Pemûes (Example 19.1).) If you have studied Chapter 4,
you know that some strategic games that lack a pure strategy Nash equìlibrium
have a 气孔ìxed strategy T、>Jash equilìbrium", in vvhìch each player randomizes.τhe
same is true oÍ extensive games wìth perfect inforn飞ation and simultaneous moves.
江owever， in this chapter 1 restrict attention almost exclusively to pure strategy
equilìbria; the only occasìon on which mixed strategy 1、>Jash 吨uìlìbri泣m appears is
Exercìse 212.1.
① Ex 巳 R Cì Sε210.2 (Extensive game wìth simu!taneous moves) Find the subgame
?衍 fect equilibria' of the following game.
First player 1 chooses eithεr A or B.
After ei 也er choice , she and player 2 simuJtaneously choose actions. If player 1
initially choo~泛s A , then she and player 2 subsequently each choose either C
or D; if player 1 chooses B initia l1y, then she and player 2 subsequently each
choose either E or F. Among the terminal histories, player 1 p妃fers (A , (C , c))
to (B , (E , E)) to (人 (D， D)) to (B , (FJ)) , and p陀fers a11 these to (A , (C , D)) ,
(A ， (D ， C片， (B ， (E ， F)) ， and (B， (F ， E)片)， between which飞 she 总
i sir州i迁ffe
如
盯r陀
e
臼I
e
prefers (A , (口旦， D)片) to ( β , (FJ)) to (A
人， (C， C)) to (B , (E , E)) , and prefers a11 these
to (A ,(C, D)) , (A ，( 后， C)) ， (B , (EJ刀， and (B , (F , E 片， between which she is
indifferen t.
? EXERCISE 210.3 (τ'w o-perìod Priso l1衍-'s Dilemma) Two people simultaneously select
actions; each person chooses 舍ither Q or F (as ìn the Prisoner's Dílemma). Then
7.1 Allowing for simultaneous moves
211
thεy simultaneously select actions again, once again each choosing either Q or
F. Each p在son' s preferencεs are rep陀sented by the payoff function that assi事lS
to the terminal hìstory ((W , X) , (γ， Z)) (where each component 怒已ither Q or F)
a payoff 叫ual to the sum of the person's payoffs to (W , X) and to 仔， Z)inthe
Prisoner's Dilemma given in Figure 15. 1. Specify this situation as an extensí飞'egame
with perfect information and simultaneous moves and find its subgame perfect
equilibria.
:t EXERCISE 211.1 (古ming claims on an investment) An amount of money accumu町
lates; in period t (= 1, 2, .… , T) its size is $2 1. 1n each period two people simultaneously decidεwhether to claim the money. If only one p时son does so, she gets
all the money; íf both people do so, 也可 split the money 叫 ually; and if neither
person does so , bofh people have the opportunity to do so in thεnext period. lf
neither person claims the money in period T , each person obtains $T. Each perω
50n cares only about the amount of money she obtains. Formulate this situation as
an extensÍve game with perfect information and simultaneous moves , and find its
subgame perfect equi 1ibria). (Start by considering the cases T ::::: 1 and T = 2.)
子 EXE 汉CIS忘 21 1. 2 (A market game)
A seller 0以rns one indivisible unit of a good ,
which she does not 飞'alue. Several potential buyers, each of whom aìtaches the
same positive value v to the good , simu1taneously offer príces they are willing to
pay for the good. After receiving the offers, the seller decidεs which, if an予 to
accep t. Jf she does not accept any offer, then no transaction takes place, and all
payoffs are O. Otherwise, the buyer whose offer the seller accepts pays the amount
P sh号。ffered and receives the good; the payoff of the seller is p, the payoff of
the buyer who obtained the good is v - p. and the payoff of every other buyer is Q.
Model this situation as an extensiv台 game with perfect information and simultaneous moves and 位ld its subgame perfect equilibrìa. (Use a εombination of intuition
and trial and error to 也飞d a strategy pro缸e that appears to be an equìlibrium, then
argue directly that it is. 丁he incentives in the game are closely related to those in
Bertrand's oligopoly game (see Exercise 68 .1)，飞Nith the roles of buyers and sellers
reversed.) Show, în particular, that in every subgame perfect equilibrium every
buy时 's payoff is zero.
MORE EXPERIMENTAL EVIDENCE ON SUBGAME PEI之FECτEQUILIBRlU M
Experiments conducted in 1989 and 1990 among college students (mainly taking
economics classes) show that the subgame perfect equilibria of the game in Exercise 211 .2 correspond closely to experimental outcomes (Roth, Prasnikar, OkunoFujiwara , and Zamir 1991) , in contrast to the subgame perfect equilibrium of the
ultimatum game (see the box on page 183).
In experimen岱 conducted at four locations (Jerusale叽Lj ub可 ana， Pi坟sburgh，
and Tokyo) , nine "buyers" simultaneously bid for the rough 吨uivalent (in terms of
local purchasing power) of U .s.$10, held by a "seller". Each experiment involved
212
Chapter7. 在xtensive
Games with Perfec:t Information: Extensions and Disc: ussion
a group of 20 participants, which was divided into two market日， each wíth one
seller and nine buyers. Each participant was ínvolved ín ten rounds of the mar挝t;
ín each round the sellers and buyers were assigned anew, and i民 any gíven round
no partícípar飞t knεw who, among the other participants, were sellers and buyers ,
and who was involved in her marke t. Ir飞 every session of the experiment the maximum proposed price was accepted by the seller, and by the seventh round of every
experimer飞t the highest bid was at least (the e电uivalent of) U .5. $9.95.
Experiments involving the ultimatum game, run in the same locations llSing a
similar design, yielded r.εsults similar to those of prevìous experiments (see the box
on page 183): proposers kept considerably less than 100% of the p栓， and nontrivial
offers were 陀iected.
τhe box on page 183 discusses two explanations for the experimenta 1 rεsults
in the u1timatum gam已 Both explanations are consistent with the results ín the
market game 鬼 One explanation is that people are concerned not only with their
own monetary payoffs, but also wíth other people's payoffs. At least somεspeci自­
cations of such preferences do not affect the subgame perfect equilibria of a market
gamεwith many buyers, which still a11 yíeld every buyer the payoff of zero. (Whe只
there are many buyers , even a seller who cares about the other players' payoffs ac阳
C叩ts the highest price offered , because accepting a lower price has little impact on
the disfríbution of monetary payoffs, a11 but 队TO of which remain zero.) Thus such
preferencεs are consistent with both sets of experimental outcomes. Another explanation is that people incorrectly recognize the ultimatum game as one in whìch
the rule of thumb "don't be a sucker" is advantageously invoked and thus reject a
poor offer, "punishing" the person who makes such an offer. In the market game ,
the players treated poorly in th伫 subgame perfect e弓uilibrium are the buyers, who
move first , and h创 lce have no opportunity to punish any other player. 丁hus the
rule of thumb is not rele飞咄飞t in this game, so that this explanation is also consÌstent
with both sets of experimental outcomes.
In the next exercise you 盯e asked to ìnvestigate subgame perfect equilibria in
which some players use mixed strategies (disc出sed in Chapter 4).
苦 EXERCISE 212 .1 (price competition)
Extend the model in Exercise 128 .1 by havíng
the sellers simultaneously choose their prices before the buyers simultaneously
choose which se11er to approach. Assume that each se11er's preferences are represented by the expεcted value of a Bernou1li payoff function in whích thεpayoff
to not trading is 0 and the payoff to trading at the price p is p. Formulate thìs
model precisely as an extensivεgame with perfect information and simultaneous
a1刀
川rr
盯阳
阳I
moves. Show that for ev 町 P 三 ~ the game has a 时g泞伊阴阴
i挝
n whi
飞让i巾
ch each 览
s el如
le
仔玄 &衍n飞n飞ounces 白
t he pric
岱
e p. (You may use the fact that if seller j' s
price is at least , se11er i's payoffin the tnixed strategy 叫副brium of the s巾gaI时
in which the buyers choose which seller to approach is decreasing iηher price Pi
when Pí > Pí')
1
7.2 lII u主tration: entry into a monopolized industry
213
7.2 lII ustration: entry into a monopolized industry
7.2.1
General model
An industry is currently monopolized by a single firm (在ε"incumbe时"). A sec例
。nd firm (the "cha l1 enger") is considering entry, which entails a positiγe cost j
in addítion to its production cos t. If the cha11部1ger stays out, then its profit is
zero , w}飞已reas jf it enters, the firms simuìtaneously choose outputs (as in Courno t' s
model of duopoly (Sec悦。n 3.1)). The cost to firrn i of producing qi units of output
is C i 忡。 If the firms' total output is 白， then the market price is 乌 (Q). (As in
Section 6.2, 1 add a subscript to P to avoid a clash with the player function of the
game.)
We can model this situation as the following extensive game with perfect information and simultaneous moves, illustrated in Figure 213.1.
Players
η飞e two fírms: the incumbent (fírm 1) and the challenger (firm 2).
Tem如alhistories
(I n, (q] , q2)) for any pair (q] ， 但) of outputs (nonnegatíve
number吟， and (0风 qd for any output ql.
Player function
P( 0) = {2} , P(I n) = {1 , 2}, and P( 0叫口 {1 }.
Actions A 2 ( 叫 = {I I1, Out}; Al (l n) , A 过 Out) ， and A 2 (1 11) are all equal to the
set of possible outputs (nonnegative numbers).
Preferences Each firm's prefereηces are repriε sented by its pro舱， which for a
terminal history 恼， 顷， 但)) is 材
q 1♂
P却d(臼号 ]+汁q2) - 巳
C 1(何
们]) 阳
q
fO
优玄忖灿
t也h
挝
巾ein
配
缸αurr
n
Cl川
1mb
悦
衍时
e
n1让ta
缸E挝
ld
P'd(q
号轻2 句
， 叭1 十 q但
ω2ρ) 一 C2(柯
白2甘制) 一 jfo
q
时rt位
he cha11沁
e吨e吼几 a肌
nd f，扣
O主 a te
时rmi加
nal his怡巧巧r (0叫
ω1ο) 岭 q们lP
q们
，乌'd(句ql) 一(问q]) for the 如
in飞cum
λ2.2
Example
Su
叩
pp
严
os优e that Ci(号如圳
d 口 cqi for a11 伤
q í (气
inverse demand function 岭
i s lin岱rwhe
盯re 让
it 旭
i spos饿
i坟ti认飞吨 gi飞ven by 马
P.d(Q) 半夜… Q for
Q ~三二 α儿， 部
as 如
iz只1 Section 3. 1.3. τo find the 阳
s ubg
伊
am!ηle 严
pQ
衍rf伦
ect 叫
e qu
旧ilibria， fírst consider
the subgame that follows the history 111. The strategic form of this subgame is the
same as the example of Courn时'5 duopoly game studied in Section 3.1.3, except
that the payo征。f the challeng盯 is rεduced by j (the fix付 cost of er飞try) regardless
of the challenger's outpu t. τhus the subgame has a unique Nash 已弓uilibrium， in
Challenger
mn nosdM
no
enγ
Figure 213.1 An entry game.
DOi
咱
p、 γ·
睛
Cournot's
duopoly game
'如们
y'吨二
214
Chapter 7. Extensive Games with Perfect Information: Extensions and Discussion
which the output of each firm is j (α- c); the incur由ent's p时itis j(α- c) 气 and
the challe吨er's profit is ~(α 一 C)2 - 1
Now consider the subgame that follows the history Out. In this subgame the
incumbent chooses an outpu t. The incumbent' s profit when it chooses the output
ql is ql ( α- ql) - cql = ql ( α- c - qd. This function is a quadratic that increases
and then decreases as ql increases , and is zero when ql = 0 and when ql = α -c.
Thus the function is maximized when ql
( α- c). We conclude that in any
s由gamepe由ct equilibrium the incur由ent chooses ql = ( α- c) in the s由game
following the history Out.
Finally, consider the challenger's action at the start of the game. If the challenger stays out, then its profit is 0, whereas if it enters, then given the actions
chosen in the resulting subgame , its profit is ~ (α- c)2 - f. Thus in any subgame pe由ct equilibrium the challenger enters if ~ (α- c)2 > 1 and stays out if
~(α c )2<f.If~(αC)2 = 1, then the game has two subgame pe白ct equilibria;
the challenger enters in one and does not in the other.
In summary, the set of subgame perfect equilibria depends on the value of f. ln
all equilibria the incumbent' s strategy is to prod山e i(αc) if the challe吨er enters
and i( αc) if it does not, and the challe吨er's strategy involves its producing
i
i
j(α- c) 证阳nters
.If /<~(αC)2 there is a u叫 ue s由game perfect equilibriur口， in which the
challenger enters. The outcome is that the challenger enters and each firm
p时uces the output j (α- c)
.If />~(αcf th町e is a unique subgame pe由ct equilibriun1, in which the
challenger stays ou t. The outcome is that the challenger stays out and the
incumbent prod山es (α- c)
i
.If /=~( α- c)2 the game has two s由gamepe巾ct equilibria: the one for the
case 1 < ~(α- c)2 and the one for the case 1 > ~ (αC)2
Why, if 1 is small , does the game have no subgame perfect equilibrium in which
the incumbent floods the market if the challenger enters, so that the challenger
optimally stays out and the incumbent obtains a profit higher than it would have
if the challenger had entered? Because the action this strategy prescribes after
the history in which the challenger enters is not the incumbent's action in a Nash
equilibrium of the subgame: the subgame has a unique Nash equilibrium, in which
each firm pr叫阳s j (α- c). P时 differently， the incur巾 ent's "threat" to flood the
market if the challenger enters is not credible.
? EXERCISE 214.1 (Bertrand's duopoly game with entry) Find the subgame perfect
equilibria of the variant of the game studied in this section in which the post-entry
competition is a game in which each firm chooses a price, as in the example of
Bertrand's duopoly game studied in Section 3.2.2, rather than an output
7.3 lII ustration: electoral competition with strat辑程 ic voters
7.3
215
lII ustration: electoral competition with strategic voters
The voters in Hotelling's model of electoral comp的ition (Section 3.3) are not pl可­
ers in the game: each citizen is assumed simply to votεfor the candidatεwhose
position she most prefers. How do the conclusions of the model change íf we
assume that each citizen chooses the candidate for whom to vote?
Consider the extensive game in which the candidates first simultaneously
choose actions, then the citizens sìmultaneously choose how to vote. As in the
variant of Hotelling's game considered on page 74, assume that each candidate
may either choose a position (as in Hote11ing's original model) or choose to stay
out of the race, an option she is assumed to rank between losing and tying for first
place with a11 the oth町 candidates.
Players
丁he candidates and the citizεns.
陀rmìnal hìstorìes
All sequences 帜，叫 wherε x is a list of the candidates' ac-
tio邸， each component of which is either a posihon (a number) or Out , and
v is a 1ist of voting decisions for the citÍ zens (i.e. a list of candidates, one for
each citizen).
Playerfunctìon P(Ø) is the sεt of a11 the candidates, and P (x) , for any list X of
positions for thεcandidates， is the set of al1 citizens
‘
Actions
τhe set of actions available to each candidate at the start of the game
consÍsts of Out and the set of possible positions. The set of actions available
to each citizen after a history x is the set of candidates.
Preferences Each candidate's preferences are represεnted by a payoff function
that assi萨lS 11 to every terminal history in which she wins outright, k to every
tεrminal history in which she ties for first place with n … k other candidates
11 - 1) , 0 to every terminal history in which she stays out of
(for 1 三 k
:s
the race, and -1 to every terminal history 加 which she loses, where 11 is thε
number of car飞didatεs. Each citizen' s preferences a陀陀presented by a payoff
function that assìgns to each terminal histo巧， the average distance from the
citizen's favorite posìtion of the set of winning candidates in that history.
First consider the game in which there are two candidates (and an arbitrary
number of citiz巳ns). Every subgame fol1owing choices of posi悦。ns by the candidates has many 民ash equilibria (as you know if you solved Exercìse 48.1). For
example, any action profile in which all citizens vote for the same candidate is
a Nash equilibrium. (A citìzen's switching her votεto another candidate has no
effect on the outcome.)
Th is plethora of Nash equilibria allows us to construct, for every pair of positions , a subgame perfect equilibrium in which the candidates choose those posí蛐
tions! Consider the strategy profile in which the candidates choose the positions
Xl and X2 , and
216
Chapter7. 在xtensive
Games with Perfect Information: Ext牵nsions 矗nd Discussion
• a11 ci tÌ zens vote for candidate 1 after a history 叫 I x~) in which x~ = Xl
• a11 c陆ens vote for ca时idate 2 after a history 叫 I x~) in which x~ 并 Xl.
The outcome is that the c在ndid在 tes choose the positions Xl and X2 and candidate 1
wins.τhe strategy pr。在le ís a subgame perfect equilibrium because for every
history 仙，川 the profile of the citizens' actions is a Nas由
h飞叫1山
1丘îl垃i池b
如
b加
riurr
吼
E
t出
he
盯r 臼
c and
出ìd
巾
at怡
ec
臼
an
只1 ind飞uce arη1 outcome she prefers by de飞v咄 tin飞屯
吝 a de
g
队啊飞
vi归
挝t挝ior只1 by
a
candidate 1 to a positiorη1 di泛ffer甸它咄
ent frorr飞 工1 leads her to lose , and a deviatìon by
candidate 2 has no effect on the outcome.
However, most of the 民ash equilibria of the voting subgames are fragile (as you
know ìf you solved Exercìsε48 .1): a citizen's voting for her less preferredcandìdate
ís weakly dominated (Definition 46 .1) by her voting for h臼 favor让e candidate. (A
citÍzen who swítches from voting for her less p陀ferred candidate to voting for h肝
favorite candid 在 te eithe1' does not affect the outcome (if hεr favorite candidate was
three or more votes behind) 0 1' causes her favorìte candidate either to tie for first
place rather than lose , or to win rather than tie.) Th us in th号。nl 1' Nash equi1ibrium
of a voting subgame in which no citizen uses a weakly dominated action, each
citi之en votes for the candidate 飞々hose position is closest to her fa vorite position.
Hotelling's model (Section 3.3) aSSU111CS that each citizen votes for the candidate
whose position is closest to h町 fa 飞!orite positìon; in its unique Nash equilibrium ,
each candidate's position is the median of the citizens' favoríte positions. Combin
ing this resuit with the result of thεpre飞'ious paragraph , we conclude that the game
we are studying has 0111y one subgame perfect e弓uilibrium in which no player's
strategy is weakly dominated: each candidate chooses the median of the citizens'
favorite positions , aηd for every pair of the candidates' positions , each CÍ tÌ zen votes
for her favorite candidate.
1n the game with three or more car飞didates， not only do many of thεvoting
subgames have many Nash equîlibria , with a variety of outcOI引es， but resh'icting
to votìng strategies that are not weakly dominated does not dramaticall 1' affect the
set of 叫出libria: a citizen's only weakly domin在ted strategy is a votεfor her least
preferred candidate (see Exercise 49.1).
Ho飞.ve飞啊~ the set of e弓uilíbrium outcome
EXERCISE 216.1 (Electoral competition with strategic voters) Assume that there
are 11 ? 3αmdidates and q citizen究， where q 之 211 is odd (50 that the n飞edian of
7.4日 lustration:
committee dedsíon-making
217
the voters' favorite positions is well defined) and divisible by n 雹 Show that the
game has a subgame perfect equilibrium in whïch no citizerγs strategy is w但kly
domìnated and every candidate enters the race and chooses the median of the
citizens' faγorite positions. (You may use the fact that every voh豆g subgame has a
(pure) Nash equilibrium in which no citizeI飞's action is weakly dominated.)
命 EXERCISE 217.1 (Electoral competition with strategic voters) Considεr the varia挝
of the game in this section in whïch 件 the set of possible positions is the set of
numbers x with 0 <工< 1, (ii) the favorite position of at least onεcitizen is 0
and the favorite position of at least one citizen is 1, and (i ii) each citize口's preferences are represeI飞ted by a payoff function that assigns to 制ch terminal history
出e distance from the citizen's favorite position to the position of thεcandidate in
the set of winners whose position is 户对加st from her favorite position. Under the
other assumptions of the previous exercise, show that ir飞机rery subgame pεrfect
equilibrium in which no citizen's action is weakly dominated , the positio口 chosen
by every candidate who enters is the median of the citizens' favorite positions. 丁o
do so, first show that in any 吨uilibrium each candidate that enters is in the set
of winners. Then show that in any r、..Jash equilibrium of any voting subgame i民
which there are more than two candidates and not a11 candidates' positions are the
same, some candidate loses. (Ar吝ue that if a11 candidates tie for first place, some
citizen can increase her payoff by changing her vote.) Fina l1y, show that in any
subgame perfect equilibrium in which either only two candÎdates enter, or a11 candidates who enter choose the same position, every entering candidates chooses the
median of the citizens' favorìte positions.
7.4 lII ustration: committee dedsion-making
How does the procedure used by a commÎttee affect the decÎsÎon it makes? One approach to this question models a decision-making procedure as an extensive game
with perfect information and simultaneous moves in which there is a sequεnce of
ballots, in each of which the committee members vote simultaneously; the result
of each ballot determìnes the choices on thεnext ballot，。乙的íentually， the decision
to bemade
Fix a set of committee members and a set of altematives over which each member has strìct preferencεs (no member is indifferent between a到Y two alternatives).
Assume that the number of committee members is odd, to avoid ties in votes. If
there are two alternatives, thεsimplest committee procedurεis that in which the
members vote simultaneously fo 1' one of the alternatives. (We may interpret the
game in Section 2.9.3 as a model of this procedu肥) In the procedure illustrated
孜
I双
n
n
um
丑挝
1让it憔
t讼
ee first votes
in Figurε 218.1 , there are three a1ternatives, x, y, and z. The cωO悦∞
whether to choose x (option
If it votes to eliminate x, it subsequently votes between y and z.
In these procedures，拍ch vote is between two options. Such procedures are
called binary agendas. We may define a binary agenda with the aid of an auxiliary
one-play时 extεnsive game with perfect ir证。rma柱。n in which the set A(的 of ac幡
11
Chapter 7. Extensive Games with Perfect Inform提ion: 在xtensions and Discussion
218
/\大
X
y飞
z
y
Figur串 218.1 A
voting procedure, or "bìnary 在genda 气
tions following any nonterminal history h has 恼地 members， and the number of
terminal histories is 挝 least the number of albεrnatives. We associate with every
ter盯lÎnal history h of this auxiliary game a只 alternative 疚。'1) in such a way that
each alternative is associated with at least one terminal histor y.
ln the binary agenda associated with the auxi1i ary game
a11 players vote
simultaneously whenever the player in G takε5 an actio孔 The options 0ηthe bal栅
lot following the nonterminal history in which a majority of committee members
choose option a1 a t the start of the game, then option a2, and so on, are the members
ofthe set A 以 I
I 向。f actions of the player in G after the l出tory 价叮 a k ). The
alternative selected after thεterminal history in which the majority choices are a1 ,
. ., ak is the alternative 叫olr ejk)associated with(alf 翁精 , é) in G. For exam
pl坠
e， in the a江 xiliary one-person game that defines the structure of the agenda in
Figure 218 .1, the single player first chooses a or b; if she chooses a the game ends,
whereas if she chooses b, she then chooses betw能丑 c and d. The alt怡e衍rnati飞vε 艾 is 附a$户
soci览&必舱
t怡
ed 悦t出
h飞 the 怡
te
衍f
江
栩mi
• • •
W1让th (仲b ， 叫d)禽
Precisely, thεbinary agenda associated with the auxiliary game G is the εxten­
sive game with p时lect information and simultaneous moves defined as follows.
PJayers
τhe set of comn飞ittee members.
1
7花旨泣韧
r疗m
狞圳lÌ劝的
ÌIl门1臼
σaJ hí必st，拍
οrie盯5 A 捋
se叫
quen
引 Cε (扫v 飞
f
.叮， 泸
V k句内) 时
0f 配
act挝iorη1 pr防
of臼出
i吕le创5 (in 飞认-vh
i沁s 川
a li怯
S仗t of the players' votes) is a terminal history if and only ìf there is a
terminal history (a 1, . . . , ak ) of G such that for eve可 j = 0,.. ., k - 1, every
eleme咐。f v1+1 is a member of A(a 1 ，.. 寸的 (A( 的 ìf j = 的 and a majority of
the players' 在ctions in v j +1 are equal to aJ 十
• •
PJayer function For every n01飞terminal history h, P(h) is the set of a11 players
Act.胎
io
∞
归
n7衍5
For刊
ev
飞v沱
eer叮
ypla
句}严
f々台町r í and e 飞v吧咄
e
衍r巧
yr孜\0
∞
时
n1让t怆e
时
加
rr
口mi
1
J) 总
set of 缸
a ct加
i始
ons 沁
i s A(归a 飞
I .吁， 的
a'f吁)， wh
挝er，ε
把 (昕
.1 1 尸， . ., 的
al吁
i s th
阳e histo
倪ry of G ín
双\ which,
for all C, a''- is the action choser飞 by the majority of players in vf!..
• •
•
Prefl阴阳5 The rank each player a蚀gns 协 the terminal history (ü 1,. . ., I向 is
equal to the rank she assigns to the alternative í\' (a 1,.. ., ak) assoεiated with
7 .4 lII ustration: committee decision-making
21 苦
the terminal history (a 1, . . . , ak ) of G ín whích , for all j , af is the action chosen
bya majority of players in v1.
Ev时y binary agenda , like every voting subgame of the model in the previous
section , has many subgame perfect equilibria. In fact , in any bìnary agenda , every
alternative is the outcome of some subgame perfect equilìbrìum, because if, in ev
ery vote, every player votes for the same option, no player can affect the outcome
by changing her strategy. However, if we restrict attention to weakly undomìnated
strategíes, we greatly reduce the set of equilibria. As we saw before (Section 2.9 .3),
in a ballot with two options , a player's action of voting for the option she prefers
weakly dominates the action of voting for thεother option. Thus in a subgarne per
fect 叫uilibriurn of.a binary agenda in which every player's vote onevery ballot is
weakly undominated, on each ballot every player votes for the option that leads,
ultirnately (given the outcomes of the later ballots) , to the a1ternatìve she prefers.
The alternative associated wíth the terminal history generated by such a subgarne
perfect equilìbrìum ís said to be the outcome of srplzisticated voting.
Which altεrnatives are the outcorn础。f sophísticated voting iηbinaηr agendas?
Say that alternative x beats alternati飞'e y if a rnajority of comrnittee rnernbers prefer
x to y. An alternative that beats every other a1tεrnati飞'e is called a Condorcet winnel二
For any preferences, there is either one Condorcct winner or no Condorcet winner
(see Exerαse 75 .3).
First suppose that the players' preferences are such that sornεalternati\吃 say
x* , is a Condorcet winner. 1 claim that x* is the outcome of sophisticated voting
in every binary agenda. The argurne肘， using backward induction , is sirnple. First
consider a subgame of length 1 in which one option leads to the alternative x.. 1n
this subgame a rnajority of the players vote for the optìo巳 that leads to x* , becau优
a majority prefers x. to every oth衍 alternative， and each playεr's onIy weakly ur卜
dominated strategy is to vote for the option that leads to the alte主native she prefers.
Thusiηat least one subgarne of length 2, at least one option leads ultìmately to the
decision (given the players' votes in the subgarnes of length 1). ln this subgame,
bythεsame 盯gument as before, the winning opti∞ leads to
. Co到伽1Uing back附
ward, we conclude that at least one option on the first ballot leads ultimately to x*
and that consequently the winning option on this ballot leads to x*.
Thus if the players' preferences are such that a Condorcεtw
220
Chapter7. 主xtensi￥e Games with Perfect Information: Extensions and Discu莘莘ìon
thε陀 is a binary agenda for which that alternative is the outcome of sophisticated
votìng.
Which altern挝i飞res are the outcomes of sophisticated voting in binary agendas
飞vhen no alternati飞'e is a Condorcet v\'inner? Consider a corγ\mittee 认吐h arbitrary
preferences (not necessarily the ones considered in the previous paragraph) that
uses the agenda in Figure 218. 1. For x to be the outcome of sophisticated voting it
must beat the winner of y and 2. It may not beat both y and 2 directly, but it must
beat them both at least "indirectly": either x beats y beats 2 , or x beats 2 beats y.
Similarly, if Y or 2 is the outcome of sophisticated voting, then it must b岱 t both of
the other alternati飞'es at least indirectly.篝
p陀cisely， say that alt合rnative x indirèctly beats alternative y if for some k 2: 1
there are alternatives 均 f .., Uk such that 工 beats Ul , Uj beats
for j 士 1 ，，， ., k
1, and Uk beats y. Thεset of alternatives x such that x beats
other alternative
either directly or indirectly is called the top cycle set. (Note that if alternative x
beats any alternative indirectly, it beats at least one alternative directly.) If there is
a Condorcet winner, then the top cycle set consists of this single alternative If there
is no Condorcet winner, then the top cycle set contains more than 0日e alternative.
•
‘
①民CISE 220.1 (Top cycle set) A committee has three members.
a. Suppose that there are three alternatives, x , y , and 2 , and th批 one member
prefers x to y to 2 , another prefers y to z to 工， and the third prefers 2 to x to y.
Fiηd thεtop cycle se t.
b. Suppose that there are fo飞\r aiternati 飞!es， "iU, X , y , and 2 , and that one member
prefers w to z to x to y, one member prefers y to 日) to z to x , and one member
prefers x to y to zυto Z. Find the top cycle se t. Show, in particular, that z is in
the top cycle set even though all committee members prefer w 刷
Rephrasing my conclusion for the agenda in Figure 218.1 , if a到 a Iternative is the
outcome of sophisticated voting, then it is in the top cycle se t. The argument for
this conclusion extends to any binary agen缸. In every subgar肘， the outcome of
sopl飞isticated voting must beat the alternative that will be selected if it is rejected.
τhus by backward induction , the outcome of sophistic在ted voting in the whole
game must beat every other alternative
directly or indirectly: the outcome
of sophisticated voting in a口y binary agenda is in the top cyde set
Now consider a converse question: for any given alternative x in the top cycle
set is there a binary agenda for which x ís the outcome of sophisticated vo板ng?
The answer is affirmative. The idea behind the construction of an appropriate
agenda is illustrated by a simple exampl巳 Suppose that there are three alternatives , x , y, and z, and x beats y beats 乙 TheÌ1 the agenda in Figure 218.1 is one
for which x is the outcome of sophisticated voting. Now suppose there are two
addit捡到al alternativ侣， u and w, and x beats u beats 挝人在len we can construct a
larger agenda in which x is the outcome of sophìsticated votìng by replacing the
alternatìve x ìn Figure 218.1 with a subgame ìr飞 whìch a vote is taken for or against
x, and , if x ìs rejected, a vote is subsequently taken between u and u人If there are
7.5 lII ustration: exit from a declining industry
221
othεr chaíns through which x beats other alternatives, we can similarly add further
subgames.
? EXERCJSE 22 川 (Designing agendas) A commìttee has three members; there are
fi\'e alternatives. One member prefers x to y to v to (俨 to 2 , anotl飞机 prefers z to x to
v to w to y, and the third prefers y to z to w to v to x. Find the top cycle set, and for
each alternative (1 in the set design a binary agenda for which a is the outcome of
sophisticated voting. Convince yourself that for no binary agenda ìs the outcome
of sophìsticated votir飞g outside the top cycle set
? EXERCISE 221.2 (An agεnda that yields an undesirable outcome) Design a binary
agenda 如r the committee in Exercise 220.1b for which th心 outcome of sophisticated
votiηg is z (which.is worse for a11 committee members than 的.
In summar严(i) for any binary agenda , the alternati飞re generated by the subgame perfect eguilibrium in \ovhich no citizen's action ín any baUot is weakly domi口ated is in tl飞e top cycle se t, and (ii) for every altεrnative in the top cycle set , there
is J binary agenda for which that alternativεis gen巳 rated by the subgame perfect
equilibrium in which no citízen's action ín any ballot is weakly dominated. 1n particular, the extent to which the procedure used by a committee aftεcts its decision
depends on tÌ1 e nature of the members' preferences. At one e>二 tremε， for prefer偏
ences such that some alternative is a Condorcet winner, the agenda isγant.
At another extreme , for preferences for whích every alternative is in the top cycle
the agenda is ínstrumental in determiníng the decision. Further, for some
preferences there are agendas for which the subgame perfect equílibrium yields
an alternative that is unambiguously undesirable in the sense that there is another
alternative that a 1l εommitt较emembεrs pref但
7.5
IIlustration: exit from a declining industry
An industry currently consists of two firms, one with a large capacity, and one
with a small capacity. Demand for the firms' output is declining steadily over
tìm已亏盯len will the firms leave the industrγ? Which firm willleave first? 00 the
firms' financial resources affect the outcome? The analysis of a model that answers
these quεstìons íllustrates a use of backward induction more sophisticated than
tha t in the previous sections of this chapter.
7.5. 1 A model
Take tìme to be a discrete variable, starting in period 1. Denote by Pt ( Q) the market
prìce in period t wher飞 the firms' total output is Q, and assume that thìs price is
declirüng ov侄 ti黯e: for every value of Q, we have Ft+ 1 ( Q) < Pt (Q) for all t ~ 1
(See Figure 223.1.) We are interes讼d in the firms' decisions to exìt, rather than
their decìsions of how much to produce in 在e event they st咛部 the market, so
we assume that fìrm j's 0到ly decision is whether to produce some fixed output,
Chapter7. E就岳阳iveGam告swith P牵rfect Information: Extensions and Discussion
2革2
denoted kυor to produce no output. (You may think of ki as firm i's capacity.)
Once a firm has stopped production, it ca时\Ot start up again. Assume that k2 < k1
(firm 2 is smaller than firm 1) and that each firm's cost of produci吨 q units of
output JS cq ,
The following extensive game with simultaneous moves models this situation.
问。yers
The t以!o firms.
Termìnal historìes All sequences (X 1, . . ., X t ) for some t 主 1 ， where XS = (5tay ,
5tay) for 1 ::; s ::; t - 1 and Xt (Exit , Exit) (both firms exit in period t) , or
x s (5 tay, 5tay) for all s wi th 1 ::; s ::; r - 1 for some 1', Xr (5 tay, Exit)
or (Exit , 5ta抖， XS = 5tay for a11 s with r + 1 三 S 三
1 ， and Xt = Exit
(one firm exits in period r and the other exits in period 叶， and all infinite
sequences (X 1, X2,.. .), where X
(5tay , 5tay) for a11 r (neither firm ever
,.
εxits).
Player function P (11 )口
{1 ， 2} after any history h in which neither firm has
1 aft<怡
白r any hi怒st如
ε
orηy 11 in
只1W
咄
h山
扰
icch飞 O
∞
nl马y 良
flrII
口
盯Eτ12 h飞.1as
部se
础机
汛x川(1此t怡
时d
e
比叫， a旦
P(扭圳
的) 口 2 aft挝
h
eran飞y history 11 红
i n wÌ山hon句 firm 1 has 仪
i
eX
刘
机i让ted
盯
e
xi
对it校
叫
ed
比 町
d
p (11
的) 江
Actions Whenever a firm moves, its set of actions is {5tny , Exi斗"
Pn芒ferences
Each fìrm's preferences are represented by a payoff functíon that
associates wÌth each terminal history the firm's total profit, where the profit
of firm i
1, 2) in period t is (Pt (kí) - c )k i if the other firm has exited and
( P， (k 1 十 k2 ) … c )kj if the other 在nnhas notεxited
λ 5.2
Subgame perfect εquilibrium
In a period in which Pt (k i ) < c, firm i makes a loss even if it is the only firm
remaining 捏1e market price for its output ìs less than its unit cost). Denote by ti
the last period in which firm i is profitable if it is the only firm in the market. That
is , ti lS the largest value of t for which P， (k i ) 主 c. (Refer to Figure 223.1.) B号cause
k1 > k2 , we 且ave t1 ::;句: the time at whìch the large firm becomes unprofitable as
a loner Ìs no later than the time at which the sma11 firm becomes unprofitable as a
loner.
The game has an infir飞ite horízon, but after period ti firm i's profit is negative
even if it is the only firm remaining in the market. Thus if fìrm i is in the market
in any period after tj , it chooses Exit in that p时iod in every subgame perfect equilibrium. ln partìcular, both firms choose Exit in every period after t2. We can use
backward induction from period t2 to find the firms' subgame p栓fect 吨uilibrium
actions in earlier periods.
If firm 1 (t检 larger firm) is in the market in any period from tl 0口， it should
exit, regardless of whether firm 2 is still operating. As a cons吨uence， if firm 2 is
still operating in any period from tl 十 1 to t2 it should stay: firm 1 wìll exit in any
such period , and in Îts absencεfirm 2'5 pr。如 is positive.
7.5 lII ustration: 卧xitfrom a dedining ìndustry
223
PS
主2
k1
白
The ínverse demand curves in a decliníng índustry. ln thís ex在 mpl哇， 11 (the last períod ín
whích firm 1 is profítable íf it is the only fírm ín the market) is 2, and 12 is 4.
的 gure223.1
So far we have concluded that in every subgarne perfect equilibriurn, firrn 1's
strategy is to exit in eve可 period frorn t1 + 1 on if it has not already done so,
and fìrrn 2' s strategy is to exit in every period frorn t2 + 1 on if it has not already
done so.
Now consider period t1 , the last perìod in whìch firrn
profit is positive if
firrn 2 is absen t. If firrn 2 exits , its profit frorn then on is zero. If it stays and firm 1
exits , then it earns a profit frorn period t1 to period t2 , after wh比如 ìt leaves. If
both firms stay, firm 2 sustains a loss in period t1 but earns a profit in the subsequent periods up to t2 , because in every subgame perfect 叫uilibrìum firrn 1 exits
in period t1 + 1. Thus if firrn 2'g one-period loss in period t1 when firm 1 stays in
that period is less than the surn of its pro自ts frorn period t1 十 1 on, then regardless
of whether 卢rm 1 stays or exits in period tl , firrn 2 stays in every subgame perfect
equilibrium. In period t1 + 1, when firrn 1 is absent frorn the industr予 the prìce is
relatìvely high, so that the assumption that firrn
one-period loss is less than its
subse气uent rnultiperiod profit is valid for a signi在cant range of parameters. From
now on, 1 assume that this condition holds.
1片段 conclude that in e飞rery subgame
equilibrium 自rm 2 stays in period t 1, so that 亘古m 1 optimally exìts (It defir让 tely exits in the next periúd , and
ìf it stays in period t1 ìt makes a loss, because fì口口 2 stays.)
时ow continue to work backw盯d‘If在rm2stays ìr飞 p但iod h 一 1 it earns a profit
ìn periods h 白rough 位， because in every subgame perfect equi1ibrium firrn 1 exits
in period t1. It may make a 1055 in period t1 … 1 世如m 1 5tays in 白at period) ,
but this loss is less than the los5 it makes in period 乌拉 the company of firm 1,
which wεhave aS5umed is outweighed by its subse气uent profi t. Thus 主egardless
of firm 1's actio抗 in period h … 1, firrn
best action is to stay in that perìod. If
t2 < 扫一 1 ， then 位m 1 makes a los5 in period t1 - 1 in the cornpany of 豆rrn 乙 and
so should exit.
‘
224
Ch矗 pter7. Extensi￥e
Games with P融 rfect Information: Extensions 萌nd DiscussÎon
Thesamεlogic applies to a11 periods back to the first period in which the 如ms
cannot profitably coexist in the industry: in every such period , in evεry subgame
perfect equilibrium firm 1 exits if it has not already donεso. Denote by to the last
period 如 which both firms can profitably coexist in the iηdustry: that is , to is the
largest value of t for which Pt (k 1 十 k 2 ) 主 C
We conclude that if firm
loss i到 period h when both firms are active is less
than the sum of its profits in periods t 1 十 1 through t2 when it alone is active , then
the game has a unique subgame perfect 吨uilibriur孔 in which the large fìrm exits
in period to 十 1 ， the first period in which both firms cannot profìtably coexist in
the industry, and the small firm continues operating until period t2 , after which it
alone becomes unprofitable.
( EXERCISE 224 .1 (Exit from a declining industry) Assume that c 口 10， k1 = 始，
k2 20, and Pt(Q) = 100 … t .- Q for a11 v明lues of t and Q for whid飞 100 - t - Q >
0, otherwise Pt ( ♀) = O. Fi nd the values of t1 and t2 and check whether 位m 2's
loss in period t 1 when both firms are acti飞'e is less than the sum of its profits in
periods t1 十 1 through t2 认lhen it alone is acti飞!e.
7.5.3
The effiθct of Q constraint on fjrm 2 :S debt
When the firms fo11ow their subgame perfect 叫ui1ibrium strategies , each firm's
profit is nonnegative in every period. 日ov何飞rer， the equilibrium dεpends on firm
2's ability to go into deb t. Fi rm 2's strategy ca11s for it to stay in the market if
firm 1, contrary to its strategy, does not exit in the first period in which thεmarket
cannot profitably sustain both firms.τhis feature of firm 2's strategy is essential to
the equilibrìum. If such a deviation by firm 1 induces firm 2 to exit, then firm 1's
strategy of exitìr飞g may not be optimal, and the equilibrium may consequently fall
apar t.
Consider an extreme case, in which firm 2 can never 伊拉to deb t. We can
incorporate this assumption into the model by making finn 2's payoff a large negative number for any terminal history in which its profit in any period is negative.
(The size of firm 2's profit depends on the contemporar飞eous actiOI飞 of 位m 1, so
we cannot easily incorporate the assumption by modifying the choices available
to firm 之) Consider a history in which firm 1 stays ìn the market after the last
period ìn which the market can profitably sustain both firms. After suεh a history
firm 2's best action is no longer to stay: if it does so its profit is negati飞吮 whereas if
it exits its profit is zero.τhus if firm 1 deviates from its equilibrium strategy in the
absence of a borrowing cor飞straint for firm 2, and stays in the first period in which
it ìs supposed to exit , then firm 2 optimally exits, and firm 1 reaps positive profits
for several periods, as the lone firm in the marke t. Consequently in this case firm 2 .
exits fir吭位m 1 stays in the market until period h.
How much debt does firm 2 need to be able to bear for the game to have a subgame perfect equilibrium in which firm 1 exits in period to and firm 2 stays until
period t2? Suppose that firm 2 can sustain losses from period to 十 1 through period
to + k, but no longer, when both firms stay in the market. For firm 1 to optimally
7.6 Allowing for exogenous un在ertainty
225
?
Profit
{o 十 l
J甘卡 k
O
11
Period •
Figure 225.-' Firm 1's profits starting ín period 10 十 1 when fírm 2 st在ys ín the market untíl períod t] 十支
部1d firm 1 stays until period 1]-
exit in p衍iod to + 1, the consequence of its st在 ying in the market must be that firm 2
a1so stays. Suppose that firm 2's strategy is to stay through p衍iod to 十左， but no
longer, ìf firm 1 does so曙Which strategy ìs best for firm 1 ìn the subgame starting
in period to 十 1? If it exits, its payoff is zero. If it stays through period 与十 k， ìts
payoff is negative (it makes a loss in every period). If it stays beyond period to 十 k
(飞，yhen firm 2 exits), it should stay until period t1 , when its payoff is thεsum of
profits that are negative from perìod to + 1 through period to 十 k and ther飞 posi静
tÌve through period t1 . (See Figure 225. 1.) If this payoff is posîtive it should stay
through period t1; otherwise it should exit immediately.
We conclude that for firm 1 toεxit in P衍iod to + 1, the period until which
firm 2 can sustain losses , which 1 have denoted to + k, must be large enough that
firm 1's total profit is nonpositive from period to 十 1 through period t1 if it shares
the market with firm 2 until period to 十 k and then has the market to itself. This
飞ralue of k determines the debt that firm 2 must be able to accumulate: the requisite
debt 吨uals its totalloss when it remains in the market with firm 1 from period to 十
1 through period to 十 k.
? EXERCISE 225.1 (Effect of borrowing constraÌI飞t of firms' exit decisions in de c1 ining
industry) Under the 邵阳mptions of Exercise 224.1 , how much debt does firm 2
1飞机d to be able to bear for the subgame perfect equ过ibrium outcome ir飞 theabsence
of a debt constraint to remain a subgame perfect equilibrium outcome?
7.6
Allowing for exogenous unceriainty
λ6.1
General model
机le model of an extensive game with perfect information (with or without simul唰
taneous moves) does not allow random events to occur during the co旧时 of play.
Chapter 7. Extensive Games wÎth Perfect Information: 主xtensions and Discussion
226
However, we ca如l easily extend the model to cover such situations. The definition
of an extensive game with perfect infonnation and chance moves is a varia址。t
thede位飞ition of an extensive ga口le with perfect information (155.1) in which
• the player function assigns "chance飞 rather than a set of players , to some
histories
• the probabilities that chance uses after any such history arεspecified
• the playε骂I preferences are defined over the set of lotteries over terminal
histories (rather than simply over the set of terminal histories).
(We may similarly add chance moves to an εxtensive game with perfect information and simultaneous moves by modifying Definition 206. 1.)丁'0 keep the analysis
simple, assume that the random event after any given history is independent of
the random event after any other history. (That is, the realization of any random
event is not affected by the realìzation of any other random even t.)
τhe definitìon of a player's strategy remains the samεas before. The outcome
of a strategy profile is now a probability distrìbution over terminal histories.τhe
definition of subgame perfect equilibrium remains the same as before.
Ex 人 MPLE 226.1 (Extensive game with ζhance moves) Consider a situation involv-
ing two players in which player 1 first chooses A or B. If she chooses A the game
ends, with (Bernoulli) payoffs 性， 1). If she chooses B, then wìth probabìlity the
game ends , with payoffs (3 , 0) , and wìth probabìlìty player 2 伊ts to choose between which yields payoffs (0 , 1) and D, which yields payoffs (1 ，的. An extensive game with perfect information and chance moves that models this situation is
shown in Fig时e 226.1. The label c denotes chance; the number beside each action
of chance is the probab i1i ty with which that action is choser飞.
We may use backward induction to find the subgame perfect equilibria of this
game. Ir飞 any equilibrium player 2 chooses C. Now consider the consequences of
player l's actìons. If she chooses A , then she obtaÍI飞s the payoff 1. If she chooses B,
怕1 she obtains 3 with probabili时 and 0 with probability ~， yielding an expected
payoff of ~丁hus the game has a unique subgame pe白ct eψilibrium r in which
player 1 choose B and player 2 chooses C.
i
i
J
1
A
Y :Z
1, 1
c
2
2
2
3，。
C
。， 1
事 igure 226.1
口
1, 0
人n extensive game with p佼佼佼 information 在nd chanζe moves.
The label c denotes
d船α; the numb{'f beside each action of chance is the probability with 耶hìch that action is chosen.
7.6 Allowing for exogenous uncertainty
227
( EXERCJSE 227.1 (Variant of ultimatum game with equit y叫 onscious pl在yers) Con明
sider a variant of the game in Exercíse 183.4 in which /)1 泣。， and thεperson 2
whom person 1 faces ís drawn randomly from a population in which the fraction
p have ß2 = 0 and the remaining fraction 1 … p have 卢2 = 1. When making her
offer, person 1 knmvs only that her opponen t' s characteristic is ß2 = 0 with probabilìty p and ß2 口 1 with probability 1 … p. Model this situation as an extεnsi飞re
game with perfect information and chance moves in which person 1 makes an offer, then chance determines the type of person 2, and finally person 2 accepts or
rej仅 ts person 1's off机 Find the subgame perfect equìlibria of this g在 me. (Use the
fact that if ß2 口 0 ， then in any subgame perfect equilibrium of the game in Exercisε183 .4， person 2 accepts a11 offers x > 0 and may accept or 叫ect the offer 0, and
if 卢2 = 1, then 51飞~ accepts a11 offers x > ~， may accept or 叫ect th ", offer ~， and
时岱 all offers x < ~.) Are there a盯 val则 of p for which an offer is 时c时 in
εquilibrium?
? EXERCJSE 227.2 (Fìrm… union bargaining) A firm knows thε 在mount by which its
re飞'enue exceeds its outlays on plant and equipment, but a union 'Vvith which it
is bargaíníng does no t. Thís "surpJus" , which is to be divided behveen the firm
and the union , ís H wíth probability p and L < H with probabílíty 1 … p. The
bargaining procedure is that of the ultímatum game: the union makes a demand ,
which the firm either accepts or rejects. We may model this sìtuation as an extensive game in which , between the union's demand and the firm's response, a move
of chance determines whether the surplus is H or L. If the size of the surplus is
:;: and the union's demand is x , then the firm's (Bernoulli) payoff is :;: x and the
union's (Bemoullí) payoff is x if the finn accepts the union's demand and each
play仔 's payoff is 0 (there is a strìke) ìf the firm 叫 ects the union's demand Find
thεsubgame perf，εct equilibría of this game for each possible value of p. Find the
probability of a strìke for εach 叫uilibrìum.
‘
? EXERCISE 227 .3 (S叫uential duel) 如 a sequential duel , two people altemately have
the opportunity to shoot each other; each has an infinite supply ofbullets. On each
of her turns, a person 血ay shoot or refraiηfrom doing so. Each of perso们 's shots
hits (and kills) its ìntended target with probabilíty PI (independently of whether
any other shots hit their targets). (lf you p陀fer to think about a
violent sit俨
uatiOTI , intεrpret the players as political candidates who alternately may launch
attacks, which may not be successfuI , against each other.) Each person cares only
about her probability of survival (not about the other person's survi飞叫). Model
this sítuation as an extensive game with perfect information and chance moves.
Sh Qì,V that the strategy pairs in which neither perso民 e飞'er shoots. and in which
each person always shoots are both subgame perfect equilibria. (泣。te that the
game does not have a finite horizon , so backward induction cannot be used.)
?! EXERCIS 犯 227.4 (Sequential truel) Each of persons A , B, and C has a gun contain-
ing a single bullet. Each person, as long as she is alive, may shoot 就 any surviving
person. First A can shoot , then B (if still alive) , then C (if still alive). (As in the
228
Chapter7. 主xtensive Game军 with
Perfect Information: Extension量 andDi章cus主ion
previous exercise, you may interpret the pl玛rers as political candidates. In this
exercise , each εandidate has a budget suffìcier飞t to launch a negative campaign to
discredit exactly one of its rivals.) Denote by Pi the probability that player i hits
her intended target; assume that 0 < Pí < 1. Assume that each player wishes
to maximize her probabílity of survival; among outcomes in which her survival
probability is the same , she wants the danger posed by any other survivors to be
as small as possible. (The last assumption is intended to capture the idea that there
is some chance that further rounds of shooting may occur, though the possibility of
such rounds is not incorporated explicitly into the game.) Model this situa tÏ on as
an extensive game with perfect information and chance moves. (Draw a diagram.
Note that the subgames following histories in which A misses her intendεd target
are the same.) Find the subgame perfect 叫uilibria of the game. (Conside主 only
cases in which PA , PB , and pc are a11 di {feren t.) Explain the logic behind A's equilibrium action. Show that "weakness is strength" for C: she is b的ter off if pc < PB
than if pc >
Now consider the variant in which each player, on her turn , has the additional
option of shooting into the air (in which case she uses a bullet but doesηot hit
anyone). Find the subgame perfect equìlibrìa of thìs game when PA < PB' Explain
the logic behind A' s 吨uilibrium action.
苦 EXERCI5E 228 .1 (Cohesion in legislatures) The following pair of games is designed
to study the in飞plications of different legislative procedures for the cohesion of a
governing coaHtion. In both gamεs a legislature consists of three members. Initially a governing coalition , consisting of two of the legislators , is given‘百lere
are two periods. At the start of each pe玄iod a member of the governìng coalition
is randomly 加州 (ì.e. each 均
le
吨
e
μgi
伊
isl
归
la
时阳
1忱t
bill, which is a partition of one unit of payoff between the three legislators.η1en
the legislators simultaneously cast votes; each legislator votes either for or against
the bill. If hvo or more legislators vote for the bill, it is accepted. Otherwise the
course of events differs between the two games. In a game that models the current
u .s. legislature, 陀
re
叫je
衍ct挝ion of a bill in pε
衍r刘iod t leads to a gi飞ve
创n飞丁哈. partition ♂
d t of the
pi挖e， 飞w
伦
d
电亨
f企捡h飞恕础
ε挝r陀
悦e 0 < 叫d; <
f扣
orμ才.i 工 1, 2， 旦3; th
怆e go咐nll吨 coal技it往ion (争t阳
h1涩es臼etfro拙 w
咄
挝
h让世州
hid
id
忧
the pr陀o宇
poωS衍 of a bill is drawr才 remains the same in period 2 following a rejection
in period 1. In a game that models the current U.K. legislature, rejectio日 of a bill
brings dovvn thεgovernment; a new governing coalìtion is determined randomly,
ar时 no legislator receÏ\咄 any payoff i到 that period. Specify each game precisely
and find its subgame perfect equilibrium outcomes. Study the degree to which the
governing coalition is cohesive (i.e. all its members vote in the same way)
i
λ6.2
Using chance mov，盯 to modeJ mistakes
A 伊me with chance moves may be used to model the possibility that players make
mìstakes. Suppo使， for example, that two people simultaneously choose actions.
7.6 Allowin雪 for exogenous uncertai戳ty
229
d
QUM
内υnυ
A川
FF
lAυAHV
, itinvfrr
I10
Figure 229.1 The players' 如rnoulli payoff旨 to the four pairs of actìons in the game studied ín Sec~
tion 7.6.2.
Each person may ch∞se either A or B. Absent the possìbility of mistakes, suppose
that the situation Ìs modeled by the strategìc game in Figure 229 .1, in which the
numbers in the boxes are Bernoulli payoffs.τhis game has two Nash equilibria,
(A , A) and (B , B).
Now suppose that 制ch person may make a mistake. 亏W饥it出
hpro
伪
ba
抽
bi山
lity 1 一 p
阶i>
i tl盯巾n 伽颐 byp
伊erso川s thεωO附 she
冶川
e们in
i附阳
t怡e盹
i让t 怡
ì s he
凹r other action. We can model this situatìon as the following extensive game
with perfect information, simultaneous moves, and chance mo飞喘.
Pfayers The two people.
元阳rmi，的门 afh山
i必st臼
oriε
巳s
Al丑Is
附
叫
e
q阳
Z are al1 eit也he盯r A 0ωr 玩B， 妇
in the history (( W
々吼， X) ， γ， Z) playe
仔r 1 chooses W,
player 2 chooses X, ar飞d then chance choosesγfor player 1 and Z for player 2.
Pfayerfunction P( 的= {1 , 2} (both players move sìmultaneously at the start
of the game) ，阳通 P(W， X) 正汽(目， X) ， γ) = {c} (chance moves twice after
the players have acted, first selecting player 1's action and then player 2's
ac tÌ on).
Actions The set of actions available to each player at the start of the game, and
to chance at each of ìts mo问s， is{A ， B}
Chance probabifities After any history (W, X) , chance chooses W with probabil-
ity 1
Pl and player 1's other action with probabìlity Pl. After any history
((W， 抖， γ) ， chance chooses X with probabilí付 1 - P2 and player 2's other
action with probability P2'
Preferences Each player's preferences are represented by the expected value
of a Bernoullí payoff functìon that assigns 1 to any history ({W, X) , A , A)
(in which chance chooses the action A for each player) , and 0 to any other
history.
在le players in this game move simultaneously, so that the subgame perfect
equilibria of the game are ìts Nash e气功libria. To 位ld the Nash 叫uilibrìa; we
construct the strategic form of the game. Suppose that each player chooses the
action A. Then the outcome is (A , A) with probability (1 … Pl) (1 P2) (the probab山ty that neither player makes a mistake).τhus each player's expεcted payoff is
(1 - Pl) 口… P2)' Similarly, if player 1 chooses A and player 2 chωses B then the
•
I
Chapter 7. Extensive Games with P母 rfect Information: Extensions and Discu部ion
230
ZYF
B
气4
问ys
、‘s''
叮E'A
fit、码，"
飞毡'ayd
ι
一问门户
r 、
穹岛
'Aτi
吁
/Ia、 PB
VFdut
扣一
司))
…一
P
匀'且
U
nv''抖ys
吁tA
扣
)1
((
、飞
A川口
,,
•• 4tt"
(1 - PdP21 (1
Pl)P2
PIP2 , PIP2
Figure 230.1 The strategic f0r m of fh巳 extensive game with chance moves that models the situ在 tío到 in
which with probability pj each player i in the game in Figu陀 229.1 chooses an action different from the
one she ìntends.
outcome is (A , A) with probability (1 - PdP2 (the probability that player 1 does
not make a mistake , whereas player 2 does). Making similar computations for thε
other two cases yields the strategic form ìn Figure 230 .1.
For Pl = 阳工 0， this game is the same as the original game (Figure 229习 ;it
has two Nash equilibr旬， (A , A) and (B , B). If at least one of the probabilities is
positi\'e then only (A , A) is a ]'、\lash equilibrium: 证
if 扣
P i > 0, 出
t h飞e创n
创飞 (口1 一 巧
ψ
p斗杜，)
7扒
Pi
7飞i > PjPi
(也gi认陀
veI
呛en飞 性白.ìat 俯
each飞 pro
时ba丛1过bi
抬也 丑li忧ty 沁
is 沁
1 e仍S附
仰s 白
tharη飞 ; ) 丁h飞a挝t 扫
i s， on
口1 由
t白
hε original game is们r‘O
巾
b飞us仗t 臼
t ot仙
he po
岱S部灿$创
si1孔b升iil扫it号y that the players make sma l1 mistakes.
In the original gan飞e， each player's action B is \,veakly don飞ìnated (Definition
46.1). Introducing the possibilìty of mistakes captures the fragility of the equilib幽
rium 侈， 的: B is optimal for a player only if she is absolutely certain th反 the otl附
player will choose B also. The slightest chance that the other player 认lill choo优 A
is enough to make A unambiguously tl飞εbest d飞。ice.
Wemay use thεidea that an equilibrium should survive when thεplayers may
makεsmall mistakes to discriminate among the Nash equilibria of ar飞y strategic
game. For two-pl町er games in 叭'hich each player has finitely many actio邸， the
叫uìlibria that satisfy this req飞lÌremel飞t are precisely those in which no player's ac俨
tion is weakly dominated. For games with more than two players , no equilíbrium
ìn which any player's actior飞 is weakly dominated satisfies the requirement, but
叫 uilibria in whìch no player's action is weakly dominated may faìl to satisfy the
requireme肘， as the following exercise shows.
① EXERClS主 230.1 (如 ash equìlibria when players may make mistakes) Consider the
three-player game in Figure 230 .2. Show that (A , A , A) is a Nash equilìbrìum in
which no player's action is weakly dominated. Now modify the gamεbyassum­
ing that the outcom巳 of any player i's choosing an action X is that X occurs with
probability 1 - Pi and thεplayer's other actìon occurs 飞vith probability Pì > O.
5ho\l气J 加t (A， A ， A) 必 nota 民ash equilibrium of the modified game when Pi < ~
for i 口 1 ， 2， 3.
A
B
A
B
1, 1, 1
1, 1, 1
。， 0， 1
1, 0 , 1
A
A
A 0, 1, 0
B 1, 1, 0
B
1, 0, 0
0, 0, 0
B
Figure 230.2 A three-player strategìc game ìn which each player has tvγo actions. Player 1 chooses a
row, player 2 chooses a column, ilnd player 3 chooses a table
7.7 Discussion: subgame perfect equilibrium and backw晶rd induction
7.7
231
Discussion: subgame perfect equilibrium and backward induction
Some of the sìtuations we have studied do not fit well i址。 the id归lized setting
for the steady state interpretation of a subgamθperfect equilibrium discussed in
Section 5.4.4, in which each player repeatedly engages in the same game with a variety of randomly selected opponεnts. 1n some cases an alternative interpretation
fits better: 拍ch player deduces her optimal strategy from an analysis of the other
players' best actions , gi飞.ren her knowledge of their preferences. Here 1 discuss a
difficulty wÎth this interpretation.
Consid肝 the game in Figure 23 1.1, in which play盯 1 moves both before and
after player 2布 This game has a unique subgame perfect equilibrium, in which
player γs strategy is (B , F) and player 2's strategy is C. Consider player 2's analysis of this game. If she deduces that the only rational action for player 1 at the start
of the game is B, then what should she conclude if player 1 chooses A? It seems
that she must conclude that something has "gone wrong l!: perhaps player 1 has
madea 飞飞istake'二 or she misundεrstands player 1's preferences , or player 1 is not
rational
If she is convinced that player 1 simply made a mistal饨， then hεr analysis of the
rest of thεgamεshould not be affected. However, if player 1's move induces her to
doubt player 1'8 motivatìon, she may need to 1时onsider her analysis of the rest of
the game. Suppose, for exarr飞ple， that A and E model sim i1 ar actions; specífically,
suppose that they both correspond to player 1's rr飞oving left，飞吗I hereas B and F both
involve her moving righ t. Then player 1's choice of A at the start of the game may
make player 2 wonder whether player 1 confuses left and right, and therefore may
choose E after the history (A, C). If so, player 2 sho时d choose D rather than C
after playεr 1 chooses A , giving player 1 an incentive to choose A rather than B at
the start of the game.
Tl飞e next two examples are richer games that more strikingly manifest the difficulty with the alternatìve interpretation of subgame perfect 叫uilibrium. The first
example is an extension of the entry game in Figure 156 .1.
EXAMPLE 23 1.1 (Chain-store game) A chail1 stor心。perates in K markets. In each
market a single challenger must decide wh仕her to compete with i t. Thεchal-
1
E
0, 0
Figur哇 231.1
1, 2
An extξnsive game ín wruch player 1 movωboth before 在nd after player 2.
Chapter 7. Extensive Games with Perfect Information: 缸tensions and Di草棚 ssion
232
Cha llenger k
111
chain store
Ac伊lesce 八 Fight
2, 1
1, 2
0, 0
Figur费 232.1 The structure of the players'εhoices in 淤arket k ín the chain-store game. The 盗 rstnumber
in each pair is challenger k's profit and the second number is the chain sto陀、 profit.
lengers make their decisions sequ仅ltially. If any challenger enters, the cham storε
may acquiesce to ìts presence (A) or fight it (F). Thus in each period k the outcome
is either Out (challenger k does not ent时， (Jn, A) (challenger k enters and the chain
store acquiesce吟， or 仰，到 (cha11enger k enters and is fought). When takmg an
action, any cha11e盯在er knows a11 the actions previously chosen. Thεprofits of cha1lenger k and the chain store in market k are shown in Figure 232.1 (cf. Figure 156.1);
the chain store's profit in the whole game is the sum of its profits ìn the K markets.
Weca口 model this situation as the followÌl飞g extensive game with perfect infor盯lation.
Players The chain store and the K challengers.
Terminal histories The set of a11 叫出附s (el ，'''， ω ， where each ej is either
O风 (1n， A) , or (I n, F)
Player function The cham store is assigr飞ed to every history that ends with 111,
challenger 1 is assigned to the empty histoτy， and challenger k (for k 2, . .寸
K) is assigned to every history (肉，.. ., ek-l) , wher巳 each ej is either Out ,
(I n, A) , or (I11 ,F).
Preferences Each player's prefe陀nces are represented by its profits.
Thisg在me has a finite horizon, so we may find its subgame perfect equilibria
by using backward induction. Every subgamεat the start of which challenger K
moves 陀sembles the game in Figure 232 .1 for k 主C; it differs on1y iηthat the chain
store's pro缸 after each of the three terminal histories is greater by an amount 吨ual
to its profit in the previous K - 1 markets. Thus ÎI飞 a subgame perfect equilibrium
challenger K chooses 1n and the incumbεnt chooses A in maτket K.
Now consider the subgame faced by challenger K 一1. We know that the outcome ìn market K is ìndependent of the actions of challenger K - 1 and the chain
store in market K 1: whatever they do, challeng肝 K enters and the chain store acquiesces to its entry. Thus the chain store should choose its action in market K 1
on the basis of its payoffs m that market alone. We conclude that the chair飞 store's
optimal action in market K - 1 is A , and challenger K ← γs optimal action is 1凡
We have now concluded that in any subgame perfect equilibrium, the outcome
in each of the last two markets is (II1, A) , regardless of the histor予 Continuing to
work backward to the start of the game we see that the game has a unique subgame
7.7Dis创 ssion: subg矗 meperfect 串quilibrium
and backward induction
233
perfect equilibrium , in which every challenger ent时s and the chain sto陀 always
acqm部ces to ent吓
CD EXERCISE 233.1 (Nash 叫uilibria of chain-store game) Find the set of r气Jash equilibrium outcomes of the game for an arbitrary value of K. (First thìnk about the case
K 1, then generalize your analysis.)
号 EXE 汶口臼 233.2 (Subgame perfect equilibrium of chain-store game)
Consider the
following strategy pair in the game for K 100. For k 1, . . ., 90, challenger k
stays out after any history in which every previous challenger that entered was
fought (or no challenger entered) , and otherwise enters; challengers 91 through
100 er飞ter.τhe chain store fights every challenger up to challεnger 90 that enters
after a history in which it fought every challenger that entered (or 到o challenger
entered) , acquiesces to any of these challengers entering after any other history.
and acquiesces to challengers 91 through 100 regardless of thεhistory. Find the
players' payoffs in this strategy pair. Show that the strategy pair is not a subgame
perfect equilibrium: find a player who can increase her payoff in some subgame.
By how much can the deviant increase its payoff?
Suppose that K 100. You are in chargεof challenger 21. You observe, contrary to th巳 subgame perfect equilibrium, that every previous challengerεntered
and that the chain store fought each one. Wh at should you do? According to
the subgam巳 perfect 吨uilibrium， the chain store will acquiesce to your entry. But
should you really regard the chain store's 19 previous decisions as "mistakes"?
You might instead read some logic into the chain store's deliberately fighting the
first 20 entrants: i f, by doing so , ìt persuades more than 20 of the remaining chal脚
lengers to stay out, then its profit will be higher than it is in the subgame perfect
equilibrium. 在lat is, you may imagine that the chain store's aggressive behavior
in the 位rlier markets is an attempt to establish a reputation for being a figl飞ter，
which, if successful, will make it better off. By such reasoning you may conclude
that your best strategy is to stay ou t.
τhus， a deviation from the subgame perfect 叫uilibrium by the chain store in
which it engages in a long series of fights may not be dismissed by challengers as
a series of mistakes, but rather may cause them to doubt the chain store's future
behavìor. This doubt may lead a challenger who ìs followed by enough future
challengers to
毛 EXAMPLE 233.3 (Centipede game)
The two-player game in Figurε234.1 is known
as a "centipede game n because of its shape. (Tl飞e game , like the arthropod , may
have fewer than 100 legs.) The players move alternately; on each move a play凹
can stop the game (句。r continue (C). On any move, a player is better off stopping
the game than continuing if the other player stops ìmmediately afterward, but is
worse off stopping than continuing if the other player continues , regardless of the
subsequent actions. After k periods , the game ends.
τhis game has a 在rute horizon, so we m句1 find its subgame perfect equilibria
by using backward induction. 在le last player to move prefers to stop the game
234
ζhapter 7. Extensive Games with Perfect Information: 军xtension幸矗 nd Discussion
8, 6
1s
Is
Is
Is
Is
Is
2, 0
1, 3
4, 2
3, 5
6, 4
5, 7
Figure 234.1 A six-period centipede game
than to continue. Given this player's action, the player who moves before her
also prefers to stop the game than to continue. Working backward, we condude
that the game has a unique subgame perfect e气uilibrium， in which each player's
strategy is to stop the game whenever it is her turn to movε. The outcome is that
player 1 stops the gamεimmediately.
? EXERcrSE 234.1 (Nash 吨uilibria of the centipede game) Show that the outcome
of every 时ash equilibrium of this game is the same as the outcome of the unique
S飞lbgame perfect eq过ilibrium (i 忍 pl可er 1 stops the game immediately).
The logic that in the onl 1' steady state player 1 stops the game immediately is
unassailable Yet this pattern of behavior is intuitively unappealing , especially íf
the number k of pcriods is large. The optimality of player 1's choosir飞g to stop
thεgame depends on her believing that if she continues , then player 2 will stop
the game in period 2. Further, player 2's decision to stop the game in period 2
depends on her belie飞ring that if she continues, then player 1 will stop the game
in period 3. Each dεcision to stop the game is based on similar considerations.
Consider a player who has to choose an action in period 21 of a 100-period game,
after each player has continued in the first 20 periods. Is she lìkely to consider the
first 20 decisions-half of which were hers-"mistakes"? Or will thesεdedsions
induce her to doubt that the other player v飞rill stop the game in th校 next period?
These questions have no easy answers; some experimental evidence is discussed
in the accompanyìng box.
‘
Exp忘 RI l\ 1ENTAL EVIDENC 巨 ON T 日在 CEN Tl PEDE GAME
1n experiments conducted in the United States in 1989，创 ch of 58 student sub仔he payoff
of pÌayer 1 is the top amount in each pai斗 Each subject played the game 9 or
10 times, faCÌI飞g a different opponent each time; 担 each play of the game, each
subject had previously played the same number of games. Each subject knew 如
advance how many times she would play the game, and knew that she would not
play against the same opponent more than once拿If each subject cared 0口lyabout
her own monηtary payoff, the game induced by the experiment was a six-period
jεcts played the game shown below (McKelvey and Palfrey 1992)
centiped巳
7.7 Dìscussion: subgame perfect equilibrium and backward induction
1
$0 .4 0
$0 .1 0
C
2
C
1
2
C
$1. 60
$0 .4 0
$0.20
$0.80
C
$0.80
$3.20
C
$6.40
$1. 60
235
2
C
$25.60
$6 .4 0
$3 ‘ 20
$12.80
The fraction of plays of tÌ飞e game that ended i口 each period is shown in the
graph below哑 (A game is counted as ending in period 7 if the last player to move
chose C. Th.e graph is computed from MεKelvey and Palfrey 1992, Table IIIA.)
Results arεbroken down according to the players' experience (first 5 rounds, la处 5
rounds). The game ended earlier when the participants were experienced, but even
among experienced participa到ts the outcomes are far from the Nash equilibrium
outcome, in which the game ends in period 1.
。.4
First 5 rounds
.,
。‘3
Last 5 rounds
0.2
0 .1
1
2
3
4
;俨
3
6
7
Period
τen plays of the game may not be enough to achieve COI飞vergence to a steady
state. But putting asidεthis limitation of the data , and supposing that convergence
was in fact achieved at the end of 10 rounds, how far does the observed behavior
differ from a Nash equilibrium (maintaining the assumption that each player cares
only about her own monetary payoff)?
百1e theory of Nash equilibrium has two components: each player optimizes,
given her beliefs about the oth时 players， and these beliefs are correct. Some decisions in MεKelvey and Palfrey's experiment 飞，气!ere patent1 y suboptimal, regardless
of the subjec怡'beliefs: a few subjects in the role of player 2 chose to continue in
period 6, obtaining $6.40 with certainty instead of $12.80 with certaint沪 To assess
t.h e departure of the other decisions from optimality we need to assign the su均ects
beHefs (which were not directly observed). An assumption consistent with the
steady state interpretation of Nash equilibrium is that a play缸 's belief is based on
her observations of the other players' actions. Even in round 10 of the experiment
each pl町er had only nine observations on wruch to base her belief, and could have
236
Chapter 7. Extensive Games with Perfect Information: Ex主ensions and Discussion
used these data in various ways. But suppose that, somehow, at the end of round 4,
each player corrεctly inferred the distribution of her opponents' strategies in the
next 5 roUI飞ds建 What strategy should she subsequently have used? From McKelvey
and Palfrey (1 992 , Table IIIB) we m"y deduce that the optimal strategy of play衍 l
stops in perìod 5 and that of player 2 stops in period 6. That is, each player's
best response to the en飞pirical distribution of the other players' strategìes differs
dramatically from her subgame perfect equìlibrium strategy. Other assumptions
about the subjects' beHefs rationalize other strategìes; the data seem too limited to
conclude that the su垮台cts were not optìmìzi口g， given the beliefs their experiencε
might 陀asonably have led them to hold.τhat is , the experimental data are not
strongly inconsistent wìth the theory of Nash equilibrium as a s饺ady state.
Are the data inconsistεnt with the theory th孔 t rational players, even those with
no experience playing the gam窍， will use backward induction to dεduce their opponents' rational actìons? 丁his theory predìcts that the first play时 immediately
stops the game, so c仅tainly the data are inconsistent with it. How inconsistent?
Onew在Y to approach this question is to consider thεimplications of each playετ's
thinking that the others are likely to be ratìonal, but are not cettainly so. If, in any period , pl句er 1 thìnks that the probability that player 2 will stop the game in the r飞εxt
perìod is less than 在 continuì 鸣 yields a higher expected payoff than stopp
平驯
pi口
Gi沁飞
v吧咀
创I盯l 位
e
t h飞elim
引lÏ 怡t 已d time the 川
s ub
问Je
吭ct沁s had to analyze t.白
h飞eg
伊
am
泯ε (扫
and 应
th
挝
ε likelihood
t由
ha
挝t 白
the
叮
y had n飞e飞ve
时r before 白
th飞O飞ught abüt川
1让ta
圳1飞1y 陀
rεlat怆
ed game) , even those who lln也
derstood the ìmplicatior飞s ofback飞Nard induction may reasonably h在ve entertaìned
the relatively small doubt about the otr飞er players' cognitive abilities required to
make stopping the game immediately an unattractive option. Or, a1tεrnatively， a
pl巧er confident of her opponents' logical abilities m可 have doubted her opponents' assessment of her OWI1 analytical skills. If player 1 be 1i eves that play衍 2
thinks that the probability th向layer 1 will continue 时eriod 3 is greater than ~，
then she should continue in period 1, because player 2 will continue 如 period 2.
That is, relati飞rely minor departures from the theor
f..I
otes
τhe id础 of regarding games with simultaneous moves as games with perfect
informa tion is due to Dubey and Kaneko (1984).
The model in Section 7.3 was first studied by Ledyard (1 981 , 1984). 丁hεap­
proach to voting in committeesïn Section 7.4 was initiated by Farquharson (1969)
(see also Niemi 1983). (The publication of Farquharson's book was delayed; the
book was completed in 1958.)τhe top cycle set was first defined by Ward (1961)
(who called it the 气najority s时"). The characterization of the outcomes of sophisticated voting in binary agendas in terms of the top cycle set is due to Miller (1 97η
(who calls the top cycle set the "Condorcet set飞 McKelvey and Niemi (1 978)，在nd
Notes
237
Moulin (1986a , pp. 283…284) 拿 Miller (1995) surveys the field (see in particular his
pp.85-86).τhe model i口 Section 7.5 is taken from Ghemawat and Nalebuff (1985);
the idea is closely related to that of Benoît (1984 , Section 1) (see Exercisε174.2). My
discussion draws on an u只published exposition of the model by Vij句r Krishna. The
idea of discri 日1inating among Nash 吨uilibria by considering the possibility that
players make mistakes, briefly discussed in Section 7.6 .2, is due to Selten (1975).
The chain叹。re game in Example 231.1 is due to Selten (1 978). The centipede game
i口 Example 233.3 is due to Rosenthal (1981).
τhe experimental results discussed in the box on page 211 arεdue to Roth,
Prasnikar, Okuno-Fujiw盯a ， and Zamir (1991). The subgame perfect equilibria of
a variant of the market garne in which each play时 's payoff depends on thεother
players' rnon专tary payoffs are analyzed by Bolton and Ockenfels (2000). The modeJ
in Exercise 212.1 is taken from Peters (1984). 在1e resuJts in Exercises 216.1 and
217.1 are due to Feddersen, Sened, and Wright (199时. The garne in Exercise 227.4
is a simplificati∞ of an exarnple due to Shubik (1954); the rnain idea appears in
Phillips (1937, 159) and Kinnaird (1946 , 246) , both of which consist rnainly of puzzles previously publìshed in newspapers. Exercise 228.1 is based on Dierrneier and
Feddersen (1998)τhe experime到t discussed in the box on page 234 is rεported in
McKelvey and Palfrey (1 992).
Coalitional Games and the Core
8. 1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.1
ζoaliticnal ga 榻的 239
The core 243
IIlustration: ownership and the distribution of wealth 247
!II ustration: exchanging homogeneous hor纪5 251
IIJ \.I stration: exchanging heterogeneous houses 256
IlI ustration: voting 260
IlI ustration: matching 2己3
Discussion: other solutionζoncepts 269
Prerequisite: ζhapter 1
Coalitional games
COA Ll TIONAL G 八 ME is a model of interacting decisio汗makers that focuses
or飞 the behavior of groups of players 刷It associatεs a set of actions wìth every
group of players , not only with individual players as do the mode!s of a strate响
gìc gamε(Definition 13.1) and an extensive game (Definition 155.1). 认le call each
group of players a coalition , and the coalition of n日出e players t检 gmnd con !i tio l1.
An outcome üf a coalitional game consists of a partition of the set of players into
groups, together with an action for each group in t七e partitio孔 (See Section 17.2
if you areηot familiar with thεnotion of a "partition" of a se t.) At one extreme,
each group in the partition may consist of a single player, who acts on her own;
at another extreme , the partition may consist of a single group containing a11 the
playε币. The most gen己ral model of a coalitional game allows players to carεabout
the action chosen by each group iηthe partitiün that defines the outcome. 1 discuss
only the widely studied class of games in which each player cares only about th台
action chosen by the member of the partition to which she belo咔s. ln such games,
each player's preferences rank thεactions of a11 possible groups of players that
contain her.
。 EFIN ITl ON 2骂9.1 (Coalitíonal gn111c) A coalitional gam台 consists of
• a set of playe主$
• for each coalition, a set of actions
• for each player, preferences over the set of a11 actions of a11 coalitions of which
sh较 is a member.
B9
ζhapter 8.
24Q
Coalitional Games and the Cor牵
1usually denote the grand coalition 件1e set of a11 the players) by N and an arbitrary coalition by S. As before , we may conveniently specify a player's preferences
by giving a payoff function that represents them.
Note that the definition do臼 not relate the actions a飞railable to a coalìtion to
the actions available to the members of the coalition. The actions available to each
coalition are si肌ply taken as gi飞'en; they are not derived from the actions available
to individual players
EXA f\ IPLE 240.1 (Tw命player unanímity game) 丁'wo people can together produce
one unit of output which they may share in any way they wish. N性ther pεrson by
herself can p1 oduce any outp叫. Each personεares or飞Iy about the amount of output she recei飞'es， and prefεrs more to less. The following coalition在1 game models
this situJtion.
‘
Players The two people (players 1 and 玛 e
Actions Each player by herself has a single action, which yields her no outpu t.
The set of actions of the coalition {1 , 2} of both players is thεset of all pairs
( .\(，立'2) of no口negatíve numbers S\l Ch that .1 1 十工2 =刀 1 (the set of di飞'isions of
one unit of output between the two piayers) 蟹
Preferences Each player's preferences are represented by the amount of output
she obtains.
In this exan飞ple and in several of the examples that follow, each coalition controls some quantity of a good , which may be distributed among its membe自由 Each
action of a coa 1i tion 5 ìn such a game is a distribution among the members of 5 of
the good that 5 controls , which I refer to as an S-allocation of the good. 1 refer to
aηN-allocation simply as an allocation
h飞 thεnext example the opportunities for producing output are richer than they
are in the t认lo-player unanimity game , and the participants are not all symmetric.
‘
Ex 人民 I PL E 240.2 (Landmv只er and workers)
A landowner's estate , when used by k
飞vorkers， produces the output f 作中 1) of food , where f is an increasing function
The total number of workers is 111 The landowner cares only a号。ut the 在mount of
output she rec价飞'es， and prefcrs more to less; the same is true for each worker. The
following coalitional game models this situation
,
,
Players The lando飞气'ner and the 111 \γorkers‘
Actions A coalition consisting soleiy of workers has a single action in "年扒ich
no member receives any outpu t.丁h号 set of actiol飞s of a coalition 5 consisting
of the landowner a1叶 k workers is the set of all S-allocations of the output
f 位十 1 )注 mong thεmembers of S.
Preferences Each player's 尹丽eferencεs are represented by the amount of outp的
she obtains.
EXA 扣M
阳
i门i俨苔飞
LE24ω0 .3贝{俨f
血
hr时-p
肘i叮e
创rmajority 伊
g ar熙
τ叭1沱附叫巳功)Tl忖
lrιre
臼
ε ?阴
∞
e
opl抬
eha飞ve acce
硝ss toonε unit
of ∞ t结
pu
时t. Anv mη1aj扣
仪o怆}汗时
ri均
t竹
y………币斗.
仰
O
ut甲
p旧
时t. Ea
u
瓦巾
C副Jh pe
臼rs始
01η1 cares ∞
0 只!句y about the amount of output she obtains ,
8.1 (oalitional games
241
We may model thìs sìt飞latìon as thεfollowìng coa Iitional game.
PIαyers
The three people.
Actions Each coalitìon consisting of a 约ngle player has a síngle actíOl1, which
yields the playεr no outpUt. Th心 set of actìons of each coalition 5 with two or
three players is the set of S心 llocations of one unit of output.
Preferences Each play町 's preferences are represented by the amount of output
she obtains.
In these examples the set of actions of each coalition 5 is the set of 5-allocations
of the output that 5 can obtain , and 己ach player's preferences arεreprεse只ted by
the amount of output she obtaìns.τhus we can summarìze each coalition's set of
actions by a single number，问ual to the total output it can obtain, and we can interpret this number as the total "payo ff" that may be distributed among the members
of the ωalition. Such a game is said to have tr(!11巾协lc payoff More generally, we
define a game wÌth transferable payoff as follows.
DEFINIτION 241.1 (CoaWi Ol wl ga /11 c wit f7 tran~fcrablc payo(f) A coalitional game has
transferable payoff if there is a collection of payoff functions , one representing
each player's preferenεes， such that for each εoaìition 5, every actiOI飞 of 5 generates
a distribution of payoffs among the members of 5 that has the same sum.
We 陀的 to the total payoff of a coaliti∞ 5 in a game with transferable payoff
as t怡 worth of 5 , and denote it v( 5). Such a game is specified by its set of players
and its worth function. The gam臼 we have studied so far are specified as fo11ows.
Two-player unanimity game N = 杠 ， 2} ， v( {1}) 工 v({2 门口 O， andv( {1， 2})
1
Landowner-worker game N = {1 ,. . .， 111 十 1}(毛巾ere the landovv时r is 1 and the
workers are 2, . . ., 111 十刀， and
(0
u 问= ~
lf 批斗 1)
•
fhre playermajority game N
1 for 巳very other coalitìon S.
if 1 is not a member of 5
if 5 consists of 1 and k workers.
{1 ， 2， 3} ，叫{i})
0 for i = 1, 2, 3, and v(S)
Thenextex盯cise ìnvites you to model as a coalitional game a situation modeled
inCh叩 ter 2 as a strategic game.
ø EXERCISE 241.2 (Stag Hunt) Consider the situation described in Section 2.5, in
which a group of n hunters can catch a stag only if every member 陀mains atten哨
位ve ， and any inattentive member may catch a hare. Assume that a coalition may
splìt its spoils in any way among its members and that each hunter's preferences
are represented by the p玛!off function α x + y, where x is the amount of hares and y
the amount of a stag she obtains, andα < 11 凡 Model the situation as a coalitional
game , verify that the game has transferable payoff, and specify the worth of each
coa1ition.
Chapter 8. Coalitional Gan将军 andthe Core
242
In the next two examples , payoff is not transferable.
EXAMPL在 242.1 (House allocation) Each member of a group of 11 people owns an
indivisible good……call it a house. Houses differ. Any subgroup may reallocate its
members' houses in any way it wishes (one house to each person). (Time-sharing
and other devices to evade the indivisibility of a house are prohibited.) Each person caτes only about the house she obtains; the values assigned to houses vary
among the people. The following coalitional gamεmodels this situation.
Players The 11 peopìe
Act;ons The set of actions of a coalition 5 is the s仗 of all assignments to mem-
bers of 5 of the houses initially owned 七y members of 5.
Preferences Each player prεfers one outcome to another accordin在 to the house
she is assigned蟹
τhis game does not have transferable payoff. For exampie , a coalition of players 1 and 2 can achieve only the two payoff distributions (叫 ， W2) and (巧， Wl) ,
where Vi is the payoff to player 1 of the house initially owned by player i, and W 1
is the payoff 始 player 2 of thls house. The game is studied iηSection 8.5.
EXAMPLE 242.2 (Marriage market) A group of men and a group of women may
be matched in pairs. Each p夺目on cares only about her partn衍， not about anyone
else's partner. A matchi l1g of the members of a coalition 5 splíts 5 into pairs, each
consisting of a man and woman , and single individuals. The following coalitional
game models this sítuation.
Players The set of all the men and all the women.
Actions The set of actions ()f a coalition S is tÌ1e set of all matchings of the
mernbeτs of S.
Preferences Each player prefers one outcorne to another according to thεpart­
ner she is assigned.
τhis game is studied in Section 8. 7.
Ac。在litional game is designed to model sit飞lations in which players are bet伊
ter off forming groups than acting indi\'iduall y. Most of the theory is oriented to
situations in which the incenti\'e to coal岱ce is extrerne , in the sense that there is
no disadvantage to the formation of the sÎngle group consisting of all the players.
In considering the action that this single group takes in such a situation, we need
to consid衍 the possibility that srnaller gro泣如 may break away on their 0飞机1; but
when looking for "equilibria" we can restrict attention to outcomes in which all the
players coalesce. Such situations are rnodeled as garnes iηwhich the grand coa1ition can achieve outcomes at least as desirable for every play校 as those achie飞rable
by any partition of the players into subgroups. We call such games "cohesive".
, DEFINITION 242.3 (Cohesive coalitional game) A coalitional game is cohesive if,
for every partition {51 ,.... 5d of the set of all players and every combination
8.2 Thecore
243
(as 1 ,..., as k ) of act阳1S， one for each coalition in the partition, the grand coalition N 均s an action at least as desirable for every player i as the action aS j of the
member Sj of the partition to which player i belongs.
The defiI飞ition implies that a coalitional game with transferable payoff is cohesive if and only if for every partition {51 , . . ., Sd of the players we have v( 51) 十
…斗 V(5k) 手叫 N).
Thεconcepts 1 subs叫uently describe may be applied to any game , cohesive or
not, but have attractive interpretations only for cohesive games. All the games 1
have described so far are cohesive. Consider, for example , the two翩player unanimity game (Example 240. 巧. The possible partitions of the sεt of players are {杠， 2} },
consisti我g of the single coalition of both players , and {{叶，扫门， in which ωch
player acts alone. We have v( {1}) 十叫 {2}) = 0 < v( {l , 2}) = 1, so the ga rri.e is
cohesive. 丁'0 give anothεr example , the house allocation game (Example 242 .1) is
cohesive because any allocation of the houses that can be achieved by the coalitions
in any partition of the set of players can also be achieved by the s心t of a11 players.
φEXERCTSE 243.1 (Cohesive games) Verify that the lando飞Nner-worker gan飞e (Ex-
ample 240.2) , the three-player majority game (Example 240巧， the 5tag Hunt (Exercise 241.2) , and the marriage market game (Example 242.2) are a11 cohesive
8.2
Thecore
Whìch action may we expect the grand coa1itìon to choose? We seek an action compatible with the pressures imposed by the opportunities of each coalìtion , rather
than simply those of indìvidual players as ìn the models of a strategic game (Chapte主 2) andaηextensi飞re game (Chapter 5). We define an action of the grand coalition
to be 飞table" if no coalìtion can br出k away and choose an actìon that a11 its members prefer. The set of a11 stable actions of the grand coalìtion is ca11ed the corc ,
d伫主ned precisely as follows.
o日IN lTION 243.2 (Core of coalitional game) 丁i飞e core of a coalitional game is the set
of actìons aN of the grand coalition N such that no coalition has an action that a11
its members prefer to aN.
If a coalìtion 5 has an action that all its members prεfer to some action aN of the
grand coalitíon, we say that 5 can improve upon aN. Thus we may alternati飞rely
define thεcore to be the set of a11 actions of the grand coalition upon which no
coalìtion can improve.
Note that the core ìs defu飞ed as a set of actions, so it always exists; a game cannot
fail to have a core, though it may be the empty set, in which εase no action of the
grand coalition is immune to deviations.
We have restricted aUention to games in which, when evaluating an outcome ,
each play仔 cares on1y about the action chosen by the coalition in the partition of
which she ìs a member. Thus the members of a coalition do not need to speαtlate
about the remaining players' behavior when considering a dεviation.
Chapter 8. Coalitional Games and the Core
244
In a garne with transferable payo菇， a coalition 5 can irnprove upon an action aN
of the grand coalition if and only if its worth v( 盯(i.e. the total payoff it can achieve
by ìtself) exceeds the total payoff of its rnernbers in aN. That is ,
aN is in the core of a coalitíonal game with tran收rable payoff if and only iffor
every coalition 5 the total payoff :可 (aN) it yields the members of 5 is at least
v(S):
xs(αN) 主 v( 5) for every coalition S.
In the next exarnple, no co注lition can irnprove upon any action of the grand
coalition, so the core consists of all actions of the grand coalition.
EXAMPLE 244.1 (Two-player unanirnity garne) Consider the two-player unanirnity
An action of the grand coalition is a pair (坷，均) with
Xl 十 X2 = 1 and 乌兰 o for i = 1, 2 (a di飞rision of the one unit of output between
the two players). 1 clairn that the core consists of alI possible divisions:
garnεin Exarnple 且0. 1.
{(X1' X2) : Xl 十 X2
1 and Xi 主 Ofori
1, 2}
Any such division is in the core because if a single player deviates, she obtains no
output, and if the grand coalitior飞 chooses a different division, then one player is
worse off.
In this exarnple no coalitìon has any action that irnposes any restrictìon on the
action of the grand coalition. 1n rnost other garnes the coalìtìons' opportunities
constrair飞 the actìons of the grand coalition.
One way to find the core is to check each action of the grand coalition in turn.
For each action , wεimpose the condition that no coalition can make all its rnembers better off; an action is a mεmber of the core if and only if it sati5fies these
conditions.
Consider, for example , a variant of the two-player unanimity game in which
player 1, by hersel f, can obtain p units of output, and player 2, by hεrself， can
obtain q unìts of outpu t. The conditìon that the cωlition consisting of pl可er 1 not
be able to improve upon the action (均 ， X2) of the grand coalitìon is xl 主 p， and
the condition that the coalition consisting of player 2 not bεable to improve upon
this action is X2 :::: q. As in the orìginal garne, the coalìtion of both players cannot
improve upon any action (冉 ， X沙， 50 the core is
{(Xl , X2) : 工1 十 X2
= l ， Xl 三 p， and X2 三 q}.
(An implication is that if p 十 q > l~in which case the garne is not cohesive-the
core is empty.)
The landowner… worker game further illustrates this method of finding the core.
EXAMPLE 244.2 (Landowner… worker game with two workers) Consider the game
Let (町，吨， X3) be an
action of the grand coalition. That is , let 帆，巧，均) be an allocation of the outp时
如 Examplε240.2 in which there are two workers 拉= 2).
8.2τhecore
245
f(3) arno只g the three players. The only coalitions that can obtain a positive arnount
of output a陀 that consisting of the landowner (play衍斗， which can obtain the
output f( 盯， those consisting of 出εlandowner and a worker二 which can obtaín
[(2) , and the grand coalition.τhus (Xl' Xz , X3) is ín the core if and only if
工1 2': f (1)
X2
2': 0
X3
2': 0
X， 十 X2 主 f(2)
Xl 十工3 主 f(2)
Xl 个 X2 + 工3 =
f(3) ,
where the last condition ensures that (Xl ，巧，均) is an allocation of f(3).
Frorn the last conditíon we have Xl 口 f(3) - X2 …工3 ， so that we may rewritε
the condítions as
O 主工2 S f(3) - f(2)
O 三 X3 三 f(3) … f(2)
X2 十 X3 S f(3) - f (1)
Xl 十工2 十 X3
= 1(3).
That ís, ìn an action in the core , each worker obtaíns at rnost the extra output 1(3) f(2) produced by the third play战 and the workers together obtaín at rnost the
extra output 1(3) - 1(1) produc时 by the second and 出ird players together.
(1)虹 XE民CISE 245.1 (Three-player majoríty garne) Show that the core of the three时
player majoríty garne (Example 240 .3) is emp年
? EXERCISE 245.2 (Variant of three-player majority game) Find the core of the variant
of the three-player majority gam号 in whìch player 1 has three votes (and players 2
and 3 each have onεvote， as in the original 伊me).
? EXERC!SE 245 .3 (Stag HU l1 t) Find the core of the Stag Hunt in Exercise 24 1.2. Con-
h'ast the coalitional game and the core with the strategic gamεand Nash equilìbria
studied ín Cha pter 2.
? EXERC!SE 245 .4 (Varíant of St智 Hμ nt) Consider the variant of the Stag Hunt ín
Exercise 30.1a. Model the sítuation as a coalitional gam‘ e and fínd the core.
The next example introduces a dass of games that model the market for an
economic good.
EXAMPLE 245 .5 (Market with one owner and two buyers) A person holds one in跚
divisible 因出 of a good and each of 以10 (potential) buyers has a large amount of
money. 1ne owner values money but not the good; each buyer values both money
Chap辈.er8.ζ:oalitional
246
Games and 在h串 Core
and the good and regards the good as 吨uivalent to one unît of money. Each coaliω
tionm在 y assign the good (if owned by one of its members) to any of its members
and allocate its members' money in any way it wishes among its members.
以乌拉 may model this situation as the follo'V飞ring coalitional game
Players τhe owner and the two buyers.
Actions The set of actions of each coalition S is the set of S-allocations of the
money and good (if any) owned by S.
ρretîθrences
The owner's preferences are represented by the amount of money
she obtains; each buyer's p陀ferences are represented by the amount of the
good (either 0 or 1) she obtains plus the amount of mo口号r shεholds‘
1 claim that for any action in the core, the owr飞er does not keep the good. Let
aN be an actioI飞 ofthεgrand coalition ir飞 which the owner keeps the good , and
let l1l i be the amount of money transrerred from potential buyεr i to the owner in
thís actiOl飞忏'ransfers of money from the buyers to the owner when the owner
keeps the good may not sound sensible, but they are feasible , so that we need to
εonsider them.) Consider thεaltemative action a~ of the grand coalition in which
the good is allocated to buyer 1, who transfers 1t1 1 十 2E" money to the owner, and
buver 2 transfers I川 ~εmoney to the owner, where 0 < E" < ~. We see that all
the players' payoffs are higher in a~ than they are in aN. (丁he owner's payoff is t
higher, bu yer l' s payoff is 1 - 2εhigher， and buyer
payoff isεhigher.) Th us aN
is not in the core.
Now consider an action aN in the cοre in which buyer 1 obtains the good. 1
claím that a λí specifies that buyer 1 pays one unit of money to the ownεr and
buyer 2 pays no money to the ownεr. If buyer 2 pays a positive amount she can
improve upon a 只 by acting by herself (and making no payment). If buyer 1 pays
morεthan one unit of monεy to the owner she too can improve upon aN by acting
by herself. Fi nally, suppose buyer 1 pays 1111 < 1 to the owner.τrhen the owner and
t ransferrin
口19
buyer 2 cωan飞 irηpro队飞ve upon a 沁N by al丑locating the good to buyer 2 ar口ld 位
~(1 十 时 L认
州
m
mit
2汀
飞üt怡s of 川
mor
m
悦I叫 t打o附r时附r2 怡川
t出
h随
10
eω
0owr
\仰
川矿1阳~ 抖
y i代
内eldin
川吨
gt仙
he川O
例川v\付刊
y吁n
阳
吼1
greater tÌ1 an 仔I 1刀川/1 削
a 1ηìd buyer 2 a positive payoff.
We conclude that the core contains exactly two actions, in
of which the
good is allocated to one of the two buy乞rs and one unit of this buyer's money is
allocated to the OWJ飞机 That is , the good is sold to a buyer at the price of 1, yielding
th
''.1. EXERCISE 246.1 (Market with one owner and two heterogeneous buyers) Consider
the variant of the game in the previous example in whìch buyer 1's valuation of
the good is 1 剖飞d buyer 2's valuation is v < 1 (i.e. buyer 2 is indifferent between
8.3 lII ustration: ownership and the distribution of wealth
247
owning the good and owning v units of money). Find the core the game that
models this situation.
ln 岳母 next exercise, the grand coalition has finitely many actions; one way of
findíng the core is to check each one ìn turn.
于 EXE 汉CISε247 .1 (Vote trading) A legìslature wìth three members decides , by ma嗣
jority vote, the fate of three bìlls , A , B, and C. Each legislator's preferences are
represented by the sum of the 飞，alues she attaches to the b i1l s that pass. The value
attached by each legislator to each bill is indicated in Figure 247. 1. For example ,
if bills A and B pass and C faíls , then the three legislators' payoffs ar巳 1 ， 3， and
0 respectively. Each majority coalition can achieve the passage of any set of bill日，
whereas each minority is powerless.
,
Legislator 1
Legislator 2
Legislator 3
A
2
1
… 1
B
C
1
2
1
一1
2
Figure 247.1 The legislatoτs' payoffs to the three bills in Ex吉rcise 247. 1.
a.
Fìnd the core of the coalitional game that models this situation.
b. Find the core of the game in which the 飞ralues t始 legislators atlach to the
pa主sage of εach bill differ from those in Figure 247.1 only in that legislator 3
values the passage of bill C at O.
ε.
8.3
Find the core of the gamεin which the values the legislators attach to the pa沪
off of each bill differ from those in Figure 247.1 only in that each 1 is replaced
by -1.
lII ustration: ownership and the distribution of wealth
The distribution and institutions of landownership differ widely across economies.
By studying the cores of coalitional games that model various institutions, we can
gain an understanding of the implications of these institutions for the distribution
of wea 1th.
Agroupofη 2:: 3 people may work thεland to producεfood. Denote t1飞eoutput
of food when k people work a11 the land by f(k). Assume that f is an increasing
fu挝tio凡 f(O) = 0, and 也e Ol均ut prod时ed by an additional pεrson decreases as
the number of workers increases: f(k) … f(k … 1) is dec附si吨 in k. An example of
such a function f Ìs shown in Figu黯 248. 1. In a11 the games 1 st叫Y in this section,
the set of players is the set of the n people and each person cares only about the
amou时 of food she obtains.
248
ξhapter
8. Coalitional Games and the Core
Output
。
k•
Figure 248.1τhe output of food as a functíon of the number k of workers , u到 der the assumption that
the output of an addìtíonal worker d仅 reases as the number of workers increases
83.7
Single landowner and landlε55 worker5
First suppose that the land is owned by a single person, the landou月的\ 1 refer to
the other people as W01如 rs. 1n this case we obtain the game in Examplε240.2. ln
this game the action aN of the grand coalition in which the landown巳r obtains a11
thεoutput f(11) is in the corε: a11 coalitions that can prod旧e any output í时lude
the 1在只downer， and none of these coalitions has any actíon that makes her better
off than she ís in aN.
Wesa"々 in Example 24,t2 that fo 1' 11 = 3 the corεcontaÍI飞s actions in whích íhe
workers receive some outpu t.在lC same is true for any 飞ralu巳 of 11: the workers need
thεlandowner to produce any output, but thεlandowner also needs the workers to
producemo陀 than f( 斗， so stable actions of the grand coalition exist in which the
workers receive some output. Take the landm飞mer to be player 1, and consìder th巳
actìon aN of the grand coalition in which each player í obtair飞s the output Xj , where
Xl +...十立 11 = f( l1) 盯 nder what conditions on (Xl"'" X/ i ) ís Íl N in the core?
骂ecausεof my assumption about the shape of the function J, .the coalit阳lS most
capable of profitably deviating from aN consist of the landowner and every workεr
bt川口已 Such a coalition can , by itself, produce f(11 … 1) , a叫 it may distríbute this
output ín any way among its members. Tbus for a deviation by 岳飞1ch a coalition not
to be profítable , the sum of Xl and any collection of 11 … 20th时 x/s must be at least
f(11 …1). That is, (Xl 十· 十几1) … Xj 二> f(11 …1) for every j 口 2，…，几 Because
X] 十
十几i 古户(吨 we conclude that Xj 三/作)… f(11 … 1) for every player j wíth
j 注 2 (i.巳. every"气rorker)‘ Tha t is, if a N is in the core , then 0 三 Xj
f( l1) - f(11 一 1)
for every player j 2: 2. ln fact , every such actíon is in the core , as you arεasked to
verify í川he followi 咔 εxercise
合 EXERCISE 248.1 (Core of landowner-worker game) C怆ck that no coalition can
improve upon any action of the grand coalition in which the output reεeived by
every 认rorker is nonnegative and at m州贝叫 - f(n -1). (Use the fad that the
form or f implies that f( 11 ì f 问 2: (n-k)(只 11) f( n 1)) for every 支主 11.)
•
8.3 lII ustratìon: owner事hìp and the dìstribution of wealth
249
We conclude that the core of thεgame is the set of all actions of the gra口d coalition in which the OUtput Xi obtained by each worker i satisfies 0 ~ Xi 芒 f(71) f(11 … 1) and the output obtained by the landowner is the difference between f( l1)
and the sum of the workers' shares. 1n economic jargon，只 11) - f(11 1) is a
worker's "marginal product". Thus in any action in the core, each worker obtains
at most her marginal product.
The workers' shares of output are driven dow到如放血。st f(n) - f(n .-1) by
competitìon between coalitions consisting of the landowner and workers. If the
output receìved by a盯 worker εxceeds 只的 f( n - 1) , then the other worke凹， m
cahoots with the landowr飞时~ can devìate ând increase their share of output. That
is， εach worker's share of output is lìmited by her comrades' attempts to obtain
more outpu t.
The fact that each worker's share of output is held down by interworkεrcom­
petition suggests that the 飞Norkεrs might be better off if they were to agree not to
join deviating coaìitions except as a group. You are asked to check this id相 ìn the
following exercì挠，
•
Q) EXERCIS 芜 249 .3 (Ur飞ionized workers in landowner-worker game) Formulate as a
coalitional game the variant of the landowner-worker game ìn which any group
of fewer than n _. 1 workers refuses to work with the landowner, and find its core.
The core of the original game is closely related to the outcomes predicted by
the economic notion of "competitive e气uìlìbrium • Suppose that the landowner
belíeves she can hire any number of workers at the fixed wage w (given as an
amount of output) , and every worker belie飞咽 that she can obtain employment at
this wage. If w 去。， then every worker wishεs to work, and if w 主 f(n) … f(n 1) ,
the landowner wishes to employ all n … 1 workers. (Red飞lcing thεnumber of workers by one reduces the outp时 by f伊) - f(11 一 1); further reducing the number of
workers reduces the output by successively larger amounts , given the shape of f.)
Hw>f(n)-f(n 一片， then the landowner wishes to employ fewer thar飞行币 1
workers, because the wage exceeds the increase in the total output that results
when the (n … 1 )th worker is employed. Thus the demand for workers is equal
to the supply if and only if 0 三 ω 三川的一 f(11 一片; every such wage w is a
"competitive equilibrium"
A different assumption about the form of f yields a dìfferent conclusion about
the cor巳 Suppose that each additional worker produces more additional output
than the previous one. An example of a function f with this form is shown in
Figu陀 250. 1. Under this assumption the εconomy has no competitive equilibrium:
for any 认!age， the landowner wishes to employ an indefinitely large number of
workers.τhe next exercise asks you to study the core of the induced coalitional
game.
ff
@EXERCIS主 249.2 (Landowner-worker game with increasing marginal products)
Consider the variant of the landowner-worker game in which each additional
worker produces more additional output than the previous one. (That îs , f( 剖 jk <
Chapter 8. Coalitional Games and the Core
250
Output
O
k•
Fígure 250.1 The output of food as a function of the number k of workers, under the assumption that
tl飞e output of an additíonal worker increases as the number of workers increases.
f(主十 l)/(k 十 1) for a 1l k.) Show that the core of this game co口tains the action of
the grand coalition in which each player obtains an equal share of the total output
‘
8.3.2 Smalllandowners
Suppose that the land is distributed equa11y between a11 n people , rather than being
concentrated in the hands of a single landowner. Assume that a group of k people
who pool their land and work together produce 仕/叫只 n) units of outpu t. (The
output producεd by half the people working half the land , for example, is half the
output produced by a11 the people working a11 the land 、)
The fo11owing specification of the set of actior飞5 available to each coalition models this situation
Actions The set of actions of a coalition 5 consisting of k players is the set of a11
S-allocations of the output (k/ 时f 圳 between the members of 5
As you might exp巳ct， one action in the core of this game is that in which every
player obtains an equal share of the total ∞ tput-that is , f 刨 / n units. Under this
action, the total amount received by each coalition is precisely the total amount
the coalition produces. In fact no other action is in the core. In any other action,
some player receives less than f(n)/ I1, and hence can improve upon the action
alone (obtaining f (n) / n for herself). That is , the core consists of the single action
in which every player obtains f( I1 )/11 UlùtS of outpu t.
8.3.3
Colleζtiveownεrship
Suppose that the land is owned coUectively and the distribution of output is d挝εr­
mined by majority vo往ng. Assume that any majority may dìstribute the output in
any way it wishes; any majority ma予 in particula飞 take all the output for itself. In
this case the set of actions a飞唱lable to each coalition is given as follows.
8.4 lII ustration: exchanging homogeneous horses
251
Actions The set of actions of a coalition S consisting of more than n/2 playεrs is
the set of a11 S-allocations of the outp时 f( 的 betwεen the members of S.τhe
set of actions of a coalition S consisting of at most n/2 players is the single
S-allocation in which no player in S receives any output
‘
Thεcorεof the coalitional game defined by this assumption is empt予 For ev-
ery action of the grand coalition, at least one playεr obtains a positive amount
of outpu t. But if player i obtains a positive amount of output , then the coalition
of the remaining players, which is a majority, m巧 improve upon the action, distributing the output f( l1) among its members (so that player i gets not恼19). Thus
every action of the grand coalition may be improved upon by some coalition; no
distribution of output is stabl役，
The core of this game is empty because of the εxtreme power of every majority
coalition. If any majority coalition may control how the land is used , b飞lt every
player owns a 飞hare" that entitles her to the fraction 1/11 of the output, then a
majorìty coalition with k members can lay claim to only the fraction k/ /1 of the
totaloutp时， and a stable distribution of output may exis t. This alternative ownershìp institution, which tempers the power of majority coalitions, does not have
interesting implications in the model ìn H飞is section because the control of land use
飞lestcd in a majority coalìtìon is inconse年lential-only one sensible pattern of use
exists (a11 the players work!). If choic创 exist-if， for εxample ， different ζropsmay
be groW1飞， and people differ in their preferences for these crops-then the outcome
of collecti飞re ownership in which each player ìs entitled to an equal share of the
output may dìffer from the outcome of ìndividual ownership.
8.4 lII ustration: exchanging homogeneous horses
M盯kets may be modeled as coalitional games in which the set of actions of each
coalition S is the set of S-allocatíons of the good initially owned by the members
of S. The core of such a game is the set of allocations of the goods available in
thεeconomy that are robust to thεtrading opportunities of a11 possible groups
of participants: if 印 is in the core , then no gro叩 of agents can sεcede from the
economy, trade among themselves , and produce a1飞 outcome thεy all prefer to aN.
In this section 1 descrìbe a sìmple example of a ma古时 in which there is money
and a single homogeneous good (all units of which are identical). 1n the next sec喘
tion I describe a maτket 如 which therεis a single heterogencous good. In both cases
the core makes a very precisεprediction about the outcome.
8.4 .1 Model
Somεpeople own one unit of an ìndivisible good , whereas others possess or飞ly
money. Some nonowners value a unit of the good more highly than some owners,
50 that mutually bene自cial trades exis t. Whl ch allocation of goods and money wìJ在
result?
Chapter 8. Coalitional Games and the Core
252
认fεmay address this question wìth the help of a coalìtional game that gen仔al­
ìzes the one in Example 245 .5. 1 refer to the goods as Ilhorses" (following the lìter伊
ature 0ηthe modet which takes off from an analysis by Euge口 von Böhm-Bawerk
(1 851… 1914)). Call each person who owns a horse simply aη oWl1er and every other
person a 110110Wnel二 Assume that a11 horses a陀 idεntìcal and that no one wishes to
own more than on巳 People value a horse differently; denote player i's valuation
by Vi. Assume that there are at least two owners and two nonowners; assume a1so
that some own时 's valuation is less than some 1飞or飞机叮1时 's valuation (i.e. for some
owner i and nonowner j we have 句 < Vi) , so th挝 some trade is mutually desirable. Assume also , 抬出'oid some special cases , that some nonowner's valuatìon is
less than some ownerls valuation (i. e. for some nOnOwner i and owner j we havε
Ví < Vj) and that no two players have the same valuation. Further assume that ev时y person has enough money to fully compensate the owner who valu岱 a horse
most highlYI so that no one's behavior is cons缸'ained by her cash balance.
As to preferences , assume that each persoηcares only about the amount of
money she has and 飞Nhether she has a horse. (In particulaζno one cares about any
other person's holdings.) Specifically, assume that each player i/s preferences are
represented by the payoff functíon
( 川川r
r
证fs挝灿邵衍叫叫叫
S刊a
川
川吨
州山
协h.10rs
…
飞旧
町
ors
町
m
r凹
5捉
se a
if she has no horse and $1' more money than she has initiall y.
(τhís assumption does not mean that people do not
,"alue the money they have
ìnitially. E电u1飞'ale口 tlv wεcould represent player i's preferenc岱 by the functìons
+ 111i jf she does not , where mí is the amount of
money she has initially.)
The following coalitional game l which 1 call a horse trading game , models the
situation.
2气十 r 十 mi if she has a horse and r
Players The group of pεop1e (owners and no且ow口ers).
Actions The set of actions of each coalition S is the set of S-allocations of the
horses and the tota1 amount of money owned by S in which each player
obtains at most one horse.
Pref合ren但s
Each player's preferences are represented by the payoff functior飞
described above.
η1Ís game ìncorporates no rεstriction on the way ìn which a coalition may dis-
trìbute its money and horses. ln partìcula几 players are not restricted to bilateral
trades of mor飞ey for horses. A coalition of two owners and two nonowners , for
examp1e, may, ìf it wishes , allocate each of the owners l horses to a nonowner and
transfer n飞oney from both nor飞。飞，yners to only or飞e owner, or from one nonowner to
th号 other.
8.4 lII ustration: exchanging homogeneous horses
8.4.2
253
The core
Number the owners in ascending order and the nonowners in descending or唰
der of the valuations they attach to a horse. Fìgure 253.1 illustrates the valuations, ordered in this wa予(在1Î s diagram should be familiar-perhaps it is a little
too familiar一if you have studied economics.) Denote owner i's valuation 叫I and
nonowner i's valuation 卢 i. Denote by k* the largest number i such tr段在>的
(so that among the owners and nonowners whose indices are k* or less , eveηr
nonowner's valu挝ion is greater than every owner's valuation).
Let aN be an actíon in the core. Denote by L* the set of owners who have no
horse in aN (the set of sellers) and by B* the set of nonowner骂 who have a horse in
aN (the set of buyers). τhese two sets must have the same number of merr飞快衍 (by
the law of conservafion of horses). Denote by ri the amount of money received by
owner i and by Pj the amount paid by nonowner j in aN.
1 claÌm that Pj
0 for every nonowner j not 如 B*. (That ÌS , a nonowner who
does not acquíre a horse neíther pays nor recεives any money.)
• If Pj > 0 for some nonowner j not in B* , then her payoff is negative, and she
can unilaterally improve upon aN by retaining her or地inal money.
• If Pj < 0 for some nonowner j not in B* , then the coalitïon of all players
other than j has Pj less money than it owned ìnitiall予 and the same number of
horses. Thus this coalition can improve upon aN by assigning horses in the
same way they are assigned in aN and giving each of its members Pj I 归一 1)
more ur飞its of money than she gets in aN (where n is the total number of
players).
•
Valuatìons -一
(ßj , (Tj)
Nonowners
Own衍$
品，
Range of 辛
values of p* 啻
σ护+1
... ßk可 1
σk"
k* Trader number •
Figure 253.1 An example of the players' valuations in a market with an indivisibl告 good. The buye怒'
valuatìons are given in black, the sellers' in gray
25唾
Chapter8.ζ倒litional
Games and t抽 Core
By a similar argument, rì 之二 o for every owner not in L* (an owner who does
not se11 her horse neither pays nor reεeives any money.)
1 now argue that in aN every se11er (member of L*) receìves the same amount
of money, every buyer (member of B*) pays tl飞e same amount of money, and these
amounts are e电ual: rj 工 Pj for every se11er i and buy衍 j. τhat is, a11 trades occur at
the same price.
S叩pose that ri < Pj for se11er j and b可eq 害 1 argue tl硝 the coalition {i , j} can
improve upon aN: í can sell her horse to j at a prìce between ri and Pj' Under aN ,
seller j's payoff is rì and buyer j's payoff is 卢1 … Pj- If i se11s her horse to j at the
price j{fi 个 Pj) ， then her payoff is !(ri 十 pj)>ri a 叫 j's payoff 始向- i(rí 个 Pj) >
ßj 一 Pj' so that both i and j are bεtter off than they are in aN. Thus ri ~三 Pj for every
S己ller j and every buyer j.
Now, the sum of a11 the amounts rí received by sellers is 吨ual to the sum of a11
the amounts Pj paid by b町的问 the law of conservation of money), and L* and
have the same number of members. Thus we havε rí
Pj for evεry se11er i in
L* and buyer j in B 气
Insummary,
for every action aN ìn the corεthere exists such that ri = Pj
every owner j inγand every nonown时 j in W, and ri = Pj
every owner not in L' and every nonowner j not in B*.
p* for
0 for
1 now argue that the common price p' at which a11 trades take place lies in a
narrow range.
1n aN , evεry owner i whose 飞raluation of a horse is less than p* must selll飞机
horse: if she did not, then the coalition consisting of herself and any 1飞onowner j
who buys a horse in 均 could improve upon aN by takìng the action in wÌ对ch j
buys i's horse at a price between the 0飞机飞机 's valuation and p*. Also, no owner
whose valuation exceeds p* trades, because her p句Toff from doing so would be
E飞 egative. Similarly, every nonowner whose valuation is greater than p* buys a
horse , and no nonowner whose valuatíon is less than p* does 骂。.
cv
EXERC1SE 254.1 (Range of prices in horse market) Show that the req飞lÎrement that
the number of owners who sell their horses must equal the number of nonowners
who buy horses, together with the previous arguments , implies that the common
trading price p* is at least (Tk勺 at least ßl川 1 ， at most ßk*' and at most crk川 1. That
is, p* 2': max{crk> , ßk*+1} and p" 主 min{ßk*'σk叫 1 }.
Finally, 1 argue that in any action in the core a player whose valuation is equal
to p* trades. Suppose that nonowner i's valuation is equal to p*. Then owner i's
飞'aluation is less than p. and owner i 十 l's 飞吸luation is grεater than p' (given my
assumption that no 队10 players have the same valuatio叫， so 位lat exactly i own时S
trade. Thus, given that every trade involves two parties, exactly í nonowners must
trade , implying that nonowner í trades. Symmetricall主， an owner whose valuation
is equal to p悠 trades.
8.4 lII ustration: exchanging homogeneou骂 hors岳5
255
In summary, in every action in the core of a horse trading game,
• every nonowner pays the same price for a horse
• the common price is at least max{Uj炉 ， ßk， +d and at most
min{ 如，。去*十 d
• every 0￥vner whose valuation is at most the prìce trades her
horse
(255.1)
• every nonown盯 whose valuation is at least the price obtains a
horse.
…
The action satisfying these conditions for the price p* yíelds the payoffs
c叫i，川
i平
圳
J
， 川本寸}
p*
阳m
飞:ìax{v
叫irP 本叮}一 p*
for every r刚刚刊时 i
τhe core does not ímpose any addítional restrìctions on the actíons of the grand
coalition: every action that satisfiεs these ξonditions is in the core.τo establish
t1飞is rεsult， 1 need to show that for any action aN that satisfies the conditions, no
coalition has an action that is better for a11 its members. When a coalition deviates,
which of its actions has the best chance of improving upon aN? The optimal action
definitely assigns the coalitíon's horses to the members who value a horse most
highly. (If Vi < 巧， then the transfer of a horse from í to j , accompanied by the
transfer from j to i of an amount of money between Vi and 7.J j makes both i and j
better off.) No transfer of money makes anyone better off without making someone
worse off, so for a coalition to improve upon aN there must be some distribution of
the total amount of money it owns that, given the optimal distribution of horses,
makes a11 its members better off than they are in aN. For every distribution of
a coalition' s mone予 the tota1 payoff of the members of the ζoalition is the same.
Thus a coalition can improve upor飞泛N if and only if thεtotal payoff of its m己mbers
undera只 is less than its total payoff when it assìgns its horses optimally.
Consider an arbitrary coalition 5. Denote by f! the total number of owners in 5 ,
by b the total number of nonowners in 5 , and by 5 * the set of f! members of 5 whose
valuations are highes t. Then 5's tota1 payoff when it assigns its horses optimally is
汇 Vj，
whereas its total payoff under aN is
汇 max{ V ú p*} - bp* =
L m拟 {Vi， p*} + 汇 max{几
where 5\5* is thεset of members of 5 not inτhe former is never higher than
the latt!时 because 5 \5* has b members , so that LES\S* max{ 叫 ， p*}-bp* 去 O.
In summary, the core of a horse trading game is the set of actions of the grand
coalition that satisfies the four conditions in (255 .1).
Ch矗 pt告 r 8.
256
Coalitional G矗mesandthe ζ。"
Ex在 RCISE 256.1 (Horse trading game with single seller) Find the core of the variant
of the horse trading game in which there is a single owner, whose valuation is less
than the highest valuation of the nonowners.
If you have studied economics you 七10W that this outcome is the same as the
competitive equilibrium气 The theories differ, howev巳r. 丁he theory of competitive equilibrium assumes that a11 tradωtake place at the same price. It defines an
equilibrium price to be one at which 飞机nand (the total number of nonowners
whose valuations exceed the price) is equal to "supply" (the total number of owners whose 飞raluations are less than the price) 羹 This equilibrium may be justified by
the argument that if demand exceεds suppl予 then the price w i11 tend to rise, and if
supply exceeds demand it will tend to falL Thus in this theory，气narket pressures"
generate an equ iIi brium price; no agent in thεmarket chooses a price
By contrast, the coalitional game we have studied models the players' actions
explicitly; each group may exchange its horses and money in any way it wishes.
The core is the s的 of actions of a11 players that survives the pressures imposed by
the trading opportunities of each possible group of players A uniform price is not
assumed , but is shown to be a necessary property of any action in the core.
11
H
‘
1. EXER Cl SE 256.2 (Horse trading game with large se11er) Consider the variant of the
horse trading game in which there is a single owner who has two horses. Assume
that the owner's payoff is (71 十 r if she keeps one of her horses and 2(71 十 r if she
keeps both of them, where r is the amount of money she receives. Assume that
there are at least two nonowners, both of whose values of a horse exceed 町. Find
the core of this 伊 me. (Dû a11 trades take place at the same price , as they do in a
competitive equilibrium?)
8.5
lII ustration: exchanging heterogeneous houses
8.5.1
Model
In Example 242.1 , the goods that may be exchanged are heterogeneous, unlikεthe
horses of the previous section. Each member of a group of 11 people owns a single
house. Any subgroup may reallocate its members' houses in any way it wish部
(one house to each person). Each person cares only about the housεshe obtains.
Assume that each person's ranking of the houses is strict (she is not indifferent
between any two houses). 1 refer to the game defined in Example 242 .1, under this
assumption, as a house exchange game.
Which assignments of houses to people are stable? You may think th挝 without
imposing any restrictions on the nature or diversity of preferences, this question is
hard to answer, and that for some sufficiεntly conflicting configurations of preferences no assignment is stable. If so, you are wrong on both counts, at least as far as
the core is concerned; remarkably，如r any preferences, a slight varìant of the core
yields a unique stable outcome.
8.5 lII ustration: 曾xchanging het，暗 rogen罄。us houses
257
8.5.2 The top trading cycle procedure and the core
One property of an action in the core is immediate: because every play盯 hasthε
option of simply keeping the house she initially owns, any player who initially
owns her favorite house obtains that house ìn any assignn飞仅ltinthεcore.
Thìs property allows us to completely analyze the simplest nontrivìal example
of thεgame， with two people. Denote the person who initially own5 player i's
favorite hm脱 by o(i).
• If at least 0时 person ìnitially owns h盯 favorite hoωe (i. e. if o( 句
1 or
。 (2) 古习， then the core contains the single assignmer飞t in which each person
keeps the house she owns.
• If each person prefers the house owned by the other person (ì.e. if 0(1) = 2
在nd 0(2) 土巧， then the core contains the single assignment in which the two
people exchange houses.
In the second case wεsay that "12 is a 2-cycle". When there are more players,
longer cycles are possible. For example , if there are three or more pl可ers and
o(i) = j, o(丹 = k, and 0 作
Î， then we say that "ijk is a 3-cycle". (If o(i) = i, we
can think of i as a "l-cycle".)
τhe case in which there are three people raises some new possibilities
• If at least two people initially own their favorite houses, then the core contains the single assignment in whìch each person keeps the house she initially
O认'n5.
• If exactly one per50n, say player i, initially own5 her favorite house, then
in any assìgnment in thecore, that person keeps her house. vVhether the
other two people exε乞ange theìr houses depends on their prefe配到ces over
these houses , ìgnoring player i's house (认七ich has already.been assigned);
the analysis is the same as that for the two-play衍 game.
• If no pεrson initially owns her favorìte house, there are two cases.
o If there is a 2-cycle (i. e. if there exist persons i and j such that j inìtially
owns i's favorite house and i initially owns i's favorite house) , then the
only assignment in the core is that in which i and j swap 乞ouses and the
remaining pl在yer keeps the house she owns initially.
。 Otherwise， suppose that o( í)
j τhen o( j)
= k, where k is the third
player (otherwise ij ìs a 2翩cycle)， and o(号= i (otherwise kj is a 2-cycle.)
百1at is,. ijk is a 3-cycle. If i gets j' s house~ j ge怜的 house， and k 伊拉 i's
house , then eve町 player i5 assigned her favoritεhouse， so this assignment is in the core. (百lÍs argument does not show that the core contains
no other assi军nments.)
This construction of an assignment in thεcore can be extended to games wÌth
any number of players. First we look for cydes among the houses at the top of
。lapter 8.
258
3
1''
1q4
h1
h}
Coalitional Game辈革 ndtheCore
Player 4
113
h2
h4
h4
113
Figure 258.1 A partial specificati♂n of the players' preferences in a game with four pl注ye币， illustrating
the top trading cycIe procedure. Each play告r's ranking is given from 快st to wor处， reading from top to
bottom. Dashes indicate irreJ.:vant parts of the rankings.
the players' rankings and assign to each member of each cyc1 e her favorit号 house.
(If therεare at most threεplayers， then only one cycle containing more than one
player can exist, but if there are more players, many cycles can exìs t.) Then we
eliminate from ζ:onsid时ation the players ìnvolved in these cycles and the houses
they are allocated , look for any cycles at the top of the remains of the players'
rankings , and assign to each member of each of these cycles her favorite house
among those remaining. We continue in the same manner until all players have
been assigned houses. 在l.Îs procedure is called the top tradìng cycle procedure.
To illustrate the procedure , consider the game with four players whose preferences satisfy the specification in Fìgure 258.1. In this figure , hi denotes the house
owned by player i and the players' rankings are listed from best to worst, starting
at the top (player 3 prefers player l's house to player 2's house to player 4's house,
for example). Dashes indicate irrelevant parts of the rankìngs. We see that 12 ìs
a 2-cycle, so at the first step players 1 and 2 are assigned their favorite houses (h 2
and Tt 1 respectively). After eliminating these players and their houses , 34 becomes
a 2-cycle, so that player 3 is assigned 114 and player 4 is assigned h3' If player 3's
ranking of h3 and h4 were reversed , then at the secor飞d stage 3 would be a l-cycle ,
so that player 3 would be assigned h3, and then at the third stage player 4 would
be assigned h4'
( EXERClSE 258.1 (House assignment with identical preferences) 'Find all the assi萨­
ments in the cor<台 of an n-player house exchange game in which every player ranks
the houses in the same way.
1 now argue that
for any (strict) prl政rences， the core of a ho泛se exchange gm时 C0 l1 tal11S tl1 e
assignment Înduced by the top tradi l1 g cycle procedure.
Every player assigned a housεin the first round receives her favorite house , so that
no coalitìon cohtaini吨 such a player can make all its members better off than they
are in aN. Now consider a coalition that contains players assi萨巴d houses in 出e
second round , but no players assigned houses in 出e first round. Such a coalition
does not own any of the houses assigned on the first round , so that its members
who were assigned in the sεcond round obtain their favorite houses among the
8.5 I/I ustration: exchanging heterogeneous houses
259
houses the coalition owns. Thus such a coalition has no action that makes all its
members better off than they are ín aN. A similar argument applies to coalitions
containing players assigned in later rounds.
8.5.3 The strong core
1 remarked that my analysis of a three-player game does not establish the existence
of a unique assignment in the core. Indeed, consider the preferences in Figure 259 .1.
W毛 see that 123 is a 3-cyde, so that the top trading cyde procedure generates the
assignment in whìch each player receives her favorite house.
1claim that the altemative assignme批 dLJnwf材1 player 1 obtai时弘 player2
obtains h1, and play红 3 obtains h3 is a150 in the core. Player 2 obtains hεr favorite
house, so no coalition containi吨 her can improve 叩OZIa;J 时either player 1 nor
丑lp
play
a 10附 car川 m
归er 3 址
泊ins the house she owns. The 0向阳 naining coalition is 杠，斗， which owns h1
obta
and h3' If it deviates and assigns hl to player 1, then she is worse off than she is in
then
aLJT飞 d if it deviates and assigns h1 to player 3, 白
a
τhus
coalition
can improve upon a~.
no
轧.
吨马
L
Although no coalition 5 can achieve any S-allocation that makes a11 of i总 mem­
bers better off than they are in a~ ， the coalitìon N of a11 th附 players can make two
of its members (players 1 and 3) better off, while keeping thεremaining member
(player 2) with the same house. 丁hat is, it can "weakly" improve upon a 忏
Defir飞e the strong core of any game to be the set of actions aN of the grand coali 伽
tion N such that no coalition 5 has an action as that some of its members prefer to
印 and al1 of Ìts members regard to be at least as good as aN.
The argument 1 have give川 hows 在at 在e action a~ is not in 出es仗。吨 co肥。 f
the game in which the players' p陀 ferences are given in Figure 259.1 , though it is
in the core. In fact,
for any (strict) preferences, the strong core of a house exchange game consists
of the sìngle assignment deJi ned hy the tap trading cycle procedure
1 omit the argurnent for this re5ul t.古le result shO\年 s that the (strong) core is a
highly successful solution for house exchange games; for any (strict) preferences , it
pinpoìnts a sìngle stable assigr黯lent ， which is the outcome of a simple, intui总 vely
appealing, procedure.
Player 1
Player 2
Player 3
h2
h3
h2
h2
h1
h3
h3
h1
Figure 259.1 The players' prefe陀nces in a g朋le with 位E哇哇 players. Each player's 阳泳ing is given
from best to worst, reading from top to bottom.
260
Ch器 pt窑r 8.
Coalitional G暑 m窑s and the Core
Unfortunately, the strengthening of the definition of the corεhas a side effect:
if we depart frorn the assurnption that all preferenεes are strict , and allow players
to be indifferent between houses, then the core rnay be ernpty.τhe next exercise
gives a只 exarnple.
EXERCISE 260.1 (Ernptiness of the strong core when preferences are not strict) 5uppose that sorne players are indifferent between sorne pairs of houses. 5peci在cally，
suppose the烈 are three players, whose preferences are given in Figure 260 .1. Find
the core and show that the strong core ìs ernpty.
PA
l aVJ
O协
-E-a
p-···A
aVJevi
VZA
h2
h1 , h3
h2
h1, h3
h2
h1 , h3
内J
-
句fh
付‘
功立
i aVJeva ••
Figure 26号.1 The players' preferenc巳 5 in the game in Exercise 260 ,1. A cell containing two houses
ind icates indifference bet讯reer飞 the优 twùhouses.
8.6
lII ustration: voting
A group of people chooses a policy by rnajority voting. How does the chosen
policy dep心nd on the individuals' preferences? In Chapter 2 we studied a strategic
garne that rnodels thís situation and fOllild that the notion of Nash equilibriurn
adrnits a 飞咽γ wide range of stable outcornes. In a Nash equilibriurn, no single
playe乙 by changing her vote, can improve the outcorne for herselt but a 自roup
of players, by coordinatíng their votes, may be able to do so. By modeling the
situation as a coalitìonal game and using the notion of the core to identi句r stable
outcomes, we can find the irnplications of group deviations for the outcome.
丁b rnodel voting as a coalitional game , the specification I have given of such a
game needs to be slìghtly rnodified ，我ecall that an outcome of a coalitional garne ìs
a partition of the set of players and an actìon for each coalitionîn thεpartition. 50
far 1 have assumed that each player carεs only about the action chosen by the coalition in the partìtiOI飞 to which she belongs τhìs assumption means that the payoff
of a coalition that devi挝es from an outcome is determined independently of the
actìon of any other coalition; when de飞liating， a coalítion does not have to consider
the action that any other coalìtion takes. In the situation I now present, a different
constellation of conditions has the sarnεirnplication: everγplayer cares onJy a bout
the action chosen by the rnajority coalition (of which there is at most one) in the
outcorne partition, and only coalitions containing a rnajority of the players have
rnorεthan one possible action. In brief, any majority may choose an action that
affects everyone, and every rninority is powerless.
Precisely, assurne that there is an odd number of players, each of whom has
preferences over a s时 of policies and prefers the outcome x to the outcome y if and
only if either 价 there are rnajority coalitions in the partitions associated with both
8.6 lII u辜tration: voting
261
x and y and she pref，εrs the action chosen by the majority coalition in x to the action
chosen by the majority coalition in y, or (Ì i) there ìs a majority coalition in x but not
in y. (If there is a majority coalition in neither x nor y , she is indifferent between
x and y.) The set of actìons available to any coalition containing a majorìty of the
players ìs the set of a11 polìcies; evεry other coalition has a single action. 1 refer to
such a 伊 me as a majority voting game.
The definition of the core of tms variant of a coalitional game is the natural
variant of Definìtion 243.2: the set of actions aN of the grand coalition N such that
no majority coa 1ition hωan action that a11 its members prefer to aN.
Suppose that the policy x is ìn the core of the game.τhen no policy is prεferred
to x by any coalition consisting of a majority of the players‘五气uivalentl严 forevery
policy Y 并 x， the set of players who either prefer x to y or regard x and y to b疗
equally good is a majority. If we assume that every player's preferences are strictno player is indifferent between any two policies-then for every policy y 乒 x， the
set of players who prefer x to y is a majority. That is , x is a Condorcet winner (see
Exercìse 75.3). For any p陀ferences， there is at most one Condorcet 认Tinner， so we
have established that
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How does the existence and character of a Condorcet wìnner depend on the
players' preferences? First suppose that a policy is a number. Assume that each
player i has a favorite po1icy xj , and that her pref，台 rences arε single peaked: if x and
x' are policìes for wmch x < x' < 巧。r x7 < x' < X , then she prefers x' to x. Then
the median of the players' fa飞rorite positions is the Condorcet winner, as you are
都ked to show in the neχtεxercìse， and hence the uruque member of the core of the
voting game. (The median is well defined becau能 the number of players is odd.)
@ EXERCISE 26 1.1 (Median voter theorem) Show th挝 when the policy space is onedimensional and the players' preferences are single peaked , the unique Condorcet
winner is the median of the players' favorite posìtions‘ (τhìs result is known as thε
median voter t11eorem.)
A one-dimensìonal space captures some polìcy choíces , but other si
Chapter 8. Coalitional Games and the Core
262
Figu 陀主岳2.1 À two-dimensìonal polícy sp在ce wìth three play告rs. The point x; is the favorìte positìon
of player i for i =之1， 2, 3. Every policy in the shaded lens is preferred by players 1 and 2 to x.
and X7 , so tha t for any value of r she is indifferent between all policies on the circle
with radius r centered at xj.
Now choose a到Y policy and ask if ìt is a Cor飞dorcet winner. The policy x in the
figure is not , because any policy in the shaded area is preferred to x by players 1
and 2, who constitute a majority. The polìcy x ìs also beaten in a majority vote by
any policy in eìther of the other lens-shaped areas defined by the intersεεtionof the
circles ce时ered at xi , xi , and x3. Is there any policy for whïch no such lens-由叩ed
area is created? By checkìng a few other policies , you can convìnce yourself that
the answer is negative. That is, no policy is a Condorcet wînner, 50 that the core of
the game is empty.
For SOme configuratìons of the players' favorite positions , a Condorcet wir飞栅
ner exists. If the positions lie on a straight line , for example , then the middle
one is a Condorcet 飞Ninner. But only very 5pecial con且gurations yield a Con幽
dorcet winner-in general there is none, so that the core is empt予扭d our analy唰
sis suggests that no policy is stable under majority rule wher飞 the policy space is
multidimensiona l.
In some situations in which policies are determined by a vote , a decision requires a positive vote by more than a simple majority. For example , some jury
verdicts in the United St挝es requirεunanimity， and changeS in some organiza唰
tio邸， and countries' ∞nstitutions requìre a 队守thirds majority. To study the implications of these alternative voting rules, fix q with 11 /2 :5 q 主 11 and consider
a variant of the majorityτule game that 1 call the q-rule game , in which the only
coalitions that can choose policìes are those containing at least q players. Roughly,
the larger is the value of 吁， the largεr is the core. You are invited to explore some
examples in the 1飞ext exercise.
? EXE 汉 ClSE 262.1 (Cores of q-rulεgames)
必.
Suppose that the set of policies is 0时-dimensional and thatεach player's
preferences are single peaked. Find the core of the q-rule game for any value
of q with 1112 三 q 三 11.
b. Find the ωre of the q-rule game when q 工 3 in the εxample in Figure 262.1
(with a two-dimensior飞al policy space and three players).
8.7 llI ustration: matching
263
8.7 lII ustration: matching
Applicants must be matched with universities, workers with firms , and football
players with teams. 00 stable matchings exist? If so, what are their properties,
and which institutions generate them?
ln this sectìon 1 analyze the model of two-sìded one-to-or飞e matching described
in Example 242.2, in which each party 011 one side must be matched with exactly
one party on the other side. Most of the main ideas that emerge apply also to
many-to-one matching problems. This model is sometimes refi心rred to as one of
ifmarriage" , though of course it captuτes only one dimension of matrimony. Some
of the language 1 use is taken from this interpretation of the model.
8.7. 1 Model
1 refer to members of the two sides as Xs and Ys. Each X may be matched with
at most one Y , and each Y may be matched with at most one X; staying single is
an option for each ìndividua l. A m日 tc1úng of any set of individuals thus splits the
set into pairs, each consisting of an X and a Y , and single individuals. 1 denote
the partner of any player i under the matching lJ byμ (i). If i and j are rr时ched ，
we thus haveμ (i) = j and /-l(j)
i; if i is single then 以 i) 口 i. Each person
cares on1y about her partner, not about anyone else s partner. Assume that every
persor飞's preferences are strict: no person is indifferent between any t:wo par位lers.
An example of possible preferences is given in Figure 263. 1, in which for instance
pl可er xl ranks y2 first , then Yl , and 在nds Y3 unacceptabl巳 1 refer to the sεt of
p盯加ers that i prefers to the option of remaining single as the set of i's acceptlαble
partners f 在 nd to the coalitional game as a two-sided one-to-one matching game.
f
f
f
8.7.2
f
The core and the deferr，εdαcceptance procedunε
A matching in the core of a two-sided one-to-one matching game has the property
that no group of players ma予 by rearranging themselves, producεa matching that
th咛 a11 like better 1 claim that when looking for a coalition that can improve
upon a matching, we may restrict attention to those consisting either of a single
individual or of one X and one Y. Preciselya matching Ìs 抬出e core if and only if
‘
Xs
Ys
Yl
Y2
Y3
X2
Xl
Y2
I Xl
Y2 I X3
Xl
X3
Y3
工2
X3
X2
Xl
X2X3
Y2
Yl
Yl
i
Yl
Figure 263.1 An example of the player毡， preferences in a two-sided one-to-one matchìng game. Each
column gives one player's r忍泳ing (from best to worst) of all the players of the other type 也atshe 位lds
acceptable
Chapter 8. Coalitional G碍mes and the Core
264
a. each player prefers her part乳白 to being singlε
b. for no pair (i , j) consisti吨。f an X and a γis it the case that i prefers j to 严(i)
and j prefers i to μ (j).
τhe following argument establishes this claim.
First, any matching 严 that does
not satisfy the conditions is not 加 the core: if 树 is violated, then some player can
improve upon 11 by staying single, and if (坊 is violated, then some pair of players
can improve uponμby matching with each other. Second, suppose thatμis not in
the core. Then for some coalition 5 there is a matching fI' of its members for which
every member i prefers fI '(i) to 州). If 5 consists of a single individual, then (a)
is vìolated. Otherwìs窍， suppose that i ìs a member of 5 , and let j = fI' ( 斗， so that
1/ (j). Then i prefers j toμ (i) anâ j p陀fers i to }I (j). Thus (b) is violated.
In the game in which the players' preferer飞ces are those given in Figure 263.L
for example , the matchi鸣 }I in which 严(马) =且， μ( 川=挠，严(月)口儿 and
li (如)
归 (i.e. X3 δnd Y3 stay single) is in the core , by the following argumer飞仁
No single pJayerεan improve upon fi because every matched play肘 's partner is
acceptable to her. Now consider pairs of players. No pair containing X3 or Y3 can
improve upon Jl oecause 工1 and X2 a陀 matched with partners they prefer to 拙， and
Yl and y2 are matched with par如ers they prefer to X3. A 出
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η2 ， with whom she is matched , to Xl;
the second cannot upon 11 because yl prefers Xl , with whom she is matched , to X2.
How may matchings in the core be found? As in the case of the market for
houses studied in Section 8.5, one member of the core is generated by an intuitively
appealing procεdure. (In contrast to the ∞re of the house market , however, the
core of a ÌìlI10翩 sided， one-to-one matching game may contain more than one action,
as we sha11 see ,)
The procedure comes in two flavor飞 one in which proposals are made by Xs ,
and one in which they are made by Ys. The d非rred acceptance procedure ωith proposals by Xs is defined as fo l1 ows. Initíally, each X proposes to her favoríte Y, and each
γeither rejects a11 the proposals she receives, if none is from an X acceptable to her,
or rejects a11 but the best proposal according to her preferences. 芜ach propos在1 that
is not rejected results in a tentatíve match between an X and a Y. If every offer is
accepted , the process eì飞ds， and the tentative matches become definite. Otherwise ,
there is a second stage in which each X whose proposaJ was 古ejected ín the first
stage proposes to the Y she ranks second , and each Y chooses am且ong the se
1. Each X whosεoffer was rejected at the previous stage and for whom some Y
is acceptable proposes to her top-ranked Y out of those who have not rejected
an offer from her二
8.7 lII ustratio n: matching
主65
2. Each γrejects the proposa1 of any X who is unacceptab1e to her, is "engaged"
to the X she 1ikes best i口 the set consisting of a 1l those who proposed to her
and the one (if any) to whom she was previous1y engaged, and rejects all
other proposals.
τhe procedure stops when the proposal of no X is rejected or when every X whose
offer was 陀jected has run out of acceptable Y s.
Consider, for example, the preferences in Figure 263 .1. The progress of the
procedure is shown in Figure 265.1 , in which •" stands for "proposes to". First
Xl proposes to Y2 , and both X2 and X3 propose to Yl; Yl rejects X2 、 proposal. Then
X2 proposes to 弛， so that Y2 may choose between X2 and Xl (with whom she was
tentati飞rely matched at the 自rst stage). Player Y2 chooses X2 , and rejects Xl , who
then proposes to Yl.. Player Yl nQW chooses between Xl and X3 (with whom she
was tentatively matched at the first stage) , and rejects X3. Finally，均 proposes to
Y2 , who rejects her offer. The final matching is thus (Xl , Yd ， 悦，而 ， X3 (a10叫， and
Y3 (alone).
11
4
X(
X2:
• Yl
工3.
→ Yl
Reject
叫究
Reject
→妇
R司ect
Figure 265.1 The pτogress of the deferred accep怡 nce procedure 轩íth proposals by Xs when the
players' preferences ar t' those given ín Figure 263. 1. Each row gives the proposals of one X.
For any preferences, the procedure evεntually stops , because there are 位让tely
many players. To show that the matching 严 it produces is in the core, we neεd to
consider deviations by coalitions of only one or two players , by an 但rlier argu日lent.
• No single player m町 improve upon 严 becausεno X ever proposes to an
unacceptable Y , and every Y always rejects every unacceptable X.
• Coηsider a coalition 卡，j} of two players, where i is an X and j is a γ. If
i prefers j to 严(i)， she must have proposed to j , and been rejected, before
propo
俯sin
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百le analogous procedure in which proposals are made by Y s generates a
matching in the core , by the same argument. For some preferences the matchings produced by the two proced旧es are the same, whereas for others they are
differen t.
量 Ex巳民CISE 265.1 (Deferred acceptance procedure wÎth proposals by Ys)
Find the
matching produced by the deferred acceptance procedure wìth proposals by Ys
for the preferences given in Figure 2已3.1.
Chapter 8. Coaiitional Games and the Core
266
1n particular, the core may contain more than one matching‘It can be shown
that the 凯atching generatεd by the deferred acceptance procedure with proposals
by Xs yields each X her most preferred partner among a11 her partners in matchings 抽出e core, and yields each Y her least preferred partner among a11 her partners in matchings in the core. 5imilarly, the matching 伊nerated by the deferred
acceptance procedure wit趴 proposals by Y s yields each Y her most preferred partner among a11 her partners in matcl飞ings in the core, and yiεlds each X her least
preferred p盯tner among a11 h仔 partners in matchings in the core.
7. EXERCISE 266.1 (Example of deferred acceptance procedure) Find the matchings
produced by the deferred acceptance procedurεboth wit乞 proposals by Xs and
with proposals by Ys for the preferences given in Figure 266. 1. Verìfy the results
in thE: prevìous paragraph. (Argue that the only matchìngs ìn the core are the two
generated by the procedu纪叫
Xl
X2
X3
Yl
Y2
Y3
Yl
yl
Yl
Xl
Xl
Xl
Y2
Y2
Y3
工2
工3
X2
Y3
Y3
Y2
X3
;主2
X3
Figure 266.1 The pl是yers' preferences ín the game in Exercise 2661.
ln summary, evεry two-sided one.士。-one matching game has a nonempty core,
which contains the matching generated by each deferred acceptance procedure.
The matchìng generated by the procedure is the best one ìn th泛 core for each play衍
。日 the side making proposals, and the 飞快xst one i口 the core for each player on
thεother side喃 (An implic放ion is that if the two procedures generate the same
matchi吨， then, given that every person's preferences are strict，出e core contains a
single matching.)
8.7.3 VarÎants
Strat只gic behavior
50 far, 1 have consìdered the deferred acceptance procedures
only as algoritl飞ms that an administrator who knows the p在rticipants' preferences
may use to find matchings in the core. 5uppose the participants' prefe陀nces are
not knowI飞. We may use the tools developed in Chapter 2 to study whether thε
partícipants' interests are serv仓d by revεaling their true preferences. Consider the
strategic game in which each player names a ranking of her possible partners and
the outcome is the matching produced by the deferred acceptance procedure with
proposals by Xs, given the announced raI水ings. One can show that in t始sgame
each Xs naming her true ranking is a dominant action, and although in a Nash
equilibrium the actior飞s of the Ys may not be their true rankings , 宿舍 Nash 吨uilib­
rium is in the core of the coalitional game defined by the playε归 ， true rankings. (I
omit an argument for thís resul t.)
8.7 lII ustration: mat命 ing
267
? ExεRCISE 267 .1 (Strategic beh出lior undεr a deferred accepta只ce procedure) Con瑞
sider the preferences in Figure 267.1. Find the matchings produced by the deferred
acceptance procedures , and show that the core contains no other matchings. Con耐
sider the strategic game described in the previous paragraph that is induced by
the procedure with proposals by X's. Take as given that each X's naming her true
ranking is a dominar飞t strategy. Show that the game has a Nash equilibrium in
which Yl names the ranking (Xl , X2 ， 马) and every other player names her true
ranking.
Figur告 267.1
Xl
X2
工3
Yl
Y2
Y3
Y2
Yl
Yl
Xl
X3
;可1
Yl
Y3
Y2
X3
Xl
X3
Y3
Y2
Y3
X2
X2
X2
The players' preferences i口 the game in Exercise 267.1
Other matchíng problems I motivated thεtopic of matching by citing the problems
of matching applicants with universities，认rorkers with firms , and football pl可ers
with teams. All these problems are many咱to-one rather than 0抗争to伊one. Under
rnild assumptions about the players' preferences , the rεsults 1 have presented for
one-to-one matching games hold , wÍth minor changes , for many吨。-one matching
games. In particular, the strong core (defined on page 259) is nonempty, and a
variar忱。 f the deferred acceptance procedure generates matchings in it.
At thls poi时 you may suspεct that the nonemptiness of the core in matching
games is a very gεneral resul t. If so , the next exercise shows that your suspicion is
incorrect-… at least, if "very ger飞eral" incluàes the "roommate problem气
ø EXERClSE 267.2 (Empty core in roommate problεm) An even numb衍 of people
has to be 叩 lit into pairs; any person may be matched with any other person. (百le
matching problem is "one-sided".) Consider an example in which therεare four
people , i, ], k, and Show that ìf the preferences of Í , ], and k are those given in
Figure 267.2 , then for any preferences of the core is empty. (Notice that is the
least f我vorite roommate of every other player.)
e.
e
]
e
]
k
k
i
k
E E E
Figure 267.2 The preferences of players i, j , and k in the game in 日xercíse 267.2
都 Ex充 RCl记 267.3 (Spatial preferences in roommate problem)
An even number of
peoplεhas to be split into pairs, to be assigned to rooms. Each perso口's charac-
2岳8
Chapt曹r 窑.
Coalitional Games 辛苦ld the Core
teristic is a number; no two characteristics are the same. 主ach person would like
to hιave a roommate whose characteristic is as close as possible to her own, and
prefers to be ffiatched even with the player whose characteristic is most remote
from hers to remaining single. Find the set of matchings in the cor已
MATC泣 ING DOCTORS WIτH HOSPITALS
Around 1900, newly trained doctors in the Unitε d States were , ，for
岛 由
t he 如
fi让rs挝t tim
芷引l已
e,
gí忖
ven the op
严tioαl叭1 of wor此
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ιs
s
t始
og
伊
ai阳
ne
似
xp
归
en扭C
岱
εi加
n clinica
址1 medicine. Initially, hospitals advertised positíons ,
for which ne\^:ly trained doctors applied. The number of positions excεedeà the
suppìy of doctors, and thεintense comp价ition between hospitals for interns led
the date at which agr时ments were finalized to get progressively earlier. By 1944,
student doctors were finalizing agreements two full years bεfore their internships were to begin. Making agreements at such an early datεwas und 创irable
for hospitals , because at that point they lacked ext巳nSÌve information about the
students.
The American Association of Medical Colleges attempted to solve the problem
by having its members agree not to release any information about students before
the end of the thi 1' d year (of a four翩year program). This change p 1'evented hospitals
from makir飞g earlier appointments, but in doing so brought to the fore the problem of coordinating offers and acceptances. 日ospitals wanted thei1' first-choice
students to acζept quicklμbut students wanted to delay as much as possible, hoping to 1'eceive better offers. In 1945, hospitals agreed to give students 10 days to
consider offers. But thε出飞.，vas pressure to reduce this pe 1'iod. In 1949 a 12…hou 1'
1'iod was rejected by thεAmerican Hospítal Association as too long; it was agreed
and hospitals could insist on
that all offe1's be made at 12:01 A.M. on November
a 1'esponse within any period. Forcing students to make decisions without having
a chance to collect offers from hospitals whose fírst-choíce students had rejected
them obviously led to inefficient matches.
τhese difficulties with efficie时ly matchíng doctors with hospitals led to the
desígn of a centralized matchíng procedurεthat combi附s hospítals' rankings of
students and students' rankings of hospitals to produce an assig时nent of students
to hospitals. It can be shown that this procedure, designed tel飞 years before gametheoretìc work on the deferred acceptance procedure, generates a matching in the
core for any stated preferences! It differs from the natural generalization of the
deferred acceptance procedure t。在 man沪怡-one matching problem, but generates
. precisely the same matching , namely the onεin the core th挝 is best for the hospitals. (T
8殖器
Discussion: other solution concepts
269
In the early years of operation of the procedure , oveτ95% of students and
hospitals participated. In the mid-1970s the participation rate fell to around 85%.
Many nonparticipants 认!ere married couples who wished to obtain positions in
the same city. The matching procedure contained a mechanism for dealing with
married couples, but, unlike the mechanism for single students, it could Jead to a
matching upon which some couple could improve. The difficulty is serious: when
couples exist who restrict themselves to accepting positions in the same cit予 for
some preferences the core of the resulting game is empty… no matching is stable.
Further problems arose. In the 1990s, associations of n飞εdical students began to
arguεthat changes were needed because the procedure was favorable to hospitals,
and possibilities for strategic behavior on the part of students existed. The game
theorist Alvin E. Roth was retained by the National Resident Matching Program
to design a new procedure to generate st击ble matchings that are as favorable as
possible to applicants. The new procedure was first used in 1998, when it matched
around 20,000 new doctors with hospitaJs.
8.8
Discussion: other solu主ion concepts
1n replacing the requirement of a Nash equilibrium that no individual player may
profitably deviate with the requirement that no group of players may profitably
deviate, the notion of the core makεs an assumption that is unnecessa巧 in the
interpretation of a Nash equilibrium. A single player who deviates from an action
profile in a strategic game can be sure of her deviant action because she unilaterally
chooses it. But a member of a group of players who chooses a deviant action must
assume that no subgroup of her comrades will deviate further, or, at least, that she
\.vill remain better off if th可 do.
Consider, for εxample， the three-player majority game (Example 240.3 and Ex0) of the grand coalition in this game is not in the
ercise 245 .1)刊e action
core because, for examplc , the coalition consìsting of players 1 and 3 can take an
action that gives player 1 an amount x with < x < 1 and player 3 加 amou挝
1 x , which leads to the payoff profile 仪， 0 ， 1 - x). But this profile itself is not
stable-the coalition consisting of players 2 and 3, for example , has an action that
gener就cs the payoff profile 忡， y， l 一抖， where 0 < y < x , in which both of them
are beìter off than they are in 杠， 0， 1 …斗 .τne fact that player 3 will be tempted
by an offer of player 2 to deviate from (x , O, 1 - x) may dampen player1 's enthus卜
asm for joining player 3 in the de由tion from
0). For simil盯 reasons， player 2
maybe 陀luctant to join in a de飞riation from this action.
Several solution conc咛 ts that take into account these considerations have been
suggested (see , for example , Osborne and Rubinstein 1994, Chapter 14) , though
none has so far had anything like the success of the core ín illuminating social and
economic phenomena.
(!' !'
!
(!' !'
270
Chapter 8. Coalitional Games and the Core
Notes
丁he notion of a coalitional game is due to von Neumann and Morgenstern (1944).
Shapley and Shubik (1 953) , Luce and Raiffa (1957, 234…235) , and Auman川nd Peleg (1960) generalized von Neumann and Morgenstern's notion. The ∞tion of the
core was introduced in th己 early 1950s by Gillies as a tool to study another solution
concεpt (his work is published in Gillies 1959); Shapley and Shubik developed it
as a solution concep 七
Edgeworth (1 881 , 35-39) poi附d out a connection between the competitive
equìlibria of a market model and the set of outcomes we 1飞ow call the core. Von
Neumann and Morgenstern (194毡， 583-584) first suggested modeling markets as
coalitional games; Shubik (1959苟照cogn泣ed the game-theoretìc content of Edgeworth's arguments and , together with Shapley (1959) , dev心loped the analysis. Sec喃
tion 8.3 is based on Shapley and Shubik (196乃. The core of the market studied in
Section 8.4 was first studied by Shapley and Shubik (1 971 /72). My discussion owes
a debt to Moulin (1995, Section 2.3).
丁hεmodel and result on the nonemptiness of the core in Section 8 .5 are due to
Shapleyand Scarf (1974) , who credit David Gale with the top trading cycle proω
cedure. The result that the strong cor巳 contains a single action is due to Roth and
Postlewaite (1977) 丁hεmodel is discussed in detail by Moulin (1995 , Section 3.2).
Voting behavior in committees (Section 8局 was first studied formally by Bl ack
(1958) (written in the miι1940s)， Black and Newing (1951) , and Arrow (1951).
Black used the ω陀 as the solution (before it had been defined geneτa l1y) and esω
tablished the mediar飞 voter theorem (Exercise 26 1. 1). He also notìced that in polìcy spaces of dimension grεater than 1, a Condorcet winner is not likely to exist, a result extended by Plott (1967) and refined by Banks (1995) and others, who
find conditions relating the number of voters, the dimension of the policy space,
and the value of q for which the core of the 牙rule game is generally empty; see
Austen-Smith and Banks (1999, Secton 6.1) for d供aìls.
The model and main results in Section 8.7 are due to Gale and Shapley (1962).
The rεsult about the strategic properties of the deferred acceptance procedures
at the end of the section is a combination of results due to Dubíns and Freεd­
man (1981) and 宜。th (1 982) , and to Roth (1984动.旦xercise 267 .1 ís based on an
example in Moulin (1995 , 113 and 116). Exercise 267.2 is taken from Gale and
Shapley (1962
FVLU
时υ
吨w
问h
「eι
Games with Imperfect Information
9
10
Bayes~an Games
273
Extensive Games with Imperfect Information
313
Bayesian Games
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
Motivational examples 273
General defìnitions 278
Two exampl仍 concerning information 282
I lIustration:ζournot's duopoly game with
imperfect information 285
IlI ustration: providing a publiεgood 289
IlI ustration: auctions 291
IlI ustration: juries 301
Appendix: auεtions with an arbitrary
distrìbution of valuations 307
Prerequisite: ζhapter 2 ahd Section 4.1.3; Section 9.7
requiresζhapter 4
N ASSUMPTION underlying the notion of Nash 叫出libri议rn 1S that each player
holds the correct belief about the other players' actions. To do so , a player
rnust know the garne she is playing; ìn partic飞dar， she rnust know the other pìayers' preferences. ìn rnany situations the partidpants are not perfectly inforrned
about their opponents' characteristics: bargainers rnay not know each others' val侧
uations of the object of negotiation, firms may not know each others' cost functions, combatants rnay not know ea在 others' strengths, and jurors may not know
their colleagues' interpretations of the evidence in a tria l. In some situations , a
participant rnay be well inforrned about her opponents' characteristics, but rnay
not know how ￥vell these opponents are informed about her own characteristics.
ln this chapter 1 describe the rnodel of a 飞ayesian garne飞 which generalizes the
notion of a strategic game to allow us to anaiyze any situatìon in which each player
is imperfec t1y informed about an aspect of her environrnent that is rel衍 ant to her
choice of an action.
9.1
Motivatìonal examples
1 st-a rt with two examples that illustrate the main ideas in the model of a Bayesian
game. 1 de位le the notion of 民ash 叫uilibrium separately for 能ch garne. In thε
next section 1 de且ne the general rnodel of a Bayesian game and the notion of 时ash
吨ui1ibrium for such a game.
命 EXAMPLE 273 .1 (Variant of Bo$ with impedεct inforrnation) Consider a variant of
the situation rnodeled by Bo$ (Figure 19 .1) in which player 1 is unsurεwhether
，、 7')
Chapter9. Bayesí器 nGames
274
1
B
S
B
2, 1
0, 0
5
。，。
2
8
B
S
1, 2
0, 2
1, 0
2 ，。
0, 1
2 ，代lishes to avoid 1
2 wishes to meet 1
Figur曹 274.1 A variant of BoS in which player 1 ís unsure whether player 2 \气瓜nts to meet her or to
avoíd he r. The frarne labeled 2 enclosìng each table indicates that player 2 knows the relevant ta七 le
The frame labeled 1 enclosíng both tables indicates t扒在 t plûyer 1 does not know the relevant table; the
probabilities she assig到s to the two t在bles appear on the frame.
player 2 prefers to go out with her or prefers to avoid her, whereas player 2, as before , knows playerγs preferer飞ces. Specifically, suppose player 1 thinks that with
probability play衍 2 wants to go out wíth her, a时 with probability player 2
wants to avoid 11er. (Presumably this ass部sment comes from player l's exp
rience: half of the time she ís involved in this situation she faces a player who
wants to go out with her, and half of the time she faces a player who wants to
a叫d her.) That i叩i巧er 1 thinks 阳t 飞飞rith probab山y she is playi吨 the game
on the 1eft of Fi gure 274.1 and \vith pr巾ability she is playing the game on the
righ t. Because probab i1 ities are involved , an ar飞alysis of tl飞e situation rε气Ulres us to
know the players' preferε口ces over lotteries, even if we are interested only in pure
strategy 吨uilibria; thus the numbers in the tables are Bernoulli payoffs.
认也 can tl飞ink of there being two states , one Ìr飞 which the players' Bεrnoulli pay翩
offs are given in the 1εft table and 0ηεin which these payoffs are gi飞毛只 in the right
table. P!a飞刊
F沧
衍r 2 kr口.10飞孔
e
w
叮's 出
t h飞e st妇
at怯
仑白……叫一啕心飞e krη10WS whe
e
沱
吐
e
etl
出
her
创rs
仙
he 飞W
飞γ罗i总sh飞ε
仍s 怕
t ο me
臼
et 臼
0 ra
队盯飞
\'01边
d
pμla
町
ye
町r 1 …叫叭rl飞泼
er陀
ea
昂spμla
咛
ye
臼r 1 doe
硝S
盯
s
肌们
no
削1c盯
时
o1让t飞忡;
p layer 1 assigns 严flr
队
)川川
m
r认
巾丁
飞O
The notion of 扒到Ja泊sh equ
山
1垃ilibr
汀r川1UI血
τ丑d
如
or a str喃3
址
at怆
egic gam
丑 e models a st饺
岱
ε
ady 纣
s tat怡
em
口
1
1
w
气叮1h飞 id飞 each play心旧r's beliefs about the oth飞ε
缸r playe
盯rs' actior飞 s are correct, and each
player acts optima lJ y, given her beliefs. We wísh to generalize this notíon to the
current situ在 tíon.
From player 1'5 point of view, player 2 has two possiblε typcs， one whose pref贮rencεs are given in the 1εft table of Fi gure 274.1 and one whose preferences are
give口 in thεright table. Player 1 does not kηow player 2 s type , so to choose an
action rationally sl飞e needs to form a belief about the action of each typε. Gi飞1en
these beliefs and her be!i ef about the likelihood of each type, she can calcuJate her
expected payoff to each of her actions. For example , if she thinks that the type who
wishes to meet her w i1l choose B and the type who wìshes to avoìd her will choose
S, then she thinks that B will yiel的era 问
p浅咛
yof
旺fof 2 认
川
w
r斗
i怕
of 0 wít位
h飞 p
伊roba抽bi
刷阳 y , so that her expected payoff is ~ .2 十 .0 = 1, and S will
yield her an expected p叩)ff of i 。十~ 1 = ~. Similar calc白tions for 阳 other
combinations of acti
i
i
i
i
f
i
.
i
•
9.1 Moti￥'ational exampl创
275
(B , B)
B
2
5
。
(B , 5)
(5 , B)
(5 , 5)
O
2
2
1
Figure 275.1 The expeζted payoffs of player 1 for the four possible paìrs of actions of the two types of
player 2 in Example 273. 1. Each row corresponds to an act lOn of player 1, and each columnζorresponds
to a pair of actions of the two types of player 2; the action of thεtype who wishes to meet player 1 is
listed first
player 2, the first member of each pair being the action of the type who wishes to
meet player 1 ênd the 使cond member being the action of the type who wishes to
avoid player 1.
For this si切 ation we define a pure strategy Nash equilibrium to be a triple of
actions, one for player 1 and one for each type of player 2, with the property that
., the action of player 1 ìs op 位mal， given the actions of the two types of player 2
(a到d player 1's be!ief about the state)
• the actîon of each type of player 2 is 叩timal， given the action of player 1.
That is, we treat the two types of player 2 as separate players and analyze the
situation as a three-player strategic game in which player l's payoffs as a function
of the actions of the two other players (i.e. thεtwo types of player 2) are given in
Figure 275.1 , and the payoff of each type of player 2 is independent of the actions
of the other type and depends on the action of play盯 1 as given in the tables in
Figure 274.1 (the left table for the type who wishes to meet player 1, and the right
table for the type \'/ho wishes to avoid player 1). In a Nash equìlìbrium, player 1 's
action is a best response in Fi阴re 275 .1 to the paìr of actions of the two types of
play的 2， the action of the type of player 2 who wishes to meet player 1 is a best
response in the left table of Figure 274.1抽出e action of player 1, and the action
of the type of player 2 who wishes to avoid player 1 is a best responsεin the right
table of Figure 274.1 to the actiOI飞 of player 1.
Why should player 2, who knows whether she wants to meet or avoid player 1;
have to plan what to do in both cases? She does not have to do so! But we, as
analysts, need to consider what she would do in both cases. The reason is that to
determine her best action , player 1, who does not know player 2's type, neεdstü
form a beHef about the action each t月守e would take，在nd we wish to impose the
equilibrium condition that th部e beliefs be correct, in the sense that for each type
of player 2 they specify a best response to player 1's equ也brium action. Thus the
equiìibriUI我 action of player 2 for each of her possible types may be interpreted as
player 1's (correct) bèlief about the action that each typεof player 2 would take,
not as a plan of action for player 2喃
1 claim that (B , (B , 5)) , wherethe 自rst compcneηt is the ac创on of player 1 and
the other component is the pair of actions of the two types of player 2, is a Nash
equilibrium. Given that the actions of the two types of player 2 are (B , 5) , playerγs
action B is optimal, from Figt
Chapter 9. 8aye革ianGames
276
for the type who wishes to meet player 1 and 5 is optimal for the type who wìshes
to avoid player 1, from Figure 274. 1. Suppose that in fact player 2 wishes to meet
player 1. Then we interpret the 吨uilibrium as follows. Both player 1 and player 2
choose B; player 1, who does not know if player 2 wants to meet her or avoid her,
believes that if player 2 wishes to meet her she will choose B, and if she wishes to
avoid her she will choose S.
(l) EXE这C!SE 276.1 (Equilibria of a variant of BoS with imperfect information) Show
th时 there is no pure strategy Nash 吨ui1ibrium of this game in which player 1
chooses S. If you have studied mixed strategy 出ash equilibrium (Chapter 屿，
如ld the mixed strategy Nash 吨uilibria of the game. (First check whether there
is an equilibrium in which both types of player 2 usεpure strategies; then look for
equilibria in which one or both of these types randomize.)
We can int世pret the actions of the two types of player 2 to reflect player 2's
intentions in the hypothetical situa泣。n bi功re she knows the state. We can tell the
划剖11与
y playe
时r 2 does not kr丑
n旧
o
ow the state; she 岛
is 讯
i nf，如
ormed of the
following story. 1n飞i让tia
state b古y a si告{注￥/，附，
an action for each poωss到ible 刽
s igna
址1. Af仕te
世r receiving the 挝
s ign旧
al she carries out 挝1 飞冶
εr
pl弘
ar归
nned action for that signal. We can tell a similar story for player 1. To be consistent with her not knowing the state when she takes an action, her signal must be
uninformatìve: it must be the same in each state. Given her signal, she is unsure
of the state; whεn choosing an action she takes into account her beHef about the
likelihood of each state, given her signal.百\e framework of states, beliefs, and
signals, wruch is unnecessarily baroque in this simple example , comes into its own
in the analysis of more complex situations.
EXAMPLE 276 .2 (Variant of BoS with 由lpεrfect information) Consider another vari睛
ant of the situation modeled by BoS , in which neither player knows whether the
other wants to go out with her. Specifically, suppose that player 1 thinks that with
probability player 2 wants to go out with her, and with p时ability player 2
wan怡 to avoid 1矶时 player 2 役1inks that wi的主伽bi1ity ~ player 1 wants to
go out with her and with probability ~ player 1 wa附 to avo时 her. As before ,
assume that each playεr knows her own preferences.
We can model this situation by introducing four states, one for each of the possible con主gurations of preferences. 1 refer to these states a
i
i
9.1 Motivational examples
277
1
2
户、
M''
HD''
穹'之3气
S
AHU1i
AHUq4
lAAUU
叮
叮Lnu
B
2
1: yl
O 0, 2
S
State yy
State yη
2: y2
EFJ
fr
j
1: 叫
OIhM
q4nu
叫LAU
户气
υ­
门υ-A
PU
叮''』
ρL
&，、
叫"
飞、
《‘-
丁 A 芥 J1111ll
门U
B
57 『
E
--巧，阳
2: 级2
B
0, 1
], 0
0, 1
qs
•
Figure277.1 A variant of BoS in which each playεr is unsure of the other pJayer's preferences. Th♀
frame 1在beled i: x encloses the states that gener 在 te the signal x for player i; the numbers appearin吕
over this frame next to each table are the probabilities assigned by type X of player i to 础。 state she
regards to be possible.
of player 1 (who wishes to go out with player 号; after she receives the signal nl
she is referred to as typεnl of player 1 机1ho wishes to avoid player 2). Similarly,
player 2 has two types , Y2 and 112.
Type yl of playe仔r 1 believes that the probabi过li让ty of each of the sta挝tes yy and yn1
f play
iβS iL
扫;川t忖ypμe 叫
11η1 时
0句
归e盯
阳r川1 bel
创1ie
始e机们悦
V陀
7喘
es 也
tha挝t 出
the叩
p时a抬
bi芷li阳f 销创圳
choft估
he 纣如a挝te硝5 叩 a川ndn
川口
i5 ; Sinm说
i丑1a盯材吨扎均哗
1马}二什type y2 of pla玛叮ye仔r2ι2b
快
beli
且1凹
e\飞ves th飞 a拭t忖削
咔
t校hep
严roba呻b
悦
b刷
il
丑li坟ty of sta挝te 纠
yy is ~伊
a衍nd
ld
that of stat怆e们加
盯叼，
lí
and 校拉阳
l泊a挝tωof
们 仗
s ta挝te们
eη附
m
1ηfω
刊h
τ
祖
旧ísm
川
Oωdel
们
oftl盯
inEx
汩
am
丑1p1沁
e27
巧
怕
t o study the 叫
e qui出
lib
忱ria of this model we cωons
冶si过
d如
缸古 thep
e
抖1a
叮
y­
岱rs' 抖
e
p 1a
川ns of actÍon before they receÍve their signa1s. 丁hat is, each player plans an
action for each of the two possib1e signals she may receive. We may think of there
being four players: the two types of player 1 and the two types of player 2. A Nash
equilibriwηconsists of four actions , one for each of these players, such that the action of each type of each original player is optima l, given her be !ief about the state
after obser飞ring her sigηal， and given the actions of each type of the otheτoriginal
i
playel二
Consider type yl of play但1. Her beliefs about the probabilities of states yy and
yn and her payoffs in these states are the same as the beHefs and payoffs of play盯 l
for the two states in Example 273 .1 .τJlUS her expected payoffs for the four pairs
of actions of the 以10 types of player 2 are giveηin Figure 275.1 , interpreting each
column to represent a pair of actions of types y2 and n2 of player 2.
@EXERCIS充 277.1 (Expected payoffs in a variant of BoS with imperfect information)
Construct tables lìke the one in Figure 275.1 for type nl of player 1, and for typ汩汩
and n2 of player 2.
218
Chapter 9. Bayesian Games
1 claim tha t (段，时，侈，的) and (( S, B) ， 伍， S)) are Nash equilibria of the game ,
where ìn each case the first component gives the actions of the two types of player 1
and the second component gives the actions of the hvo types of player 2. You may
use Fí gure 275.1 to verify that B is a best response of type Yl of player 1 to the pair
(B , S) of actions of player 2，在nd 5 is a best response to the paìr of actìons (5 , 5).
You may use your answer to Exercìse 277.1 to verìfy that in each of the claimed
Nash equìlìbria the actìon of type nl of player 1 and the action of each type of
player 2 is a best response to the other players' actions 畅
1n each of these examples a Nash equilibrium is a 1ist of actions , onεfor each
type of each playe乙 such that the actìon of each type of each player is a best re俐
sponse to the actions of all tbe types of thεother pl在yer， given the player's beliefs
about the state after she observes her signal. The actions planned by the various
types of pl在yer i are not relevant to the decision problem of any type of player Î ,
but there is no harm in taking them, as well as the actions of the types of the other
player, as given when player ì is choosing an action.τhus we may defìne a Nash
叫uìlìbrium in 臼 ch example to be a Nash 叫uilibrium of the strategic game in
which the set of players is the set of all types of a l1 players in the original situation.
ln the next section 1 define the general notion of a Bayesian garηe and the notion
of Nash equilibrium in such a game. These defìnitions require significant theoretical developmen t. If you find the theory in the next section heavy going, you may
be able to skim the section and then study the subs何时nt il1 ustrations, relying on
the intuition developed in the examples in this section , and returning to the theory
only as necessary for c1arification.
9.2 General definitions
9.2.1 Bayesian games
A strategic game with imperfect information is called a UBayesian game". (The
reason for this nomenc1 ature will become apparer飞t.) As în a strategic game, the
decìsion-makers are called players, and each playεr is endowed wîth a s的 ofactions.
A key component in the specification of the imperfect information is the set of
states. Each state is a con飞plεte description of one collectíon of the players' relevant
characteristics, inc1 uding both their preferencεs and their information. For every
collection of characteristics that some player believes to be possible , there must
be a state. For instance, suppose in Example 273 .1 that player 2 wishes to meet
player 1. ln this case, the reason for inc1 uding in the model the state in which
player 2 wishes to avoid player 1 is that player 1 believes such a prefεrence to be
possibl巳a
At the start of the game a state is realized. The players do not observe this state.
Rather, each player receives a sigl1 al that may give her some ìnformation about the
state 口enote the signal player i receìves in state ωby τ)(ω). The fu时tion 句 is
called player i's signal functio l1. (Note that the signal ìs a deterministic function of
thεstate: for each state, a definitεsignal is receiv巳d.) The states that generate any
9.2 Ge衍 eral de俞 nitions
279
given signal ti are said to be consistent v年ith ti' The sizes of the sets of states consistent with each of player ì's signals reflect the quality of player i's information.
证， for example , Ti (ω) is different for each value ofω， then player i knows , given
her signal , the statεthat has occurred; after receiving her signal , she is pεrfectly
informed about a11 the players' rele飞rant characteristics. At the other ext陀me， if
Ti( ω) is the same for all states, then player i's signal conveys no information about
the state. If τi( w) is constant over some subsets of the set of states , but is not the
same for all states , then player i's signal conveys partial information. For example, if there are three states， ω1 ， ω2， and 仅如 and Ti(W1) 并飞 (ω7.)口 τ1 (ω升， then
when the state isω1 player i 如10WS that it isω1 ， whereas when it is eith仔 ω20 r ω3
she knows only that ìt is one of these two states. (In Figures 274 .1 and 277 .1, each
frame encloses a set of states that yield the same signal for one of the players.)
L句e refer to player i in the event that she receives the signal ti as type ti of
player i. Each type of each player holds a beli吃f about the likelihood of the states
consiste时 with her signal. If, for example, ti
飞 (ωd 口 Ti(ω址， then type ti
of player í assigns probabilities toω1 and ω2. (A player who receives 在 sìgnal
consistent with 0只ly one state n批urally assígns probability 1 to that st挝咛
Each player may care about the actíons chosen by the other players , as ìn a
strategic game with perfect information, and also about the state. The players
may be uncertain about the sta役， so we need to specìfy their preferences regarding probabìlìty distributions over pairs (a , w) consistìng of an action profile a and
a state ω. 1 assume that each player's preferences over such probabìlìty distributions are represented by the expected value of a Bernoulli payofffunctio l1. Thus 1
specify each player i's preferences by giving a Bernoulli payoff function Uj over
pairs (a， ω). (~臼te that in Examples 273 .1 and 276.2, both players care only about
the other pl町肘 's action, not independently about the state.)
In summary, a Bayesian game is defined as follows.
DEFINITION 279.1 (Bayesian g仰的 A Bayesian game consists of
创
刻
rbgar&'A
OO
时
ug
P
b
<
h
c3
争 a&a
--'aaee YM
QJM
and for each player
.. a set of actions
.. a set of signals that she may receíve and a signal function that associates a
signal with each state
.. for each signal that she may receive, a belief abou! the states consistent wìth
the signal (a probability distrìbution over the set of states with which the
signal is associated)
.. aBe盯棚
moullip
严
ayof任妇
f付func
吐tio川ve古叩
pai始
Y臼s (机
GιF ω
叫)， wher较 ai捡sa
衍I川 ctio叩r陀'Ofi
韶le
付
an
ωis a state, the expected value of which represents the player's preferences
among lotteries over the set of such pairs.
Chapt壁 r9. 移ayesian
280
Games
Note that the set of actions of each player is indεpendent of the sta祀 Each
pl咛挝 may care about the state , but the set of actions available to her is the same
m every state 攀
Theeponyrr飞ousThorr飞浅s Bayes (1702…61) 自rst showed how probabilities should
be changed in the light of new information. His formula (discuss伫d in Section
17.6 .5) is needed when one works with a variant of Definition 279 .1 in which each
player is endowed with a "prior" belief about the states, fro黯 which the beHef of
创ch of her types is derived. For the purposes of this chapter, the beHef of each
type of each pl巧er is more conveniεntly taken as a primitive , rath但 than beìng
derìved from a prior belie f.
The game in Examplε273.1 fits into this general definition as follows.
Players
τhe pair of people.
5tates The set of sta tes is {meet , a拟d}
Actions The set of actions of each player is {B , S}
Signals Player 1 may receive a single signal, say z; her signal functionτi satisfies T1 (meet) T1 (avoi的 z 乙 Player 2 receives one of two signals, say m and
v; her signal function T2 铅tìsfiesτ2(meet) 口 mand τ'2 (avoid) 二 v.
蚓
M
加
8
eliefs 凹
Pla叮
ye盯r 1μa臼g
伊
部$叩
n
pr叫
袖
O
b旧咄
abi血
i且it咛yi
丁 卡to
悦e份ad
出
h刊s归
t丑at怡ea拙
f丘t时们E挫夺仪Cω'in
吨
g the
陇e川$句m
Play衍 2a
踹ss刽ign冶s pr陀
ob冶
抬
a
bilit守
Y 1 tωo the state m
η1ε
衍
εt after reζei衬
ving the 剑
s ign冶
al m ,
and probabìlity 1 to the statε avoid after receiving the sìgnal 队
Payo的
The payoffs Ll i 归 ， meet) of each player i for all possible action pairs are
given in the left panel of Fi gure 274.1 , and the payoffs Ll í(a , avoid) are given
ir飞 the right panel.
Similarly, the game in Ex但nple 276.2 fits into the definition as follows.
Players The pair of people.
States The set of st在tes is {yy , yn ， ηy， m}.
Actio盯
The set of actions of each player is {B IS}
,
Signals Player 1 rεceives one of two signals, Yl and 111; her signal function τl
satisfies τ'1 (yy)
τ1 (yn) 二 y1 and 写(叩)平均 (1111) 二叫. Player 2 receives one of two 咆nals， y2 and 112; her 匈nal functìonτ2 satisfies T2 (yy)
T2(ny) = y2 and T2(y叫 Z τ2(nl1) = 112
制
M
如
8
e/i，ε台
d
Pla
町
ye
衍r1a
硝岱
i
创rj吨 t仙
he 鸣叫民 and probabìlity to each of thεstates 均 and 11n after
receiving the signal 111. Player 2 部signs P时ability ~ to the state yy and
p时ability ~ to the state 呼 after re优iving the signal y川ndp协abi1ity ~ to
the state yηand probability ~ to the state I1n 础er receiving the signal 112
Payo活 τhe payoffs Uì 怡， ω) of each player i for all possible action pairs and
states are given in Fi gure 277 .1.
9.2 General definitions
281
9.2.2 Nash equilibrium
In a strategic game , each player chooses an action. In a Bayesian game, each player
chooses a collection of actions , one for each signal she may receive. That is , i抗 a
Bayesian game each lype of each player chooses an actìon. 1n a Nash equilibrium of
such a game, the action chosen by each type of each player is optima l, given the
actions chosen by every type of every other player. (1口 a steady state, each player's
experie口ce teaches h盯 these actions.) Any given type of player Î ís not affected by
the actions chosen by the other types of player Î, so there is no hanτ1 in thinking
that player i takes as givεn these actions, as well as those of the other players. Thus
we may define a Nash equilibrium of a Bayesian game to be a Nasl飞 equiIibrium
of a strategic game in which each player is one of the types of one of the players in
the Bayesian game; 叭That is each player's payoff fur飞往ion in this strategic gdme?
Consid肝 type tí of player Î. For each state ωshεknows every other player's
type (i. e. she knows the signal received by every other player). 百üs information,
together with her beliεf about the states, allows her to calculate her expected payoff for each of her actions and each collection of actions for the various types of the
other players. For instance, 如 Example 273.1 , player l's belief is that the probabìlity of each state is and she knows that player 2 is 可pe m in the s城附们nd
type v in the state avoid. Thus if typε m of player 2 chooses B and type V of player 2
chooses 5, piayer 1 thinks that if she chooses B then her expected payoff is
!,
伊 1 (( B, B) , meet) 十 !Ul ((B, S) ,avoid),
where Ul ís her payoff function in the Bayesian game. (In general her payoff may
depend on the state, though in this example it does not.) The top box of th较 second
columniηFigure 275.1 gives thls payoff; the other boxes give player }'s payoffs for
heτother actíon and the other combinations of actions for the two types of player 2.
1n a general game, denote the pro切bility assigned by the be !i ef of type ti of
player Î to state w by Pr(ωI ti)' Denote the action taken by each type tj of each
player j by a(j, tj). Player j's signal in state ωis Tj(叫， so h时 action in thls statε
is 功，写 (ω)). For each stateωand each player j, let âj(w)
a (j， 写(叫). Then the
expεcted payoff of type ti of player i when she chooses the action aj is
nM
<
nHH
-u
ω
，，
8‘、飞
u
，，
av‘
g飞
·、
，公膏'伽
铲A
'''l
飞飞
·
叫
Z
PA 'u
(281.1)
where 0 is the set of states and (aj， â_í(的) is the action profile in which player i
chooses the action ai and e刊ry other player j chooses âj( 叫‘ (Note that thls ex掏
pected payoff does not depend on the actions of any other types of player i, but
only on the actions of the 飞rarious types of the other players.)
We may now define preci使ly a Nash equilibrium of a 如yesian game.
j>- DEFINITION 28 1. 2
(Nash e，号 uilibrium of Bayesian game) A N在sh equilibrium of a
Bayesian g挫le is a Nash equilibrium of the strategic game (with vNM prefer唰
ences) de位led as fo l1ows.
Chapter 9. Bayesian Gam部
282
Players Th e set of a11 pairs (川í) in which i ís a player 如 the Bayesian game and tí
is one of the signals that i may 肥ceive.
Actions The set of actions of each player (i, ti) is the set of actions of player i in the
Bayesian game.
Preferences The Bernoulli payoff function of 出ch player (( t i) is given by (28 1. 1).
r EXERCISE 282.1 (Fi ghting an opponent of unknown strength) Two people are involved in a díspute. Person 1 does not know whether p时son 2 ís strong or weak;
she assigns probabilityαto person 2's being strong. Person 2 is fully informed
Each person can either fíght or yield. Each person's preferences are represented by
the expected value of a Bernoulli payoff function that assigns the payoff of 0 if she
yields (regardless of the other person'::; action) and a payoff of 1 if she fights and
her opponent yields; if both pεople fight, then their payoffs are (… 1, 1) if persor飞 2
is strong and (1 ，一片 if P时son 2 is wea k. Formulate this situation as a Bayesian
game and find i怡油sh equilibria if il: < i 在IIdifa:>i
‘
o EXERCISE 282.2 (A口 exchange game) Each of two ìndi，叫 uals receives a tìcket on
\yhich there is an integer from 1 to m i贝dica他19 the sìze of a príze she may recεive.
The individuals' tickets are assigned randomly and indεpendent1y; the probability of an individual's receiving each possible number is positive. Each indi飞rid­
ual ìs g Ì\-en the option of exchanging her prize for the other局 individual's prize;
the individuals are gi飞ren this option simultaneousl予If both individuals wish to
exchar飞ge ， then the prizes 旺e exchanged; otherwise each individual receives her
own prìze Each individual's obj机 tive is to maximize her expected monetary pay啕
off. Model this sítuation as a Bayesian game and show that in any Nash 吨uílib­
rium the highest prize that eith时 individual is willing to exchange is the smallest
possible priz巳
‘
? EXERCIS 仨 282.3 (Adverse selectio时Firm A (the 飞 εquirer") is considering ta挝ng
over firm T (theμtarget"). ìt does ηot know firm T' s 飞咄飞le; it be1ieves that this
飞'alue， when firm T is controlled by its own management, is at least $0 and at most
$100 , and assigns 巳qu在 1 probability to each of the 101 dollar values in this rang巳
Firm T will be 飞North 50'Yo more under firm A's mana 。2;ement than it is under its
own managemen t. Suppose that fìrm A bids y to take ov佼 firm T , and firm T îs
worth x (under its own management). Then jf T 注汇cepts A' s offer, A's payoff is
~x - y and T' s payoff is y; if T 叫ects A's offer, A's payoff is 0 and T' s payoff is
X. Model this situation as a Bayesian gamεin which firm A chooses how much
to offer and finn T decides the lowest offer to accep t. Find the Nash eguilibrium
(equilibria?) of thís game. Explain why the logic behînd theεquilibrium is called
adverse se/ection.
9.3τwo examples concerning information
Th号 notion of a Bayesian game may be used to study hO\年 information patterns
affect the outcOl飞飞e of strategic interactiOl飞 Here are two examples.
9 .3 Two examples concerning information
L
T
B
1 ， 2ε
2, 2
M
R
1, 0
0, 0
1 ， 3ε
0, 3
283
2
T
8
L
1, 2f
2, 2
1,
0,
State w1
Figure 283.1 The first Bayesian game consid告red in Sectíon 9.3.1.
9.3. 7 More information may hurt
A decision咽makεr in a single-person decision problern cannot be worse off if she
has more information: if she wíshes, she can ignore the inforrnation. In a game the
sarne ís not true: if a player has more inforrnation and the other players know that
she has more inforrnation , then she may be worse off.
Consider, for example , the two-player Bayesían game in Figure 283 .1, where
0<ε<
In this game there a时wo st舵s， ar由leither player knows the state
Player 2's unique best response to each action of player 1 ìs L (If play时 1 chooses
T, L yields 2E, whereas M and Rξach yield ~E; if player 1 chooses B, L yields
2, whereas M and R each yìeld ~.) Fu巾er， player l's uniquεbest response to
L is B. Thus (B , L) is the unique N册h equilibriurn of the game; it yields each
player a payoff of 2. (If you have studied Chapter 4, you will be able to verify that,
moreover, the garne has no other rnixed strategy equilibriurn.)
Nowconsid盯 the variant of this game in which player 2 必 informed of the state:
player 2's sìgnal function T2 satisfies τz(的)并写 (ω2)' In this garne (T , ( 兵， M)) is
the unique Nash equilibrium. (Each type of player 2 has a strictly dominant action ,
to which T is player l's unique best response.)
Player 2's payoff in the unique Nash equilibriurn of the original game is 2,
whereas her payoff in the unique Nash 吨uílibriurn of the garne i只 which she
knows the statεis 3εín each state. Thus she ís worse off when she knows thεstate
than when she doεs not. To understand this result, notice that arnong player 2's 缸­
tions, R is good only íri state ω1 ， M is good OI让y ín state w2 , and L is a cornprornise.
Wh en she does not know the state she chooses L, inducing player 1 to choose B.
When she Ìs fully informed she tailors her action to the state, choosing R in stateωl
and M ìn stateω2， inducing player 1 to choose T. The garne has no steady state in
which she igr飞oresh仅 inforrnation and chooses L because this action leads player 1
to choose B, making R better for player 2 in state w1 and M betler in stateω2.
i.
9.3.2
Infection
ηle notion of a Bayesian garne rnay be used to rnodel not only situations in
which players are uncertain about each others' preferenc时， but also situations in
which they are unce主tain about each others' knowledge七 Conside乙 for ex沼我ple， the
Bayesian game in Figure 284 .1.
Ch昌 pter 9.
284
3
Z
Bayesian Games
Z
1
L
R
2, 2
2
R
2, 2
0, 0
L
0, 0
L
L
2, 2
R 3, 0
1, 1
R 0, 0 1, 1
j己
0, 0 1, 1
L
5tate α
5tate 卢
。，。
5tate γ
Figure 284.1 The first Bayesiar飞在ame in Sectìon 9.3.2. In the unique Nash equilìbrium of th臼 gan1e ，
each type of each player chooses R 崎
Notice that player 2's preferences are the same i口 all three states , and player l's
preferences are the same in states 卢 andγ. ln particula飞 111 state γ， each play衍
knows the other play缸 's prεferences， ar飞d player 2 knows that player 1 knows her
preferenc巳s. The shortcorning in the players' information in state γis that playe川
does not kno 'vv that player 2 knows her preferences: player 1 kno飞tVS or飞ly that the
st毒犯 is either 异 orγ， and in state 卢 player 2 does 110t know whether the state is 夜 or
卢， and hence does not know pJayer l's preferences (because player l's preferences
in these two states differ).
This imperfection in player l's knowledge of player 2's information si在nifí如
cantly affects the equilibria oÍ the game蕉If ir飞formation were perfect in state /了，
then both (L , L) and (R , R) would be Nash equilìbria. However, the whole game
hasa 议
ω}η
叫
1
asked to sh飞ow in the n飞乞xt exerClse. 丁he argument shûws that tl飞e incentives faced
by player 1 in state 0: "infect" thc remainder of the game
EXERCISE 284.1 (I nfection) 5how that the Bayesian game in Fìgure 284.1 has a
unique Nash equilibrìum , in which each player chooses R regardless of her sîgna 1. (5tart by considering player l's action in statel:-: 剖ext consider player 2's
action when she gets the sigr飞a1 that the state isαorβThe口 consider player l's
action when she gets the signal that the state is 卢 orγ. Fìnally consider player 2's
action in state γ)
Now extend thεgan飞巳 as in Figure 285. 1. Consider state ð. In this state , player 2
knows player 1'5 prefe陀nces (because she knows that the state is either γorð， and
in both states player l's preferences are the sam的. 飞离Ihat player 2 does not know is
whether player 1 kno飞"，s that player 2 knows player l's prεferences.τhe reason is
that play时 2 does not know whether the statεisγor ð; and in state γplayer 1 does
not know that player 2 knows her preferences, because she does not know whether
the state is ß or 1 , and in state 卢 player 2 (飞"，ho does not kl飞ow whether the state
IS 夜时向 does not know her preferences. Thus thηlevel of the shortεomiηg in the
players' information is higher than it is in the game in Figure 284. 1. Nevertheless,
the incentives faced by pl可er 1 in state ít again 气nfect" the remainder of the game ,
and in the only Nash 叫uilibrium every type of each player choos部R.
Th巳 game may be further extended. As it is extended , the level of the imperfection in the play巳rs' information in the last state increases. When the number of
9.4 lII ustration: Cournot's duopoly game with imperfect information
l
2
L
L
2, 2
Z →一飞
1
4
3革
R
ι
0, 0
七tate
只
2, 2
0, 0
0, 0
1, 1
State 卢
0:
285
2
L
R
是3
L
2, 2
0, 0
1
R
1
4
。， 0
L
L
2, 2
1, 1
R
。，。
State γ
R
0, 0
1, 1
St吕 te ð
Figure 285.1 The second Bayesian game i只 Section 9.3.2.
states is large , the players' information in the last state is ∞ly very slightly imperfec t. Nevertheless , the incentives of player 1 in state 交 still cause the game to have
aur飞ique Nash equilibrium, 如 which every type of each player chooses R.
In each of these examples , the e气uilibrium induces an outcome in every s相 te
that is worse for both players than another 0的come (namely (L , L)); in a11 states
but thε 估价， the alternative outcome is a 民ash equilibrium in the game with per揣
fect informatior飞 For some other specifications of the payoffs in state 夜 and the
players' belìefs , the game has a unique equilibrium in which the 飞。od" outcome
(L , L) occurs in every state; the poi时 is only that 0时 of the two Nash equilibria is
selected , not that the "bad" 吟出librium isηecessarily selected. (Modify the payoffs of player 1 in state 次 so that L strictly dominates R, a1飞d change the beliefs to
a附$佣si培
gn
叩
伊r时 剧制
p
abbil过Iit守
y 柏
tO 制阳C巾
h 仗阳削
a挝t怡θ盯C∞
∞
O
叫
m1汗pμ?
冶收
a批t巾
i山bl怡ewi佼白
h
飞1e白achs
飞刊$吨n冶叫
a1.)
!
9.4
lII ustration: Cournot's duopoly game with imperfect information
9.4.1
Imperfect information about cost
τ\vo firms compete in selling a good; one firm does not know the other firm' s cost
function. How does this lack of information affect the firms' behavior?
Assume that both firms can produce the good at constant unit cost. Assume
a150 that they both know that firm 1's unit cost is c, but only 如m 2 knows its own
unit cost; fìrm 1 believes that firm 2's cost is cL with probability and cH with
probability 1 …在 w here 0 < < 1 and CL < CH.
认fe may model this situation as a Bayesian game that is a variant of Cournot's
game (Section 3.1).
e
e
Players Fi rm 1 and firm 2.
5tates {C H}
Actions Each firm's set of actions is the set of its possible outputs (no口只egative
numb岱s).
5ignals Firm l's signal function 町 satisfies 写归)
= Tl(L) 加 signal is the
部meiηboth states); firm 2's signal function T2 satls阳 τ2(H) 乒 T2(L) (its
sigηal is perfectly informatîve of the state).
Chapter 9. 8aye量 ianGa然毒草
286
1
凡。
-1
日
"FL
→止
…
L
Figure 286.1τhe information structure for the model in the variant of Courno t'岳 model in Section 9.4. 1,
in which firm 1 does not know firm 2'5 cost. The frarne labeled 2: X , for x = L and 芷= H, en c\ oses the
f主 tate that generates th世 signal x for firm '2.
e
Beliefs The single type of firm 1 assigns probability to state L and probability 1 … to state H. Each type of firm 2 assigns probability 1 to the single
state consistent with its signa l.
e
Payoff functions The firms' Bernoulli payoffs are their profits; if the actions
chosen are (价，如) and the stδte is 1 (either L or H) , then firm 1's profit is
ql (P(q1 个 q2) … C) and firm 2's profit is (P( 们才但) - C 叶， where P(Q1 十但)
is the market pri优 when the firms' outputs are '71 and Q2.
The information structure in this game is similar to that in Example 273 .1; ít is
ilI ustrated in Fi gure 286 .1.
A Nash equi 1i brium of this game is a triple (剖 ， qL， q 'H)， 飞vhere qi is the outp肘
。f firm 1, q1: is the output of type L of firm 2 (i. e. firm 2 when it 陀cei飞咄 the signal
τ云山， and q'H is the output of type H of firm 2 (i.e. firm 2 when it receives the
signal 写(川口， such that
.. qi ma刘红山es firm 1's profit given the output qt. of type L of firm 2 and the
output qÎ7 of type H of fírm 2
.. q1. maxÌmizes the profit of type L of firm 2 given the output qî of firm 1
.. q'H maximizes the profit of type H of fírm 2 given the outp时 qî of firm 1.
τ'0 find an equilibrium , we first find the firms' best response functions. Given
fírm ]'s beliefs, its best response b] 坷 L ， qH) to (qL , qH) solve京
-rx
lO(jJ 恃 i 十 qd - C)q1 牛仔…的( P(q1 斗守 H) … C) 号 1] .
Firm 2's best response bdq1) to q1 when its cost is CL soJvεs
气?xlW(q1 十 qd
cdqd ,
a l1d its best response bH (q1 ) to q1 wh巳n its cost is CH solves
甲ax [(P(q1 十 qH) … CH)qH]
吁H
A Nash equilibrium is a triple 句 î ， qL' q'H) such that
号î = bJ(qLq 'H), qÎ.
bdqî)' and q'H
bH (ql').
9 .4 lII ustration:ζournot￥ duopolyg 矗 me with imperfect informatìon
287
ø EXERCJSE 287.1 (C∞rnot's duopoly game with imperfect information) Consider
thεgame when thεìnverse demand function is given by P( Q) =α …♀ for Q 三 a
and P(Q) = 0 for Q >次 (see (56.2)). For values of cH and CL close enough that
there is a Nash equílibrium in whìch all outputs are positive , find this 吨uilibrium.
Compare this equilibrium with the Nash equilibrium of the game in which firm 1
knows that firm 2's unit cost is CL , and with the l'、Jash equilìbrium of the game in
which firm 1 knows that firm 2's unit cost isεH
9.4.2
Imperfect informaUon about both cost and information
Now suppose that firm 2 does not know whether firm 1 knows firm 2's cos t. That
lS , supposεtha t one circurr飞 stance 出at firm 2 belìeves to be possiblc is that firm 1
knows its cost (although in fact it does no斗，在ecause firm 2 thinks this circumstance
to be possible , we 时ed four states to model the situation , which 1 call LO , HO , L1,
and H1 , with the fol1 o，叫 ng interpretations.
LO: firm 2's cost is low and firrn 1 does not I∞ ow whether it is low or high
HO: firm 2's cost is high and firm 1 does not know whether it is low or high
L1: firm 2's cost is low and firm 1 knows it is 10认7
H l: fìrm 2's cost is high and firm 1 knows it is high.
Firm 1 receives one of th陀e possible signals, 0, L, and H. The states LO and HO
generate the signal 0 (firm 1 does not know firm 2's cost) , the statε L1 generates
the signal L (fìrrn 1 knows firm 2's cost is low) , and the state H1 generates the
si京1al H (firm 1 knO\vs firm
cost is high). Firm 2 receives one of two possible
signals, L, in states Lü and L1, and H , in states HO and H l. Denote by e(as before)
the probability assig口时 by type 0 of firm 1 to f让m 2's cost being CL , and by πthe
probability assigned by each type of firm 2 to firm 1's knowìng firm 2's cos t. (The
case 7I = 0 is equivalent to the one considered in Section 9.4.1.) A Bayesian garne
that models the situation is defined as follows.
Playeís Fi rm 1 and 自rm2.
States {LO， 后 ， HO ， Hl} ， wher伫 the first letter in the name of the state indicates
firm 2's cost and the second letter indicates whether firm 1 does (匀。r does
not (0) know firm 2's cos t.
Actions Each firrn's set of actions is the set of its possiblεoutp飞Its(nonnegative
numbers).
Signa往
Firm 1 gets one of the sì伊1als 0, L, and H , and her signal functìon τ1
satisfies 罚性的口飞 (HO) = 0, Tl( L1)
L, and 飞 (Hl)
H. Firm 2 gets the
si伊al L or H and her signal function 写 satisfies T2 (L的口时间= Land
τ2(HO)
巧 (H1) = H.
Chapter 趴在ayesian Games
2事8
Figure 288.1τi飞e ínformation structure for the modd ìn Section 号 .4.2， in which firm 2 does not know
whetl飞苦 r firm 1 kno飞vs its cos t. The fτame labeled ì: x endoses the st在 t告s that generale the signal x for
fìrm i.
Belíefs Fírm 1: typεo assìgns probability e to state LO and probability 1 - e to
stδt公 HO; type L assígns probability 1 to state L1; type H assigns probabílity 1
to state H. Firm 2: type L assigns probability 7[ to state L1 and probability 1
πto state LO; type H assigns probabilityπto state Hl and probability 1 … π
to statεHO
pαyof疗ffiυ门 ctions
The firms' Be了n飞ou11i payof旺fsa盯fεtheir prof且its句; if the actior丑飞5cho­
S臼
sel
创1川ìa
缸E挖
ε (问阳
们1 ，斗吁2β) ， 出
q
th飞.1er
创仙
nf扣
i订了rm
Q2(P(q们1 + (如
12ω2ρ) … C红tJ 凶
i ns仗ta挝t已臼5 LO a时 LL and Q2(Pl 们今但
cd in state5 HO
and H 1.
The 出
i nforma
挝tion str唱飞uctur陀巳 in 出
thi衍5 gamτ
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e.
CD EXERCISE 288.1 (Cournot's duopoly gamewith imperfect informatio时 Writedown
the maximizatioll problems that determ加e the best response function of εach type
of each player. (Denote by qo , qi , and qll 出
t h飞e outψpu
时t沁50f
泛f 咛
t yp
严
岱50
θ
ar只ld by qL and qf付-! 白
the out甲
pu
时f沁5 of type5 L and H of firm 2.) 挝t、斗Jo飞w suppos拇
e 仙
t ha
仗t the
111飞ve
臼r5ε de
创mar只ld function 巳
i sg
♂i飞F吧e
创咄咄
l趴
仰ì by P( 安划) = 庆 … Qfιo厅f ♀ 豆 仪 and P( Q) = 0 for
Q> 乱 For 兀ralues of CH aI飞d CL close enough that there is a Nash equilibrium in
which all outp江 ts arεpositive， find this equilibrium. Check that when lT = 0 the
叫uilibrium output of type 0 of firm 1 is equal to the equilibrium output of firm 1
you found in Exerci使 287.1 ， and that the equílibrium outputs of the two types of
firm 2 are the same as the ones you fOUl飞d in th时 εxercíse. Check also that when
7[乙二 1 th巳吟uilibrium outputs of type t of firn飞 1 ar飞d type L of 在rm 2 are the same
as the 叫l1 ilibríum outputs when there is perfect information and the COSt5 are c
and CL , and that the equilibrium outputs of type h of firm 1 and type H of firm 2
are the same as the equilibrium o u. tputs when there Îs perfect information and the
costs arεεand CH. Show that for 0 <π< 1, the equilibrium outputs of types L
and H of firm 2 lie between their values when 汀= Oand when π= 1.
9.5 lIIustr就ion: providing 器 publicgood
9.5
主 89
lII ustration: providing a public good
Suppose that a public good is provided to a group of peoplεif at least one person
is willing to pay the cost of the good (as in the model of crime reporting in Sec币
tion 4剧. Assume that the people differ in their valuations of the good , and each
person knO\l飞IS only her own valuation. Who, if anyone, will pay tl飞e cost?
Denote the number of individuals by 11 , the cost of the good by c > 0, and
individual i's payoff ìf the good ìs provided by Vi. If the good is not provided ,
then each individua l' s payoff ìs O. Each individual i knows her own valuation Vi.
She does not know anyone else's valuation, but she knows that all valuations arε
at least 里 and at most 否， where 0 ::;空< c < V. She believes that the probability
that any one indì功dual's valuation is at most v is F(吟， ind叩 endent of all other
individuals' 飞raluations， wherεF is a continuous increasing function. The fact that
F is increasing means that the individual does not assign zero probability to any
range of values between 旦 and v; the fact that it is continuous means that she does
not assign positive probability to any single valuation. (An example of the function
F ìs shown in Fi gure 289.1.)
The following mechanism deterrnines whether the good is provided. All l1拍拍
di飞liduals simultaneously submit envelopes; the envelope of any indìvidual i may
contain either a contribution of c or nothing (no intermediate contributions are
allowed). If all individuals submit 0, then the good is not provided and each individual's payoff is O. If at least one individual submìts c, then the good is provided,
each individual i who submits c obtains the payoff Vi … c, and each individual i
who submits 0 obtains the payoff Vi. (The pure strategy Nash equilibria of a varìant of this model, in whích more than one contrìbution is needed to provide the
good, are considered in Exercise 33 .1.)
We can formulate this situation as a Bayesian game as follows.
Players
τhe set of 11 individuals.
States The set of all profiles (凡…， V n ) of 飞!aluations， where Q 三句三百 for
all i.
Actio盯
Each player's set of actions is 栓， c}
1
c
v
Figur窑茹苦.1 An example of the function F for 趴e model in Section 9 丘 Foreachγalueofv， F(v) 总 th号
probability that any given indivìdual's valuatìon ìs at most v.
ζhapter 9.
290
Bayesian G矗 mes
Signals The set of signals thatεach player may observe is the sεt of possible
valuatìons. The signaì fU !1ction 巧。f each player i is given by τj(Vlt.. ., V n}
vd岱 ch player knows her own valuatìon).
assi言ns probability
F(vdF(U2)'" F(Vi … })x
F(Vi十 :}F(z川 to the eve时 that the valuati∞ of every other player j ís
at most V j-
Beliefs Every type of player
Payo疗句 nctions
内υUU
户
L
JIll-t飞
(
Player i's 如 rnOl山 payoff in state (Vl ,' . ., L' 11 ì is
if no one contributes
if í does not contribute but some other player does
if í contributes.
EXERCISE 290 .1 (Nash 叫出libria of game of ω时命的ing to a pl仙c good) Find
conditions under which for each 飞'aluεof i this game has a pure strategy Nash
equilibrìum in which each type z斗 of play臼 i with Z' i > c contribut臼， whereas
every other type of player Î, and all types of e飞'ery other player, do not contributε
ln addition to thεNash equilibria identified in thísεxercise， the game has a
symmetric Nash equilibrium in ..气rhich e飞毛ry player contributes if and only if her
\'aluation exceeds 50n飞巳 Cl立ical amount (';. For such a strategy profile to be aI飞
equilibri江m， a player 飞vhose \'aluation ìs
th汶口 V must optirr飞ally not contrìbute , and a playεr whose valuation is at least [1' must optimally contributε­
Consider player i. Suppose that 心 \'ery other playεr contributes if and only if heτ
valuation is at least 扩 The probability th3t at least one of the other players contributδ5 is the probability that 3t least one of tl1伫 other playeI￥ valuations is at
least (;川 whích is 1ψ(F( i:'*))卜 1 ， (Note that (F( LI 药片卜 1 is the probability that all
the other valuations are at most v'.) Thus if player i's valuation is u打 her expected
ifshedoesnotco山-ïb时ea叫 Vi -- c if shεdoes conω
payoff is (1 一 (F( 扩 ))'/-1
tribute‘ H巳ncεthe conditions for player i to optìmally not contribute wher飞 Vi < v*
ilnd optimally contribute when Vi 主旷 a 四 (1 一 (F(v女 )γJ 斗 )z气注 Vi - (if Vi < 扩 r
and 门… (F(V'))11-1 ) 阿三 V i - c if z' i ::;> 7'ψ ， or equivalently
",\ F( ，'~) )í1 1 二( ifl 与<: Il
v ，(F(y 喝口1l -1 ::;> (
if Vj 兰 v 二
(290 2)
‘
If these inequalities are satisfied , then
(F( _ ))11 … 1 二二 t\
(290.3)
Conversely, if 扩 satisfies (290 .3), then it satisfies the two inequalities in (290.2).
τhus the game has a Nash equilibrium in which every player contributes wheneveτher valuation is at least v* if and only if u产 satisfies (290.3).
Notεthat because F( 叫= 1 only if v 主苔， and 言 > c, we have 扩 > c. That is,
every player's cutoff for COI飞 tributi鸣 exceeds the cost of the public good. Wh en
费忍lIIustration:
auctions
291
at least one player's 飞raluatíon exceeds c, all players are better off if the public
good is provided and the high-valuation player contributes than if the good is not
provided. 日ut in the equilibrium , the good is provided only if at least one player's
曰luation exceeds v' , which exceeds c.
As the num已er of individuals increases, is the good more or 1εss likely to be
provided in this equilibrium? The probability that the good is provided is the
probability that at least one player's 飞.r aluation is at least v 气 w hich is equal to 1
(F 作家) fl. (Note that 扭扭叮 )11 is the probability that every playe内 valuation is
less than .) From (290.3) this probability is equal to 1 - c F(v* ) h广，江ow does
t户 vary with 11? As 11 i时reases， for any given value of 扩 the value of (F(v*))II-l
decreases, and thus the val时 of v 气 F( 叫)川 decreases τhus to mai时ain the
equali专Y (290.3) , the 飞ralue of v* must increase as l'í increases We conclude that
as 11 increases, the change in the probability that the good is provided depends
on the change in F (V~) 1 们不 as v* increases: the probability increases if F(v*)lv*
is a decre附加g function of 扩， whereas it decreases if F( γ) Iv' is an increasing
functíon of 扩If F is uniform and Q > 0, for example,
) 11)* is an increasing
functíon of '1'*, so that the probability that the good is pr们vided decreases as the
population size increases.
丁he notion of a Bayesian game may be used to model a situation in which each
player is ul飞certain of the number of otlìer players. 1n the next exercise you are
asked to study another variant of the crir抗争犯porting model of Section 4.8 iηwhich
each of 吐ìe two players does not know whether she is the only \气ritness or whεther
there is anothεr witness (in which case she knows that 飞Nitness's valuatíon). (The
εxercise requirεs a knowledge of mixed strategy 手Jash equilibrium (Chapter 4).)
‘
EXERCISE 29 1.1 (Reporting a cr出le with an unknown numbεr of witnesses) Co扣
sider the variant of the model of SectÌOI飞毛8 in which each of tvvo players does not
knovv whether she is the only wi红less or whethεr 泣lere is another wÌtr飞ess. De喃
note by πthεprobability each player assigns to beìng the sole witness. Model thìs
situation as a Bayesian game with three states: one in which player 1 is the only
witness, one in ,,,,hich player 2 is the only witness, and one in which both players
are wìh飞 esses. Find a condition on 泛泛nder "vhich the game has a pure Nash equilibrium in \Nhich each player chooses Call (given the signal that she is a witness) ,
When the condition is 飞'iolat时， find the symmetríc mixed strategy Nash equilibrium of the game, a1飞d check that when 汀 o this equilibríum coincides \气rith thε
one found in Section 4.8 forη2.
9.6 Jilustration: auctions
9.6.1
Introduction
In the analysis of auctions in Section 3.5, every bìdder knows evεry other bidder's
valuation of the object for sale. Here I use the notion of a Bayesian game to analyze
auctions in which bidders are not perfectly inforrned about each otl飞棚'兀raluations.
292
Chapter 认 Bayesian
Games
Assume that a single object is for sa讼， and thatεach bidder independently receives some information-a "signal"… about the value of the object to her. If each
bidder's si伊al is simply her valuation of the object, as assumed in Section 3.5,
we say that the bidders' valuations are prívate. If each bidder's valuation depends on oth盯 bidders' sì伊1als as well as her own, we say that the valuations
are com 抒1on.
The assumption of private values is appropriate, for example, for a work of art
whose beauty rather than resale value interests the buyers. Each bidder knows
her valuation of the 0坤ct， b时 not thatof a町 other bidder; the other bidders' valuations have no bearing on her valuation. The assumption of common 飞址ues is
appropriate, for example, for an 0注 tract containing u时mown reserves on which
each bidder has conducted a tes t. Each bidder í's test resu1t gives her some infor1主lation about the size of the reserves, and hence her 飞ralu时ion of thεse reserves ,
but the other bidders' test results , if known to bidder i, would typica l1 y improve
this information.
As ín the analysis of auctions in which the bidders are perfectly informed about
each others' valuations, 1study models in which bìds for a single object are submitted simultaneously (bids are seale的， and the participant who submits the highest
bid obtaíns thεobject. As before 1 consid时 botl飞卢rst-price auctions, in which the
winner pays the price she bid , and second-price auctions, in which the winner pays
the highest of the remaining bids.
(1n Section 3.5 1 argue that the 如st-price rule models an open descending
("Dutch") auction, and the second-prìce rule models an open ascending ("Englìsh") auction. Note that the ar在ument that the second-price rule corresponds
to an open ascending auction depends upon the bidders' valuations being prìvate.
If a bidder is ur飞certain of her valuatìon, which is related to that of other bidders,
then in an open ascending auction she may obtain in如rmatìon about her valuation
from other participants' bids, informatìon not available in a sealed七id auction.)
1 first consider the case in which the bidders' valuations are private，出en the
case in which they arεcommon.
9.6.2 Independent private values
ln the case in which the bidders' valuations are prìvate, the assumptions about
these 飞raluations are similar to those 如 the previous section (on the provision of
a public good). Each bidder knows that a11 other bidders' valuations are at least
三 0， and at most v. She believes that the probability that any given
bidde内 valuation ís at 1110St v is F( v) , independent of a11 other bidders' valuations ,
where F is a continuous increasing function (as in Figure 289.1).
丁11εpreferεnces of a bidder whose 飞raluation is v are represented by the expected value of the Bernoulli payoff function that assigns 0 to the outcome in which
she does not win the object and v … P to the outcome in which she wins the object
and pays the pricε p. (That is, each bidder is risk neutral.) 1 assume that the εx­
pected payoff of a bidder whose bid is tied for first place is (v .- p) / m, where m 总
旦， where y
9.6 lII ustration: auctions
293
the number of tied winning bids. (τhe assumptìon about the outcome when bids
are tied for first place has mainly 气echnical" significance; 如$茫茫 tion
it was
convenient to make an assumption different from the one here.)
冶
e a1旧t抗ior只1 飞w
矿吨
f
Denote by P( 的 the price paid by the wγ inne眩r of the
bids is b. For a first-price auction P(b) is the winning bid (the largest 均， whereas
for a second-price auction it is the highεst bid made by a bidd时 different from the
winn白. Given the appropriate specification of P, the fo11owing Ba归sian game
models 在rst- and second-price auctions with independent private valuations
(and imperfεct information about valuations).
Play芒rs
The set of bidders, say 1, . . . ，我.
States The set òf a11 profiles (Vl ,' . ., Vn ) of valuations , where 旦主 Vj :S:否 for
a11 i.
Actions Each player's set of actions is the set of possible bids (nonnegative
numbers).
Signals The set of signals that each player may observe is the set of possible
valuations. The signal function Ti of each player i is given by Tj( V }, . . "V I1 )
Vj (each player knows her Qwn valuation).
Beliefs Every type of player í assigns probability F(Vl)F( 时. ,. F( 1.0 -1) x
F(Vi+l) . . . F(v l1 ) to the eve川 that the valuation of every other player j is
at most vJ
‘
Payo仔 functions
Player i' s Bernoulli payoff in state (Vl' . . . , Vn ) is 0 if her bid bi
is not the highest bid , and (Vi P(b))lm if no bid is higher than 占i and m
bids (including bi ) are equal to bi :
γA
QU
J
r?A
机
u 萨弄
弄 m 幻
dp~
MVAmnRJal
EPKO
h--h
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rτhUE
bh 勺 b
<一>
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,,,,,m
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-
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U
Nash equili扣切m in α second翩price sealed-bid auction As in a second-price sealed-bid
auction in wruch
bidder knows e\'ery otl飞机 bidder's valuation,
in a second-price sealed-bid auctio l1 with impe求cti收rmation abo时 valu­
atíolls , a player's bià eq切 1 to her valuation weakly dominates all her other
bids.
Precisel予 consid仔 some type Vj of some player í, and let bí bεa bid not equal to Vi.
Then for all bids by all types of a挂在e other players, thεexpected payoff of type Vi
of player i is at least as high when she bids 句 as it is when she bids 鸟， and for some
bids by the various 守pes of the other players, her expected payoff ìs greater when
she bìds Vj than ít is when she bids 坏
294
Chapter 9. BayesÎ矗nGame军
古1εargument for this result is similar to the argument in Section 3 .5 .2 in the
case in which the players know each oth町 's valuations. The main difference between the arguments arises because in the case in which the players do not know
each others' valuations, any given bids for every type of every player but Í leave
player i uncertain about the highest of the remaining bìds, because she is uncertain
of the other players' types. (The difference in the tie附breaking rules between the
two cases also necessitates a small change in the argumen t.) 1n the next exercÍse
you are asked to fill in the details.
( EXERCISE 294.1 (Weak domination in a second-price sealed-bid auction) Show that
for each type V 1 of each player i in a second-price sealed-bid auction v.rith imperfect
informatìon about 飞叶飞lations the bid v ， 飞气!eakly dominates all other bids.
Vle conclude, in partìcular, that a second-price sealed-bid auction wíth imperfect information about valuatìons has a Nash 吨 uilibrium in which every type of
ev臼y player bids her valuation. The game has also other equilibrìa , som号。fwhich
vou are asked to find in the next exercise.
? EXERCISE 294.2 (Nash equilibrîa of a second-price sealed-bid auction) For every
player Í , fínd a Nashεquilibrium of a seco口d-price sealed-bid auction in which
player i 认rins. (Think about the Nash equilibria when the players know each
oth但s' valuations , studied in Section 3.5.)
Nash eqllilíbl切 m Î I1 afirst俨户'íce sealed-bíd auctíon As when the players are perfectly
informed about each oth啊 's v叶uations， the bid of Vi by type Vi of player í weakly
dominates any bid great町 than 町， does not weakly dominate bids less than 巧， and
is itself weakly domìnated by any such lower bid. (lf type Vi of player i bids V ù her
payoff is certainly 0 (either she wins and pays her valuation , or she loses) , whereas
ìf she bids less than 町， she may win and obtain a positive payoff.)
These facts suggest that the game may have a Nash equilibrium in which each
player bids less than her valuatìon. An analysis of the game for an arbitr着ary disf川bution F of valuations requires calcuJusδ口d isτelegated tO an appendix (Section 9.8). Hεre J consider the case in 飞.vhich th舍re are two bídders a1飞d each player's
valuatíon is distribut创 "uniform厅， behveen 0 and 1. This assumption ∞ the distribution of valuations means th挝 the fraction of 飞呗luations less than v is exactly
v, so that F 的 = v for a lJ V with 0 s:万三 1
Denote by 以叫 the bid of type V of player i. 1 claim that if there are two bidders and the distributíon of 飞raluatíons is uniform betw臼n 0 and 1, the game has a
(symmetric) Nash equilibrium 如 which the function ßí is the samεfor both playe也 with 趴叫= ìv for aHz人 That is , each type of each player bids exactly half her
valuation.
To verify this claim, suppose that each type of player 2 bids in this way.τhen
as far as player 1 is concerned, player 2's bids are distributed uniformly between
o and ~. Thus if player 1 bíds mo陀 than ~ she surely 陆观灿ereas if she bids
9.6 lII ustration: 矗划在 tions
295
-AR4
U
'EA
-A句4
O
//jJV
事
Player 1's
expected ,
payoff
V1
b 1 一→
figure 2吉5.1 PI吕 yer 1 、牛 expected payoff 4S a function of its bid ín 击 first-price sealcd 七 id auction in
there are two bídders a 叫 the vâlu旧at民10郎 ar，e们淤飞Ul1l证fωorml与yd 犯耐t材τib趴1附 df行亏艾扣om
bids ~v归2
w头ich
&鸟1 旦
i t出he proba站b弘ìl且ìt咛Y 佼出出加削
1冶挝
a硝t sh飞冶e们w
叮v'ì出
n
邸s拟怒
ìS 阳 P
伊roba抽bil
刷制
让lì让ty 出
tha挝t叩
playe凹r 2's 刊
v alua阳
les臼s 仙
t ha
衍
ar盯12岛句di出只 飞w
岛矿飞
γF斗hich case play衍 2 bids less than 岛
t/1让}， which is 2b1 . Consequently
her payoff as a function of her bid b1 is
(叫一 bIJ 毗叫
Vl …
b1
if b1 >
!
τhis function is shown in Fìgure 295. 1. Its max挝zer is !Vl (see Exercise 497.n
so that player 1's optimal bid is half her valuation. Both players are identicat so
tlús argument shows also that given ßl 忡 zjtYFPlay时 2's optimal bid is half her
valuation. 在lUS，部 clairned， the game has a Nash equìlibrium in wlúch each type
of each player bids half h衍 valuation.
When the number n of bidders exceeds 2, a simìlar analysis shows that the
game has a (symmetric) N在sh equilibrium in which every player bids the frac翻
tion 1 … 1/ n of her valuation: 自 (υ) = (1 … 1/n)v for every player i and every valuation v. (You are asked to verify a claim more 伊neral than t"his one in
Exercise 296.1.)
In this example-and, it turns out, for any distribution F satisfying fue conditions given at the start of this section-the players' common bidding functìon in a
symmetric Nasl飞 equilibrium may be given an i1luminating interpretation. Choose
n - 1 valuations randomly and independently, each according to the cumulative
dìstribution function F. The lúghest of fuese n - 1 valu拭ions is a 吁andom variable": its value depends on the n -1 valuations that were chosen. Dεnote it by
X. Fix a valuation v. Some values of X are less than v; others are greater than v.
Consider the distribution of X in those cases in wlúch it is less than v. The ex幡
pected value of this distribution is denoted E[X I X < v]: the expected value of X
conditional on X being less than v. We may prove the following result. (See the
appendix, Section 9.8, for suggestive arguments.)
Chapter 9. Bayesian Games
296
扩 each l收1纭t旷r:云s 归
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刃
'al阳
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如 往
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tll与
y 卢 om 斤
t h陀ε 5蚓d
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and 力i1η1盯
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at
巾咐
lt
tiz陀
7炫e dís纣tri仇ltíOI1， a 卢 rst-price sealed-bid auctio l1 印Jitll
impeJfect i哈rmatíoll a以tt valuations) 1117s a (symmetric) Nash equilibrium
íl1 wh灿 each type v Of each pl叩1'1' bíds E[X I X <叶 ， the expected val !ll' of
tlle highest of the other players' bids conditional 011 v beíng h拉her tlum all the
other valua ti0 l1 5.
Put differently, each bidder asks the fo11owing questìon: Over a11 the cases in
v气rhich my valuation is the hìghest , what is thεexpectation of the highest of the
。ther players' valuations? Thìs expectation is the amount she bids.
If F is uniform from 0 to 1 and n 2, we may verify that indeed the equilibrium
we four飞d may be expressed ìn this 认lay. For any valuation 专 of player 1, the cases
in which player 2's 飞'aluation is less than v are distributed uniformly frorr飞 o to v,
so that the expected value of player 2's 飞raluation co只ditional on its being less than
v is
vvhich is equal to the equilìbrium bidding function that we found.
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0111η1严?泊
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f
we saw 白
t ha
川t 岛
fi扫1γ疆 S
纣t- an飞d sec
∞
ond-price 缸
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a
u1κ
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CtiOI只飞s are 飞
U re
队飞vεenl江le equr口飞7沼aler
吼 时
1让t'η， when the
players know each others' 'aluations: their distinguished e电uilibria yield the same
outcome. 1 now argue that the same is true , under thεassumptions of this section ,
飞vhen the players are uncertai r. of εach others' valuations.
Consider the equilibrium of a second-price auction in which every player bids
her valuation. 1n this equilibrium , tl飞役 expeεted pr忖.才ice paid by a bidder 飞与叽:让th 飞valua­
tion 衍z? 飞认飞γ'ho 毛飞γ'ins i吉 th 心 expect抬atio口 οf the 七
11igh飞e
部st of thε oth飞 er 11 一 1 飞v 在 luatior只lS丸， con飞一
d
We ha肌飞兀v它eJ严
盯5处t s付n that a first伽price auction has d symmetric Nash 叫uilibrium ìn
u
which this amount is p陀cisely the bid of a player with valuation v, and hence the
amount paid by suc11 a playe r. Thus a winning bidder pays the same expected
price in theηquilibrium of each auctio l1. In both cases , the player 飞vith the highest
valuation submits thεwinning bid , so botl飞 auctions yield the auctioneer the same
revenue:
,
‘
If ench bidda is risk静 11cutml nnd ench bidde/'泣 'ualllatio l1 is drawn indef丑'11dentlyfro111 thc S:7l11♂ con tÍ11l10 1I 5 nnd increasÎng cll l11 ulative distrilnttio刀 ， tÍil'l1
the Nash eqlliJi briu lIl of a second-prícc sealed-ùid nuctiol1 巾，可fh Î111pel:如t in …
户 1'111 at 1Ll/l β bout î Ja luatio l1 s) Î!1 wlIÌc l1 eacl1 playcr bids hcr valuation yil' lds
tl1 e same reve l1 UC as f!1 e sym111ctric Na革h cqui/i briu l1l of th l' correspondÍ1喀
卢/'st-p 阳: seale川id auctio l1
This result depends on the assumption that e剖h player's preferences are repre …
sented by the expεcted value of a risk-neutral Bernoullí payoff function The next
exercise asks you to study an example in which each player is risk a飞rerse. (See
page 103 for a discussion of risk neutrality and risk aversion.)
‘
苦 EXERCISE 296.1 (Auctions with risk-averse bidders)
Consider a 飞!ariant of thε
Bayesian gam在 defined earlier in this section in which the pl冯rers are risk averse.
9.岳lII ustration: 昌 uctions
297
Specifically, suppose each of the 11 players' preferences are represented by the
expected value of the Bernoulli payoff function X 111 飞 where x is the player's
monεtary payoff and m > 1. Suppose also that each player's valuation is dis伺
tributed uniformly between 0 and 1, as in the example on page 294. Show that the
Bayesian game that models a first-price sealed-bid auction under these assumptions has a (symmetric) Nash equ过ibrium in which each type Vi of each player i
bids (1 -l/[m 何一 1) + 1])Vì' (You need to use the mathematical fact that the solution of the problem maxb[bk(v - b)e] ís kv/(k + f).) Comp在re the auctioneer's
revenue in this equilibrium with her revenue in the symmetric Nash equilibrium
of a second-prìce sealed-bid auction in which each player bids her valuation. (Note
that the equilibrium of the second-price auction does not depend on the players'
payoff functions.) 、
9.6.3
Common valuatio门5
In an auction with common valuations, each player's valuation depends on the
other players' signals as well as hεr own. (As before , 1 assume that the players'
signals are independent.) 1 denote the function that gives playεr í's valuation by
g Ít and assume that it is increasing in a11 the signals. Gi飞ven飞 t白
h飞e appropr泣iat怡
e spe
配c­
ific
缸缸
叫a
C
剖
创耐
t阳
ion
O
a function of the 严
p Eοofi出
i沁
e b of bi
让id
巾s， the following Bayesian game models first- and
second伽price auctions with common valuations (and imperfect ir飞如rmation about
valuations).
Pfayers The set of bidders , say {1,. . ., n
States The set of a11 profiles (t 1,. . ., t l1 ) of signals that the players may receive
Actions Each player's set of actio邸 is the set of possible bids (nonnegative
numbers).
Signafs The sigr飞alfu时tion 飞 of each player í is given by Ti (t 1, . . ., t l1 ) 口 tf
(each player observes her own signal).
Beliefs Each type of each player bεlieves that the signal of every type of εvery
other player is independent of a11 the other players' si伊1als.
Payo疗 functions
Player j's Ber∞ulli payoff in state (t 1,.. ., tn) is 0 ifl町 bid bi
ìs 丑ot the hìghest bid, and (品的，.. ., tn) 一坷的) / m ìf no bid is hìgher than
bi and m bid叫including bì ) are equal to b(
JFtva
M ‘
叫
…
ZAU它
斗rm
44
叫内 JtT
b
问U
V·A-hU 俨A
b' 句
<一>
fz dh
m ·I·,,
jj OO
r宁'AFTA
f'',
'HU'
、 Bg ， f
'HU
、‘ 2 ， r
F
/22 飞
f
D
豆
咽'A
'φ
，。
J''t
飞
f--J飞 ls 飞
2
一一
持
f
‘、
22'''
A
,
.a
、、毛，J
⑨
咆2'A
'''1、、
44E‘、
，
u
LU a#Tt
~wo t )
1}i
VACD
Chapter 9. Bayesian Games
298
Nash e伊ilibrium in a second-price sealed-bid ωction The main ideas 如 the analysis
of sεaled心id common value auctions are i1lustrated by an example in which there
arehωbidders， each bidder's signal is unifo口nly distributed from 0 to 1, and the
valuation of each bidder i is given by Vj 二 IXt j 十 γ马， where j is the other player
and tX 三 γ~ O. The case in which 此日 1 andγ 艺工 o is exactly the one studied in
Section 9.6.2: in this case, the bidders' valuations are private. If a ::=:γ， then for
any given signals, each bidder's valuation is the same-a case of pure COrn...'Uon
飞raluations气If， for example, the signal tj is the number of barrels of oil in a tract,
then the expected 飞也lation of a bidder i who kno认叫削ignals t i and tj is P . t j +
ti) , where p is the monetary worth of a barrel of oil. Our assumption, of course ,
is that a bidder does not know any other player's signa l. However, a key point in
the analysis of common value auctions is that the other players' bids contain some
information about the other players' signals-information that may pro自 tably be
used.
1 clairn that under these assumptions a second-price sealed-bid auction has a
Nash 叫uilibrium in which each type tj of each player í bids (α÷γ) ti
To verify this claim, suppose that each type of player 2 bids in this way and
type f 1 of player 1 bids b1 . To determine the expected payoff of type tl of player 1,
we need to fu飞d the probability with which she wins, and both the expected price
she pays and the expected 飞ralue of player 2's signal if she wins.
t1
!(
Probability that player 1 wins: Given that player 2's bidding function is
(夜十 γ) 句， player l's bid of b1 wins only if b 1 兰以十 γ) 句 ， or if t2 三
b1 / (α 十 γ). Now，与 is distrib毗d uniformly from 0 to 1, so the probability that it is at most bd (α 十 γ) is bd 杜十 γ). Thus a bid of b1 by play校 1
wins with probability b 1 1 抖 +γ) .
Expected price player 1 pays if she wins: The price she pays is equal to
player 2's bid , which, condítionalon its being less than 屿， is distributed uni幽
formly from 0 to b1. Thus the expected value of player 2's bid , gíven that it is
less tf1 an b1, is b1.
!
Expected value of player 2's signal if player 1 wins: Player 2's bid, given her
sígnal t2 , is (α 十 γ) 巧， so th飞a挝t the 帜
e xpecte
时
d value of signals that yield a bid
1e
恬剧$部s 出
t han
of 沁
oft与
ω2ρ) ..
Now, player l's expected payoff if she bids b1 is the difference between her
expected valuation, given her signal fl and the fact that shεwins， and the expected
price she pays , multiplied by her probabìlity of winning. Combining the previ
一 a 一τ . (2(a+ γ)tl - b1 ) b1 .
4 飞此寸 γ 广
This function is maximized at b 1 口(夜十 γ )tl. Thatis, ifeach type t2 ofplayer2 bids
(α 十竹马! any type tl of player 1 optimally bids 件十 γ) t1 Symmetrically, if each
‘
9.6 lII ustration: auctions
299
type t1 of player 1 bids (a + γ )h ， any type t2 of player 2 optimally bids (α÷γ )t2.
Hence, as c1aimed , the game has a Nash equilibrium in which each type ti of each
player i bids (α÷γ ) ti
<D EXERCISE 299.1 (Asymm的ric Nash 叫uilibria of 日econd-price sealed-bid common
value auctions) Show that whenαz γ= 1, for any value of λ> 0 the game has
an (asymmet改)到ash equilibrium in which each type t1 of player 1 bids (1 +λ ) t 1
and each type t2 of pl町er 2 bids (1 个 1/λ )t2.
Note that when player 1 calculates her expected 飞'alue of t枪 object， she finds
the expected value of player 2's sìgnal given that her bid wins. She should not base
her bid sìmply on an estimate of the valuation derived from her own signdl and the
(uncondìtional) expectation of the oth在 play肘 's signa l. She wins precisely when
her bid exceeds 役lose of the other players, so if she bids in this wa予 then over all
the cases ìn which shεwins， she more likely than not overvalues the 0均ect. A bidder who incorrectly behaves in thìs way ìs said to suffer from the wínner's curse
(Bidders 出 real auctions know this problem: whe口 a contractor gives you a quotation to renovate your house, she does not base h衍 price simply on an unbiased
estimate out how much it will cost her to do the job; ra白er， she takes into account
that you will select her only if her competitors' estimates are all higher than hers,
如 which case he主 estìmate may be suspiciously low.)
Nash e伊 ilibriU1泛 的 α 卢rst-price sealed白bid auction 1 c1 aim that under the assumptions on the players' signals and valuations ín Section 9.6.2, a first附price 日ealed­
bid auction has a Nash equilibrium in whích each type ti of each player i bids
i 识 +γ )t卜 This c1 aim m咛 be verified by arguments like those in that section. In
the next exercise, you are asked to supply the details.
<D EXERCl SE 299.2 (First-price sealed-bid auction with common 曰luatio的 Verify
that under the assumptions on signals and valuations in Section 9.6.2, a f让st-price
sealed-bid auction has a Nash equilibrium in which the bid of each type ti of each
player i ìs (庆十 γ)ti
i
Comparing equilibria of卢刚- and second-price auctions We see that the r轩enue 吨uiv­
alence of 在rst呵 and second-price auctions that holds when valuations are private
holds also for the symmetric equilibria of the examples in which the val飞lations
are common. That is , the expected price paid by a player of any given type is the
same 拍出e symmetric equilibrium of 出e first-price auctìon as it is in the symmetric eq山libri旧n of the second-price auction: in each case type ti of player i pays
j 何个 γ )ti 迁
i fshe
隙ewi白
盹吉轧
ns
s， and 附
wk
m
如
飞1S川1叫~
In 妇
fact， 白
the 陀
revenue εqui忖
valenc
使
e pri王挝
1泛
ci增
pl怡
e holds much m
泯10
创re geε
缸创
n隙阳
时r阳
e
all悖
予稽 S
y
缸uψp­
严
P
os
妮
e由
t ha
低te
岱
ac
巾
h bidder is risk neutral and ind
岱
ep
严
3咒
endently receives a signal trom the
same distribution, and that this distribution satisfies the conditions in Sectíon 9.6.2.
在len the expected payment of a bidder of any gìven type is 出esamεin the s严n­
metric Nash equilibrium of a second-price sealed-bid auction as it is in the symmetric Nash e电站librium of a fust-price sealed-bid auction. Furthe室~ this revenue
300
Chapter9. 防毒yesíanGam告S
equivalence is not restricted to first- and second-price auctions; a general result ,
encompassing a wider range of auction forms , is stated at the end of the appendix
(Sectìon 9.8).
AUCTIONS OF THE RADIO SPECTRUM
In the 1990s several countrìes started auctÌoning the right to use p在rts of the radio
spectrum used for wireless communication (by mobile telephones , forεxample).
Spectrum lîcenses in the United States were originally allocated on the basis of
hearings by the Federal Communications Commìssion (FCC) .ηùs procedure was
tíme-consuming , and a large backlog developed , prompting a switch to lotteri创­
Li censes awarded by the lott臼iεs could be resold at high prices , attracting many
participants. 1n one case that drew attention, the winner of a license to run cellular
M叩hones in Cape Cod sold it to Southwestem Bell for $4 1.5 million (Neω York
Times , May 30, 1991 , page A1). 1n the ea r1 y 1990s, the U.S. govemment 队as per咿
suaded that auctioning licenses would allocate them more efficiently and might
raise nontrivial rθvenue.
For each interval of the spectrum , many licenses were available , each covering
a geographic area. A buye了 's valuation of a licεm巳 ζould be expected to depend
on the other licenses it owned , 50 many interdependent goods were for sale. 1n designing an auction mechanism, the FCC had many choices: for example , the biι
ding could be open, or it could be sωled， with the price equal to either the highest
bid or the second-high创t bid; the licenses could be sold sequentially, or simult在­
neously, in 飞Nhich case par吃icipants could submit bids for individual licenses or
for combinations of licenses. Experts in auction theory w伫re consulted on the design of the mechanism. John McMillan (who ad飞lised the FC C), writes that "When
theorists met the policy-makers, 汇oncepts like B町res-Nash equilibrium, incenti飞哈
compatibilíty constraìn 怡， and order-statìstic theorems camεto be discussed in the
ζorridors of power" (1994 , 146). No theoretìcal analysi5 fitted the environment
of the auction well , but the experts appealed to some principles from the existíng
theory, the results of Iaboratory experimen饭， and experience in auctions held in
New Zealand and A ustralìa in the early 1990s in maklng their recornmendations.
τhe mechanism adopted ín 1994 v气'as an open ascending auεtion for whìch bids
were accepted simultaneously for alllicenses in each round. Experts argued that
the open (as opposed to sealed 七id) format and the simultat飞eity of the auctions
prorr飞oted an efficient outcome becausεat each stage the bidders could see their
rivals' pr
9.7 lII ustration: juries
主 01
ternet traffic will move to wireless dεvices， in 2000 President Bill Clinton ordered
further auctions of large parts of the spectrum (New York Tim归， October 14, 2000).
Whether the auctions that have been held have allocated licenses 时ficiently is hard
to telt though it appears that the wínners were able to obtain the sets of lícenses
they wanted Certainly the auctions have been successful in generating revenue:
the first four generated over $18 billion.
‘
9.7
lII ustration: juries
9.7.1
Model
In a triat jurors are presented wíth evídence concernir飞 g the guilt or innocence of
a defer飞dant. Their interpretations of the evidence may differ. Each j 盯恨那sesses
the evidence, and on the basis of her interpretatior飞 votes eìther to convict or acquit
the defenda时. Assume that a una时mous verdict is required for conviction: the
defendant is convicted if and only if eve巧， juror votes to convict her. (This rule is
used in the United States and Canada , for example.) What ca口 we say about the
chance of an innocent defenda时's being convicted and a gu ì1 ty defendant's being
acquitted?
In deciding how to vote, each juror must consider the costs of convicting an
innocent person and of acquitting a guilty perso口. She must consider also the
likely effect of her vote on the outcome, which depends on the other juro路， votes.
For example, a juror who thìnks that at least one of her colleagues is likely to
vote for acquittal may act differently from one who is sure that all her colleagues
will vote for convìction.τhus an answer to the question requires us to consider
the strategic ìnteractìon between the jurors, which we may model as a Bayesìan
game.
Assume that each juror comes to the trial with the belief that the defendant
is guìlty with probabilìty 7τ(the same for every juror), a belief modified by the
ε飞ridence presented. We model the possibility that jurors 加terpret the evidence
differently by assuming that for each of the defendant's true statusεs (伊边tyand
innocentì，相C趴 juror interprets the evidence to point to gu ì1 t with positive probability, and to innocence with positive probabilit予 and that the jurors' interpret公
tions are independent (no juror's intε主pretation depends on any oth盯 juror's interpretation). 1 assume that the probabilities are the same for a11 jurors and denote
the probabìlity of any given j泣如r's interpreting the evidence to point to guilt when
the defer飞dant is guilty by p, and the probabilìty of her interpreting the evidence to
衍
ο
point to innocence when the defendant is innocent by q. 1 assume also 拉
t ha
挝taj扣
旧
u
ro
附
r
i总smo陀削l且ik
挝鸣均
i与yt
甘往
h飞l衍
a
arnn
飞
and her抗飞ce in pa盯rt悦icular p > 1 … q.
Each juror wishes to convict a 伊卸 defendant and acquit an innocent one.
She is indifferent between these two outcomes and prefers each of them to one
in which an irmocent defendant ís convícted or a 伊ilty defendant is acqui忧ed.
Chapter9. Bayesian Games
302
A茄sume speci在cally that each j旧时 's Bemoulli payoffs are:
5
/iiiiJ3111L
?叩
一一
if guilty defendant convicted or
innocent defendant acquitted
if innocent defendant convicted
ifg工lÍlty defendant acquitted.
(302.1)
、 eBP ，r
7阳
咱，.命
/6 飞、
The parameter z may be gi飞ren an appealing interpretation. Oenote by r the probability a juror assigns to the defendan t' s being guilty, gìven all her information.
Then her expected payoff if the defendant is acquitted is -r(l- z) 十 (1 … r).O =
-r(l - z) and her expected payoff if the defendant is convicted is r . 0 … (1 … r)z =
斗 1 一中 .τhus she prefers the defendant to be acquitted if … r(l … z) >… (1 … r)z ,
or r < z, and convicted if r > z. 如lat is, z is equal to the probability of guilt required for the juror to want the defendant to be convictεd. Put differently, for any
Juror
acquittal is at least as good as conviction if and only if
(302.2)
Pr 恒的ndant is guilty, given juror's infor部ation) 三 z.
We 囚a叮ynow fo臼rmul恒at怡衍 a Ba玛尹
yF々守部s挝iarη飞 ga
扭
m飞ε t白
ha
挝t models the s Ìtuation. τhe players
are the jurors, and each player's action is a vote to convict (C) or to ac电uit (Q). We
need one state for each configuration of the players' preferences and information.
Each player's preferences depend on whether the defendant is guilty or i贝贝ocent，
and each player's infc川，'mation consists of her interpretation of the evidence. 丁hus
we define a state to be a list (X， 句， . . .， ，5向'1)' wher陀eXd已not怡es the def
巾
t伦
enda
衍
ant'
处ta
s
挝tus
鸟， guilty (G) or innωoc
优
er时
1忱t (1η) ， and Si represents player i's interpretatio日 ofthe
evidence , which may point to guilt (g) or innocence (b). (1 do not use i for "innocence" because 1 use it to index the players; b stands for "blameless".) The signal
that each player i receives is her interpretation of the 机忖ence， Si. 1n any state in
which X = G (i忌. the defendant is guilt抖， each player assigns the probabilìty p to
any other player's 陀ceivíng thεsignal g , and the probability 1 … P to her receiv
口als. Similarly, in any state
ing the signal b, ind巳pendently of all other players'
in which X 口 l 书.e. ìhe defendant is innocen叶， each play衍部signs the probability q to any other player's receiving the signal b, and the probabilìty 1 号 to her
receiving the signal g, indepeηdent1y of all other players' signals.
Each player cares about the verdict, which depends on the players' actions and
on the defendan t' s true status. Given the assumption that unanimity is required to
convìct the defendant, only the action profile (C ,. . ., C) leads to convictìon. Thus
(302 .1) implies that player i's payoff function in the Bayesian game is defined as
follows:
7ι~
§'J
飞毛
气
、、
咆2·A
Zri
flit-飞/
2332
。一一
Ui(a， ω)
if a 长 (C ，...，C) a时的= lor
if a (C .. . , C) and Wl = G
ifa (C... , C)and ω1 = 1
if a 乒 (C，...，C) andWl =G ,
(302.3)
wher役时1 必 the first component of the state , giving the defendant' s true status.
9.7 lII ustration: juries
303
In summary, the Bayesian game that models the situation has the following
components.
Players A set of 11 jurors.
States The set of states is the set of alllists (X ， 坷，... , 5 /1) where Xε {G ， l}and
Sjε {g， b} for every juror j , where X = G if the defendant is 伊ilty， X = 1
if she is innoce时， 5j = g ìf player j receives the signal that she is guilt予 and
5í 口 b if player j receives the signal that she is innocen t.
Actions The set of actions of each player is {C , Q} , where C means vo松柏
convict, and Q means vote to acqui t.
Signals The set üf signals that each pl巧er may receive is 毡，叶 and player j' s
signal function is defined by 写 (X ， Sl ，...， S I1) = Sj (eachjuror is in如rmed
only of her own signal).
8elie台
Type g of any player i believes that the state is (G , 51 ,. . . ， Sn) 认付1 probability Pr( G I g) pk … 1 (1… P)1l -k and (I, 51 ,. . ., 5n ) with probability Pr (I I g) x
性的卜 1ql1 斗， wherε k is the numbεr of players j (including i) for whom
sj 工 g ir飞 each case. Type b of any player i believes that the state is
(G， 51 尸. ., 5 11 ) wìth probability Pr(G I 抖 p气 1 … p) 卜川 and (1, 51 ," ., 5F口112斗斗))
with probab i1ity Pl乓~(I I 的
b )(1 … t的
j拉)k与号矿俨/1←
1
for 飞W矿叶homs斗/ 口 gi妇
nead
巾
h飞 Cαas
优
已.
e
Payo疗 functions
9.7.2
The Bernoulli payoff function of each player i is given in (302.3).
Nash εquilibrium
Start by considering the very simplest case, in which there is a single
juro汇 Suppose that her signal is 扛 τb determine whether she prefers con飞riction or
acquittal , we need to fir飞d the probability Pr( G \ b) she assi伊s to the defer飞dar泣's
being guilty, given her signal. We can find this probability by using Bayes' rule
(see Section 17.6.5, in particular (504 .3)), as follows:
One juror
Pr(G I b)
pr(b I G) Pr( G)
Pr(b I G) Pr( G) 十 Pr(b I 1) Pr (I)
(1 … p) π
一 (1 - p) π 十 q(1 一的-
Thus by (302.2) , acquittal yields an expect较d payoff at least as high as does conviction if and only if
之>
1- p) π
… (1 - p) π +q 口一 π)'
在lat is, after getting the 创伊al that the defendant is innocen t, the juror chooses
acqui加1 部 long as z is not too small-as long as she is too concerned about ac-
quitting a guilty defendan t. If her signal is g, then a similar calculation leads to
Ch毒pter 9.
304
Bayesian Games
tl飞e con cJ usion that convìction yields an expected payoff at least as high as does
acquittal if
之<
Thus if
p 7f 十
(1 … p) π/_/
(30令 1)
(1 … p) π÷ 号召… π) -，--，严十
then the juror optimally acts according to her sigr飞往1， acquitting the defendant
when her signal is b and εonvicting her wl况n it is g. (A bit of algebra shows that
the term on the left of (304.1) is less than the term on the rigl飞仁 given p > 1 q.)
Two jurors
Novγsupposethere are two jurors. Are there values for z such that the
game has a Nash equilibrium in which each juror votes according to her signal?
Suppose that juror 2 acts in this way: type b votes to acquit, ar飞d type g votes to
convic t. Consider type b of juror 1. If juror 2's signal is b, juror l's vote has no effect on the ou tcOIτ足， because juror 2 votes to acquit and unanimity is required for
conviction. Thus when deciding how to vo饺， juror 1 should ignore the possibility
that juror 2's signal is b and assume it is g. 在飞at is , juror 1 should take as evidence
her signal and the fact that juror 2's signal is g. Hence , given (302.2) , for type b of
juror 1 acquittal is at least as good as conviction if the probability that thεdefen­
da时 is guilty, given that j 口ror l's signal is b and juror 2's signal is g, is at most z.
This probability is
Pr(G I 岛， g) =
Pτ (b ， g
I G) Pr( G)
Pr(b , g I G) Pr(G
(1… p)p π
(1 - p) ρ + q(l ← q)(l
π)
丁}飞U5 type b of juror 1 optimally votes for acquittal if
z > -:-------'-1- p) 严
'-- (1 … p) 严十 q(l q)(l 一对于
Bya similar argument , for type g of juror 1 conviction is at least as good as acquîttal
if
立<
Thus when there are two jurors, the game h在sa 时ash equilibrium in which each
juror acts according to her signa l, voting to acquit the defendant when her signal
is 主 and to convict her when it is g, if
(1- p)p 汀/时/
(1 … p)p π + q(l … q) (1 一功一-'
(304.2)
Consider the expressions on the left of (304.1) and (304.2). Divide the numerator and denominator of the expression on the left of (304.1) by 1 … p and the numerator and denominator of thεexpression on the left of (304.2) by (1 - P)p. Then ,
9.7 JIl祖 stratÎon: juries
305
given p > 1 … q, we see that the expression on the left of (304.2) is greater than 出e
expression on the left of (304.1). That is , the lowεst value of z for which an equilibrium exists in which each juror votes according to her signal is higher when therε
are two jurors than when there is only one jur时. Why? 如cause a juror who receives the signal b, knowing that her vote makes a difference only if the other juror
votes tocon飞lict， makes 1飞时 decision on the assumption that the other juror's signal
is g, and so is less worried about convicting an innocent defendant than is a single
juror in isolation.
Many jurors Now suppose the number of jurors is arbitrary, equal 怡 11. Suppose
that every juror other than juror 1 votes to acquit when her signal is b and to convict
whenh时 signal is g. Consider type b of juror 1. As in the case of two jurors,
juror l's vote has no effect on the outcome unless every other juror's signal is g.
在1US when deciding how to vote, juror 1 should assume that all the other sigr冶沁
are g. Hence, given (302 .2), for type b of juror 1 acquittal is at least as good as
conviction if the probability that the defendant is guilty, given that juror l's signal
is b and every oth时 juror's signal is g, is at most z. 百lÎs probability is
‘
Pr(G I b, g, … , g)
Pr (b , g,..., g I G)Pr(G)
I G) Pr(G) 十 Pr( 岛， g，.. ., g I I)Pr (I)
Pr(b， g尸 • ., g
穹i
Thus type b of juror 1 optimally votes for acquittal if
z >
1-n
7τ
NoV\巳 given that p
> 1 - q, the denomìnator唱 deCf'εases to 1 as 泛 increases. Thus
the lower bound on z for which type 告。f juror 1 votes for acquittal apF寻roaches
1 as n incrεases. (You may check that if p = q 0.8 , π 之二 0.5, and 11 工 12, the
lower bound on z exceeds 0.999999.) ln particula毛 in a large jUl)二 if jurors care
even slightly about acquitting a guìlty defenda时f then a juror who interprets the
e飞Tidencεto point to innocence will nevertheless vote for conviction. Th e reason is
that the vote of a juror who interprets the evidenc公 to point to innocence makes a
difference to the outcome onl y if every other juror interprets the evider飞ce to point
to guilt, in which case the probability that the defendant is in fact guilty is very
high.
We condude 在at the model of a large jury in which the jurors are concerned
about acquitting a guilty defendant has no Nash equ过ibrim工1 in which eve巧1 Juror
votes acco古ding to her signal. What are its equilibria? You are asked to 位叫也e
conditions for two equilibrìa in the next exercise.
Chapter 弘 Bayesian
3号岳
Games
'1 EXERCISE 306 .1 (Signal晦independent eguilibria in a model of a jury) Find conditions under which the game , for an arbitrary number of jurors, has a Nash
equilibrium in which every juror votes for acquittal regardless of her signa l, and
conditíons under which every juror votes for conviction regardless of her signal.
Under some conditions on z the game has in additio民 a symmetric mixed stratω
egy Nash equilibrium in which each type g juror votes for con飞riction， ar飞d each
type b juror votes for ac王luittal and c。我 viction each with positive probability. Dem伦 by 户he mixed strategy of each ìuror of type b. As before, a j旧时 's vote affec怡
the outcome only if a11 other jurors vote for conviction, so when choosing an action,
a juror should assume that all other jurors vote for conviction.
Each type b jllror must be indifferent between voting for conviction and voting
for acquittat because she takes each action with positive probabilîty. By (302.2) wε
thus need the mixed strategy ßto be such that the probability that the defendant is
guilty, given that a11 other jurors vote for conviction, is e弓ual to z. Now, the probability of any given juror's voti只g for convic拉伯 is P + 口 - p) 民 C) if the defendant
is guilty and 1 一号叫声 C) if she is innocent.在1US
Pr( G I signal b and n 一 1 votes for C)
Pr(b I G)(Pr(vote C I G))'l-l Pr(G)
Pr(b I G)(Pr(voteC I G))11-1Pr(G) 十 Pr( b I I)(Pr( vote C I I))卜 1 Pr (I)
(1 - p)(p 十 (1
(1
p) 卢(C) )11 … lπ
1')(1'+(1-1') 卢(C))← 1π 个 q(1 …号十 qß(C)) 刃 1(1 -的.
百le cor飞dition that this probability 吨uals z implies
(1 - p) 如?十 (1
p)弄 (C))11-1 π(1 …二)口号 (1 …号十号ß(C)r- 1 (1 一 η泣，
(306.2)
and hence
where X = [π(1… P )(J … z)/((1-7τ )qZ) ]l /(11 一川. For a range of parameter values,
O 三 ß(C) :; 1, so that ß( C) is indeed a probability. Notice th放 when n is large, X
is dose to 1, and hen叹声 C) is close to 1: a juror 飞在rho ìnterprets the evidence as
pointing to innoεenc较 very likely nonεi:heless votes for conviction.
Each type g juror votes for conviction, and so must 伊t an expected payoff at
least as high from con飞riction as from ac伊 ittal. From an analysis like that for each
type b juror, this condition is
p(p + (1- PJß(C)) ← 1π (I一 z) 三 (1 - q)(1 … q+q卢 (C) )月一 1 (1 - π)z
Given p >
! and q > !' this condition follows from (306.2).
Ar只1 in
时附
t怡
er.
坦
es
纣
sti
刮ir吨 P
严玄rop衍t咛
Y of 拉
tU
趾归i岭
恼s 吨
e 号Ul亘:画豆um 沁
is 出
tha
挝t the probability that an in-
nocent defendant is convicted increases as n increases: the larger the jury, the more
likely an innocent defendant is to be convìcted. (The proof of this result is not
simple.)
9.8 Appendix: auctions with <l n arbitrary distribution of valuations
307
始了iaηts The key point behind the results is that under unanimity rule a juror's
vote makes a difference to the Ot此ome only if every 。在er juror vo俊s for conviction. Conseque时i予 a juror, when deciding hm年 to vote , rationally assesses the defendan t' s probability of guilt under the assumption that every otherjuror votes for
conviction. The fact that this implication of unanimity rule drives the results suggests that the Nash equilibria might be quite different if less than unanimity were
required for conviction.τhe analysis of such rules is difficult, but indeed the Nash
吨uilibria th町 generate differ significantly from the Nash equilibria under un仕
nimity r飞Ile. 1n partiζula乙 the analo事le of the mixed strategy Nash 叫uilibria considered earlier generates a probability th挝 an Ìrmocent defendant is convicted that
approaches zero as 由εjury size Ìncreases , as Feddersen and P台sendorfer (1 998)
show.
The idea behind the equilibria of the model in the next exercise is related to the
ideas in this section, though the model is differen t.
? EXERCISE 3仰.1 (Swing voter's curse) Whether candidate 1 or candidate 2 is elected
depends on the votes of two citizens. The economy may be in one of two states, A
and B. τhe citÍ zens agree that candidate 1 ìs best if the state is A and candìdate 2
is best if the statεis B. Each citìzen's preferεnces are 陀presented by the expected
value of a Bernoulli payoff function that assigns a payoff of 1 if the bεst candidate
for the st协a挝t怆e wins (怡obt切ains
时th
O
阳
1er candωid
白创
a时t怡ewi
吁ins鸟， a 时 P
阳ayof任f Odf j i证ft由
he ω
th忧
e state, whereas citizen 2 believes it is A with probability 0.9 and B with probability 0. 1. Each citizen may either vote for candidate 1, vote for candidate 2, or not
飞rote禽
a.
Formula挖出is situation as a Bayesìan game. (Construct the table of payoffs
for each state.)
b. Show that the game has exactly two pure Nash equilibria, in onεof which
citizen 2 does not vote and ìn the other of which
votes for 1.
C.
Show that an action of one of the players ìn the second 叫uilibrium is weakly
domìnated.
d.
以Ihy is "swìng voter's c飞lrseJ/ an appropriate name for the determìnant of
citizen 2's decision in the fìrstεquilibrium?
9.8
Appendix: auctions with an arbitrary distribution of valuations
9.8.1
First-price seafed翩bid auctions
1n tlús section 1 construct a syrrunetric equilibrium of a first-price sealed-bìd auction for a distribution F of valuations that satisfies the assumptions ìn Section 9.6 .2
and is differentiable ∞{茹苦). (Unlike the remainder of the book，出e section uses
ca1culus.)
As before, del飞ote the bid of type V of player i (i.e. player i when her valuation is
v)by 禹作). In a s严到metric e吨uilibri出几 every play时 uses the same biddìng 知nc-
Chapter 9. Ba￥esíanGames
308
tion: for some function ß we have 卢 í
卢 for every player i. A reasonable 事less
is that if such an 叫uìlibrium exists，卢 is increasing: the higher a player's valuatio风 the more she bids. Under the additional assumption that 卢 is differentiable, 1
derive a condition th挝 it ll1ust satisfy in any symm政ric 吨的librium. Exactly one
function satisfies this condition, and this function is in fact incrεasing (as you are
asked to show in an exercise).
Suppose that a11 n - 1 players other than i bid according to the increasing differer飞tiable function ß. Then, given the assumptions on F, the probability of a tie is
zero , and hence for any bid b, the expected payoff of player í when her valuation
is v and she bids b is
作… b) Pr(Highest bid is b)
伊- b) Pr(All n - 1 other bids :::; b).
(308 .1)
Now, a player bidding according to the function 卢 bids at most 争， for 卢 (Q) 三 b 三
民的， if her valuation is at most ß 气的 (the inverse of 卢 evaluated at 峙. Th us the
probabilíty that the bids of the n 1 other players are a11 at most b is the probability that the I:üghest of n … 1 randomly selected 飞raluations-a random variable denoted X in Section 9.6.2-Îs at most 卢气的. Denoting the cumula ti飞re distribution
fl.1 nction of X by H , the expected payoff in (308 .1) is thus
(v
b) 日(卢-} (b)) if 卢(空)三 b 三卢{号
(and 0 if b < 卢(剖 ， v bifb>ß(v))
1 now claim thãt in a symmetric equilibrìum in wlüch every player bids accord栅
ing to 乱 we have ß( 叶三 v if v > 空 and 卢也)工卫. If V
Q and 卢 (v) > V , then
a play校 with valuation v wins with positive probability (players with valuations
less than v bid less tha口卢{吟， because 卢 is i时reasing) and obtains a negative payoff if she does so. She obtains a payoff of zero by bidàing v, so for e气uilibrium we
阳台dß 作
v whenever v > Q. Given that ß satisfies this condition, if ß 也) > Q
then a player with valuation Q wins with positive probability, and obtains a neg公
tive payoff if sh巳 does so. Thus 卢(空)三 Q. If卢 (Q) < Q., then players with valt时ions
slightly greatεr than Q also bià less than Q (because 卢 is continuous) , so that a player
with valuation Q who increases her bid slìghtly wins with pos技ive probability and
obtains a positive payoff if she àoes so, rather than obtaining the payoff of zero.
We conclude that ß(引以空
Now, the expected payoff of a player of type v when every other player uses
the bidding function 卢 is differentiable on (时间) (given that 卢 is incre战略 and
differentiable, and 卢(到=到 and， if v > 乱 is increas归g at Q. Thus the derivative
of this expected payoff with respect to b is zero at any best response less than 卢(码:
1(1.\\
,{
… H(ß-l(b)) 十
v
b)H'(卢… l(b))
ß'(ß-l(b))
(308 .2)
(The derivative of ß-l at the point b is 1/ 卢 '(ß-l(b) ).)
ln a symmetric e气uilibrium in which every player bids according to 卢I the best
response of type v of any given player to the other players' strategies is ß(咛. Beca use ß is increasi吨， we have ß(v) < 卢(百) for v < 苔， so 卢(叶 must satisfy (308 .2)
9.8 Appendix: 梢ζtìons wìth an arbitrary distrìbution of valu然ions
309
whenever 旦 < v
V. If b = 卢(叶， then ß~l ( 的工 V， so that substituting b = 卢 (v)
ìnto (308.2) 川d multip1yi吨 by 卢'( v) yie1ds
卢 (v)H l'o) 十卢 (υ )H'(v) = vH'(v) for 空 < v < V.
The left-hand side of this differentia1 equatior飞 is the derivative with respect to
v of 卢 (v)H( 叫‘ τnus for some constant C we have
州州)
lV xH'(x)d
The function 卢 is bOUl飞ded ， so considering the limit as v approaches 旦， we deduce
that C = O.
We conclude that íf the game has a symm时ric Nash equi1ibrium in whiCh
each p1ay时 's bìdding function is increaslng and differentiab1e on (~芮， then this
fUl飞ction is defined bv
JU
O
<
衍
,
a'E··
VA
<•
j
hν
…
叫
U­
工一飞
MEmxft飞、…
一­
、μ
剖e
mλ 一
fhz
U
and ß'(:Q)
在. Now, the function H is the cumu1ative distributìon function of X,
the hîghest of n 1 i叫ependently drawn va1uations. Th旧 ß*( 玛 is the expected
va1川 of X conditiona1 on its bei鸣 1ess than v: ß* ( 斗
E[X I X < ul , as claîmed in
5ection 9.6.2
We may a1t盯native1y express the nurr飞erator in theεxpression for ß* 扫) as
vH( 叶 H(x)dx (using integration by parts) , so that giver飞 H(v)
(F 例 y~l
(the probability that iî - 1 va1uations are at most u) , vve have
h:'
Ft-
Tλ­
pjJ一
ß*(v)=v 一
U
vdr
……
ori
< U <- -u
(309.1)
In particu1ar, ß* (咛 < v for~ < v 三五
⑦ EXERCIS廷 309.2 (Property of the bidding function in a first栅price auction) 5how
that the bidding function ß* defined ìn (309.1) is increasing.
(
309 .3 (Example of Nash equilibrium in a first-price auction) Verify that
扣r the distributíon F unìform from 0 to 1 the bìdding functìon defined by (309.1)
is (1 - 1/11) 认
9.8.2 Revenue equivalence of auctions
I argue at the end of Section 9.6.2 that, under the assumptions io that sectioo, the
expected price paid by the winner is the same in a symmetric equilibriurr飞 of a
first-price auction as ìt is in the equilibrium of a second附price auction in which
each player bìds her valuatìon. A more general res过lt may be established.
Consider an auction that differs from an auction with independent private va1uations as defined in Section 9丘2 on1y in that the price is an arbitrary continuous
Chapter 9. Bayesian G矗mes
310
function of the bids (抗ot necessarily the highest Or second highest bid). Suppose
that the auction has a symmetric equilibrium in which the common biddir飞g functÌon 卢 is increasìng. Assume that every player other than i adheres to 卢布 τhen as
ì's bid increases, the probabi1i ty that it exceeds a11 the other players' bìds increases
contínuously from 0 to 1. Thus i's choosing a bíd is tantamount to her choosing a
probability of winni吨.D
挠
εnωot饺
eb
坷
ye叫
(p)川
h飞er εX
净
pe
佼ct削
时
e
dp
阳
aymαm
引附
1况e
附
m
衍
础挝n飞泛t 认
w斗挝?
ìty of winninη
吨
gi怡s p. n飞en her probability of winning ìn the 吨 uilibrìum solvεs the
problem maxp(p . v … e(p 日， where v is her valuatìor飞 Assume that this problem
has a unique solution , denoted p* (v). If e is differer阳已le， the first-order condition
irnplies that v = e'(p*(v)) if 0 < p*(v) < 1. Thus if 0 < p'(v) < 1 for 在11 v, we
may integr在te both sides of this 叫uation， obtaining
e(p. 附 =e 川
(310.1)
Now, the common e气uilibrium bidding function is increasi吨， so the object is solà
to the bidder with thεhighest 飞'aluation. Thus p'
Pr(X < v) and the expected
payment e(p 气空)) of a bidder with the lowest possible valuation is zero. Hence
(310.1) implies that the expected payment e 印象 (v 月 of a bidder with any given val仙
uationvisequaltoPr(X < v)E[X I X <叫， iηdependent of the price determination
rule.
This result is a special case of the more general reve 1lt lC eqllivale J1 cc principle,
\vhich applies to a class of common value auctions , as well as private 飞ralue auctíons, and mav be stated as follows 必
Suppose thllt each bidder (i) Ís rísk ηeutral， (iì) indepe l1 dc lI tly receÎl'es a 三 ignal
户mll tlJe same conti 1HlO llS and Ìn
ηlCαj它
m斤Sil1宫
gC l/1η11门111σ
a t1汽[1(' di必ι山
I们Iï巾dí沁
∞)]儿
[ο
j儿F 疗
仰
ω
1ηJd (i íí) 1111ωι
a 即v泛i衍μω汗atiOl月1 阶
t hσ
at 丛
d L伊;吟 Cω ωωlηη1'
tJ， ，阳
r
ωεω0;η1 抒
I l肘ec
仇
ha
肌
ω
阳
1η
1四η
邢sm
污1 均
t he 吕叩叩
沪111
y
η1111η etr
ηicl扣\呼lash 叫 uilibri仪 of whích tl1 e object ís
80ld to the bidder with the highest signal and t11e expected payoff of a bidder
with the lowest possible valuation is ze ro. 1n tiI e sy l1l metric Nash equilibriu l1l
ofa 呵 such mechanis吼 the expected payme时 of a bidder of any gi'ven type is
the s{/m己的ld hence the auctiol1 eer ￥ expected revenue Ìs thc sam伫
9.8.3月esεrvepríces
The followíng ex创，'cise (whích requirescalculus) asks you to show that, in an exam…
ple , a seller in a second …príce sealed-bid auction can increase the expected selling
price by imposing a positive "reserve p1'Íce 飞 (Note that the revenue e气uivalence
principle does not apply to an auction with a positivεreser飞'e price because whc口
the highest valuation is less than the reserve price the object is not sold, and in
particular is r飞。t sold to the bidder with the highest 飞raluation.)
育 EXERCISE 310.2 (Reserv巳 prices in second-price sealed-bid auction)
Consider a
second-price sealed七id auction with two bidders in which each bidder's valu扣
tion is drawn independ创ltly from the distribution uniform from 0 to 1. Suppose
Not告$
311
that the seller ìmposes the reserve price r. That i在， if both bids are less than /', the
object is not sold (and neither bidder makes any paymer飞t)， if one bid is less than
r and the other is at least /', the object is sold at the price /', and if both bids are at
least r, the object is sold at a pric已 equal to the secor飞 d hìghest bid. Show that for
each bìdder (and any \'alue of r) , a bid equal to her valuation weakly dominates all
her other bids. For the Nash 号quílíbrium in whichεach bìdder submits her valuation , find the reserve price r that maximizεs the expected pricεat which the obj机t
is sold.
Notes
The notion of a general 3ayesian gamεwas defined and studied by Harsanyi
(1 967/68). Thεformulation I describe here is taken (with a minor change) from
Osborne and Rubinstein (1 994, Sectioη2.6).
The origin of the obser飞'ation that more information may hurt (Sectíon 9.3.1)
is unclεar. 丁he idea of "infection" in Section 9 .3 .2 was fírst studied by Rubinstein (1 989). The game in Figure 284.1 is a \'ariant suggεsted by Eddie Oekel of
the one aηalyzed by Morris , Rob , and Shìn (1 995).
Games modeli口 g voluntaryζontributions to a pllblic good were first considered by Olson (1965 , Sectíon 1. 0) and ha飞'e beεns口bsequently much studied. The
model in Section 9.5 is 在飞!ariant of one in an unpublished paper of Wìlliam F.
Samuelson dated 1984.
VKKrey(1961}initiated thestudy of alieuo11s dcscribed in secti0119.6.Firstprice common value auctions (Section 9.6.3) were first studied by 飞快Ison (1967,
1969 , 1977). The "飞川口口时 'sεurse" appears to ha飞'e been fìrst articulated by Cape只r
Cl app , and Campbell (1 971). The general rεvenue equivalεnce principle at the
end of Section 9.8.2 is dueωMyerson (1981). The equilibria in Exercise 299 .1 are
described by Milgrom (1981 , Theo陀m 6.3). The literature is survεyed by Klemperer (1999). The box on spectr飞1m auctions 0只归 ge 300 is based on McMillan
(1994) , Cramton (1 995 , 1997, 1998), and McAfee and McMillan (1996)
Section 9.7 is based 011 Austen-Smith and Banks (1 996) and Feddεrsen a口d
Pesendorfer (1 998).
Exercise 282.1 was suggested by Ariel Rubinstein. Exercise 282.2 is based on
Brams, Kilgour, and 0肝is (1993). A model of adverse selection was first stud翻
ied by Akerlof (1970); thεmodel in Exercise 282.3 is taken from Samuelson and
Bazerman (1 985). Exercise 307 .1 ìs based on Feddersen and Pesendorfer (1 996).
g
Extensive Games wÎth Imperfect Information
10.lEx始nsive games with imperfect information 313
10.2 Strategies 3.17
10.3 Nash equilibrium 318
10冯 Beliefs and sequentialεquilibrium 323
10.5 Signaling games 331
10.6 lIIustration:ζonspicuous expenditure as a signal
of quality 336
10.7 lII ustration: education as a signal of ability 340
10.8 lII ustration: strategic information transmission 343
10.9 lII ustration: agenda 仨ontrol with imperfect information
P陀requisite: ζhapters 4 and 5, and Section 7.6
351
贝克沁OTION of an extensive game with perfect informatíon defined in Chap-
ter 5 models situations in which each player, whenever taking an action, is
informed of the actions chosen previously by all players. 1n thìs chapter 1 discuss a generalization of this notion that allows us to model situations in which
each playe乙 when choosing 始r action, may not be informed of tl飞e other players'
previous actions.
10.1
Extensive games with imperfe岱 information
To describ较 an extensive game with pe改ct ìnformation , we need to specify the
set of players, the set of terminal historìes, the player functior飞， and the players'
preferences (see DefÌ1ùtion 155.1). To describe an extensive game with impelfect information, we need to add a single ìtem to this lis t: a specìfication of each player's
information about the history at evεry point at whích she moves. Denote by Hi the
set of histories after which player i moves. We specify player i's ìnformation by
partitionir飞g (dividing up) Hi into a col1ection of information sets. (See Section 17.2
for the notion of a partition.) This collection is called player i's i咕了mation partiω
tion. When making her decision, player i is informed of the information set that
has occurred but not of which history withín that set has occurred.
Suppose, for example, that player i moves after the histories C 0 , and E (i.e.
Hi = {CO , E}) and is informed only that the history is C or that it is either 0
or E. That is，证 the history C has occurred , then she is informed that C has occurred, whereas if either 0 or E has occurred, then sl飞e is informed only that either
o or E has occurred ar飞d not C-she ís not informed w怆ther the history 白.at has
飞n
314
Chap雪er
10. Extensive Games with 1m perfec在 Information
occurred was D OrιTnen player i's information partition consists of two information sets: {C} and {D , E}. If instead she is not informed at a11 about which
history has occurr时， then he主 information par出ion consists of a single informa哪
位on set, {C， 口， E }, 州ereas ìf she is infor盯时 preci辑ly about the 挝
hìs纣to
叹E予 t出
hen
informat悦íorη1 part往it延ion consi岭st怡s of 位
th
如
1泣主附 i垃n
时
担1forma
挝t挝io叭川$使
挝t沁s， {C} , {D} , and {盯 .
ε
As before , denotεthe set of actìons available to the playεr who moves after the
history h by A(h) (see (156.1)). We allow two histories h and h' to be in the same
information set only if A 拉) = A( 的Why? The player who moves a丘er any
history must know the set of actions available after that histo巧~ so if h and h' are
in the same i只formation set and A:h) 并 A(h') ， then the player who moves at this
information set can deduce which of these two histories has occurred by looking at
the actions available to her. On1y if A(h) 口 A(的 is the player's knowledgεof the
set of actions available to her consistent with her not kr飞owing whether the history
is h or h'. If the information set that cor飞tains h and h' is IÌl the common value of
A(h) and A(h') is denoted A (I J That is , A (I i) is the set of actions available to
player i at her information set Ij.
Many int!εresting extensìve games with imperfect information contain a move
of chance , so my definition of an extensive gam己un1ìke my original de也ition
of an extensive gamεwith perfect information (155.1) , a110ws for such moves.
Gìven the presence of such moves , an outcome is a lottery over the set of terminal
historiεs， so each player's preferences must be defined over such lotteries.
D旦 FINITION 314.1 (Extensive game)
An extensÎve game (with imperfect informa-
tion and chance moves) consists of
• a set of players
• a set of sequences (terminal his协ries) having the property that no s吨uence
is a proper subhistory of any other se弓uence
• a furtction (the player function) 出at assigns either a player or char飞ce" to
every sequence that is a proper subhistory of some terminal history
ff
• a function 设lat assi伊s to each history that the player function assigns to
chance a probability distribution over the actions available aft叹 that history,
where each sllch distribution is independent of every other such distribution
• for each player, a partition (the player's infonnation partition) of the s的 of
historìes assigned to that player by the player function such that for every hìs
tory h 担 any gìven member of th
<
10.1 Extensive games with imperfect information
315
of Stravinsky, and neither person, when choosing a concert, knows the one chosen
by the other person. Example 18.2 models this situation as a strategic game. We
may alternatively model it as the following extensive game, in which the play栅
ers choose their actions sequentiall予 but the second 咱mover is not informed of the
choice made by the first-mover.
Players The two people, say 1 and 2.
Terminal historiε5 (B ， 韵，但，町，币，剖， and 币， 5) .
Player function P (0) 二 1 ， P(B) = P(S)
2.
Chance moves None.
Information partitions Player 1's information partition contains a single info卜
m挝ion se乞 o (player 1 has a singlεmove， and when moving she is informed
that the game is beginning); player 2's information partition also contains a
si吨le information set {B , S} (player 2 has a single move, a时 wher只飞 movi加
n飞
she is not 拉
iE负lfi
扣
O主m
泯1εd whet应
h飞e
缸r the history is B 优
0玄 5
幻).
Preferεnces
As gìven in the description of the situation
‘
Tl飞is game is illustrated in Figure 315. 1. As before, the small circle at the top
of the figure represents the start of the game and each line represe口ts an action;
the numbers below each terminal history are Bernoulli payoffs whose expected
values represent the players' preferences over lotteries. The dotted lìne connect唰
ing the histories B and 5 indicates that these histories are in the same information
set of pl可er 2. 时ote that the condìtion 出at player 2's set of actions at every history withir飞 her information set be the same is satisfied: after each such history,
player 2's set of actions is {B , S}.
丁he next example shows how an extensive game may model a situation havìng
a richer information structure and random events. The situation considered LT1. the
example is an extremely simple version of the card game poke乙 in which there are
two players, one of whom is dealt a single card.
EXAMPL 巨 315.1 (Card game) Each of two players begins by putting a doHar in
the po t. Player 1 is then dealt a card that is equally likely to be High or LOI.o; shε
observes her card , but play盯 2 does no t. Player 1 may see or raise. If she sees, she
1
5 nu nuu
r
nu
r
nυ
2, 1
1, 2
Figure 315.1 羽毛e extensive game in Example 314.2. The dotted line (with 寸'拢。时刘延presents the
fact that when moving, plaYi衍 2 is not informed whether the histo巧I ìs B or S. That is, B and S are h'1
the same inform在 tion s位.
Chapter 10. Extensive Games with Imperfect Information
316
Chance
Loω(~)
1,
1
1
λ1eet
1,… 1
2, -.2
See
-1 , 1
且在eet
1, -1
-2 , 2
Figure 316.1 An 告xt告nsive game that models the c在rd g在me in Example 315. 1. The number in parentheses beside e在ch 1在站在1 of the moves of d飞ance is the probability with which that mo飞'êoccurs
shows her card to player 2. (Player 2 does not have any card for player 1 to see ,
but you can imagine her holding a fixed card with value between High and Low. In
Exercise 316.1 you are asked to consider a situation in which player 2, like player 1,
is dealt a card.) If playerγs card is High she takes the money in the pot, and if it is
Low player 2 takes the money in the pot; in both cases the game ends. If player 1
rms邸， she adds a dollar to the pot and player 2 chooses whether to pass or meet. If
player 2 passes , player 1 takes the money in the p肘. If player 2 mee恕， she adds a
dollar to the pot, and player 1 shows her card. If the card is High , player 1 takes
the money in the pot , while if it is Low player 2 does so.
An extensive game that models this situation is shown in Figure 316.1. Player 1
has two information sets , one containing the single history H告11 and one containing the single history Low. Player 2 has one ÍI飞formation set , consisting of the two
histories tH结/1， Raise) and (Lm护， Rl山码. This information set rεflects tl飞e fact that
player 2 cannot observe player l's car丛 Note th挝 again the requirement that the
set of actions at every history within an information set be thεsame is satisfied at
pl在 yer 2's information se t.
Q) EXE 这CIS 毛 316.1 (Variant of card game) Consider the 如llowiI飞g variant of the card
game in the previous example. Initially each player puts a doll盯 in the po t. Then
each player ìs dealt a card; each player's card is equally likely to be Hìgh or Lozυ， Înd栏pendent of the other player's card. Each player sees only her own card. Player 1
may see or raise. If she sees , then the players compare theìr cards. The one wìth the
higher card wins the pot; if the cards are the same , then each player takes back the
dollar she had put in the po t. If player 1 raises , then she adds $k to the pot (where
k is a fixed positive number) , and player 2 may pass or meet. If player 2 passes, then
player 1 takes the money in the po t. If player 2 meets , then she adds $k to the pot
and the players compare cards , the one with the higher card WiIU1ing the pot; if
the cards are the same, then each player takes back the $口十刮 she had p时 in the
po t. Model this card game as an extensive game. (Drawing a diagram is sufficient;
you can avoid the need for information sets to cross histories by put由19 the initial
move in the center of your diagram.)
10.2 Strategies
317
Challeng时
且/
Iunrea沪伽
Incumbent
Acquiesce /
3, 3
、\之:ght A呵!úesce /
1 1
f
4, 3
\
Fight
2, 4
0, 2
Figure317.1 A只吉xtensive game that models th日 entry game in Example 317.1 .τhe dotted line indicates
that the hístories RClldy and Unrelldy are in the same information se t. The challenger's payoff is listed
first , the incumben t's second.
In these exampl岱， one of the players has an information set that cor飞tains more
than one history. A game in which every information set of every player contains
a single history is equìvalent to an extensi飞regamεwith perfect information. Thus ,
for instance, the entry game of Example 154.1 臼n be regarded as a general extensive game in which the challenger has a single information set consisting of the
empty history, ar飞d the incumbent has a sing1e ir飞formation set consisting of the
history 111.
τhe next examp1e is a 飞rariant of the entry game in which the challeng町" before
entering, takes an action that the incumbent doεs not observ已
EXAlvlPLE 31 7. 1 (Entry gam功 An incumbent faces the possibi1ity of entry by a
challenger, as in Examp1e 154.1. The challenger has three choices: it may stay
。时， prepare itself for combat and enter, or enter without making preparations.
Preparation is cost1y but reduces the 10ss from a figh t. The incurnbent may either
fight or acquiesce to entry. A fight is less cost1y to the incumbent if the entrant
is unprepared; but regard1ess of the entrant's readiness , the incumbent prefers to
acquiesce than to fight.在le incumbent observes whether the challenger enters but
not whether an entrant is prepared.
An extensive game with imperfect information that mode1s this sìtuation, given
some additiona1 assumptions about the p1ayers' preferences, is shown ìn Figure
317. 1.主ach p1ayer has a sing1e information set: the challenger's information 使t
consists of the empty rustor予 and the inζumbent' s informa tion set consists of the
two histories Rea句 (the challenger enters , prepared for combat) and Unready (the
challenger entεrs without having made preparations).
10.2 Strategies
As in an extensive game with perfect information, a strategy for a p1ayεr in a genera1 extensive game specifies the action the player takes whenever it is her turn to
move. In a genεra1 game, a play权 takes an action at each of her information sets,
so a strategy is deCined as follows.
Chapter 10. 在xtensive Games with Imperfect Information
318
如 DEFINIτION 318.1 (Strategy in extensive gnme) A (pure) strategy of player i in an
extensive game is a function that assi驴1S to each of i's information sets Ii an action
ìn A( I í ) (the set of actions available to player i at the information set IJ
Consider, for example , the game in Figure 315.1 (BoS as an extensive game).
Each player has a single information set at which two actions, B and S, are available. Thus each player has two strategies, B and S. In the game in Figure 316.1 (the
card game) , player 1 has 队ro information 优抠， at each of which she has two aζtions，
Raise and See. Th us she has four stratεgies: (Raise， 其必然咛 (raìse whether her card
is H告h or Loωt (Ra ise, See) (raise ifthεcard is High , see if it is Lor吟， (See , Rais吟，
and (See , See). Player 2 has a single information set, at which she has two actions ,
namely Meet and Pass; 在us she has two strategies, Meet and Pass.
( EXERCISE 318.2 (Strategies in variants of card game and entry game) Find the
players' strategies in the games in Exercìse 316.1 and Example 317.1.
ln some of the games we study, we wish to allow players to choose their actions randomly. One way of doing so is to allow each player to choose a mixed
strategy-a probability distribution over her pur已 strategies-as we did for s位ate­
gic games (see Definition 107.1).
DEFINITION 318.3 (Mixed strat喀y 的仅tensive game) A mixed strategy of a player
inanεxtensìve game 1S a probabìlity distribution over the pl町er's pure strategies.
As before, a pure strategy may be identified with a mixed strategy that assigns
probability 1 to the pure strategy. Given this identification, a player's set of mixed
strategìes contains hεrp飞lre stratεgies.
10.3
Nash equilibrium
Having defined a strategy, the definition of a Nash equilibrium is straightforward:
a strategy profile is a Nash 叫uilibrium if no player has an alternati飞le strategy that
increases her payoff, given the other players' strategies. The fQllowing definitior飞
closely follows Definition 108. 1.
DEFINIτION 318 .4 (1\ fash equilibrium 々f extensi衍 gam办 τhe mixed strategy profile
tX* in an extensive game is a (mixed str拭egy) Nash 吨ui1ibrium if, for each player i
and εvery mixed strat仓gy tX í of player i, player i's expected payoff to 在* is at
as large as her expected payoff to (次川 ~j) accordi吨 to a payoff functi Ol飞 whose
expected value represεnts player i's preferences over lotteries.
As before, I refer to an equilibrium in which no player's strategy entails any randomization (i.e. every player's strategy assigns probability 1 to a single action at
each information set) as a pure Nash 叫uilibrium.
One way to find a Nash equilibrium of an extensive game is to construct the
strategic form of the game and analyze it as a strategic game , as we did for exter子
sive games with perfect information (see Section 5.3).
10.3 Nash equilibrium
31 曹
EXAMPL 五 319.1 悟。5 as an extensìve game) Each player 如 Example 314.2 has two
strategiεS， B and S. 在1e strategic form of the game is exactly the strategic game
B05 , as given in Figure 19. 1. Thus the game has two pure Nash equilibria , (B , B)
and (5，町， and a mixed equilibrium in which player 1 uses B with probability ~
and player 2 uses B wìth probability :1.
In this example player 2, when taking an action , is not informed of the action
chosen by player 1. This lacuna in player 2's information is reflected in her information set, whìch contains both the history B and the history 5. However, even
though player 2 is not informed of playεr l's action , her experience playing the
game te11s her the history (or probability distributio只 over histories) to expεct. ln
the steady state in which every perso只 who plays thεrole of eith校 player chooses
B, forexample , each 抖咛er knows that thεother player will choose B. She is not
illformed of this fact, but her long experience playi咔 the game leads her to the
(correct) conclusion about the othεr player's action. Similarly, in the steady state
in which B is chosen by a third of the people who play the role of player 2, the
experience of each person who plays the role of player 1 leads her to expect her
叫versa川() choose B with probability ~. The point is that a player's inform批ion
partition reflects the information obtained from her observations of the other players' actions during a play of the game; her experiεnce playing the game may yiεld
her more information about the other players' steady state actìons.
EXAMPLE 319.2 (Card game)τhe strategic form of the card game in Example 315.1
ìs gi飞'en in Figure 320. 1. By inspection, we see that the game has no 叫uilibrium
iηwhìch either player's strategy is pur巳 Player l's strategy (See , See) is strictly
dominat时 by her mixed strategy that a附 gns pro
时b加a由bi
制让li让t y ; tωo (R
阳α 1凡 阳
Raαω1巧叫
se
阳e) a 口
probabìlity to (Rí
'\ai
恒
αaηi1S
♂se巳， 5仅
N川h 叫u
山ili池
br口lUm
口1. 凹
P la
玛ye
盯r 1γF、s strategy (臼
5eωt巳
川，卢R
ι
1a
mη
旧
1让i以
xed strategy of player 2 白
tha
址t 削
a ss
创i伊1冶s pos
剑iti\飞ve probabi江li挝ty 如
tO 扎
M
命r仪
ω
et ， so given the
absence of an equilibrium in pure strategies, in a11 Nash equilibria player 1 randomizes between (Raise , Raise) and (Raise , 5ee). The condition that each player
receive the same expected payoff from each strategy to which she assigns positive
probability ge附atεs the condìtions 号口扫一号) and -p = -1 (1…抖， where p
is the proba bility player 1 assigns to (Raíse , Raise) and 号 is t
!
320
Chapter 10. 在xt母nsíve Games with Imperf，捏在t Information
Pass
Raise, Raise
Figu陀 32也 1
Meet
1 ，一 1
。，。
Raise, See
0, 0
1
See , Raise
1, -1
See, See
。，。
言'
1
2
1 1
2'2
。，。
The strategic fo罚我 of the card game in Example 315.1
xγ
X.3 , 2
1, 1
γ
2, 4
去， 3
Figure 320.2τhe payoffs in the situation considered in Example 320.2
? EXERCISE 320.1 (Nash equilibria of variant of card game) Find the Nash 叫uilibria
of the game in Exercise 316.1 for 0 < k < 1 and for k > 1. How does player l's
propensity to bluH depend on the value of k?
EXAMPLE 320.2 (Commitment and observability) Two people each have two actions, X and Y. Their payoffs to the four action pairs are shown in Figure 320.2.
First suppose 血at they choose actions simultaneously. The strategic game that
models this situation has êl unique Nasr飞 equilibrium， in which both players choose
γ(Note that X is strictly dominated by Y for player 1.)
Now suppose 巾的 the players choosεtheir actions sequεntially， player 1 first ,
and that player 2 observes player 1's action before choosing her action. That is ,
play时 1 commits to her action bεfore player 2 chooses her action. Thεextensive
game with perfect information that modεIs this situation has a singl台 subgame
perfεct equilibrium, in which both players choose X. Player 1 is better off in this
equilíbrium than s挝 is in the equilibrium of the simultaneous-move game.
Fî nally suppose that player 1 moves first , but her action is not perfεctly observed by player 2. If player 1. chooses X , player 2 may think she chose γ， and
vice versa. We may model 出is situation as an extensive game with imperfect information in which player 1's move is followed by a move of chancεthat selects a
signal observed by player 2. Player 2 observes the signa l, but not player l's action.
Assume that the probability that the signal is correct is the same for both actions ,
and less than 1. Denote this probability by 1.一 ε. (Thus if player 1 chooses X , the
signal is X with probability J … εand Y with probabilityε， and if player 1 chooses
Y, it is Y wi问灿ability 1.…f: and X with p时abilityε) Assume that 0 主 ε<
This extensive game , its strategic form , and its Nash equílibria are shown 出
Fîgure 321. 1. (τhe strategy 1J of player 2, wheriε 1 and J are both eìther X or γr
means choose 1 after the sig口al X and J after the signalγ. The mixed strategy
equilibria may be found by using the methods of Section 4.10.) We see , in particular, that 役le game has a pure strategy Nash equilibrium (γ， γ 盯 for a11 values ofε
with 0 <ε 〈;
!
10.3 Nash equilibrium
321
Extensive game:
3, 2
1, 1
\x
Y
1-E
Chance
F
2
4, 3
2, 4
X
Y
Xly
oChance
EIY
2
1-EIY
YYY 飞
3, 2
1, 1
4, 3
2, 4
Stréltegic form
X
Y
XX
3, 2
4, 3
XY
YX
YY
3 - 2ε， 2- 1" 1+2ε， 1 +ε1 ， 1
2 +2ε， 4ε4-2ε ， 3+ε2 ， 4
Nash equilibrium strategy pairs:
Player 1
Y
X
1
O
E
1- 1"
1 - 1"
E
xx
O
O
1-4ε
2-~ε
Player 2
XY
YX
yγ
O
。
1
1
214ε
。
1-4ε
2-~ε
。
O
2-~t-
Figure 32 1.1 An extensive game (top) that models a situation in which the payoffs are those in Figure 320.2, player 1 moves first , and player 2 observes an imperfect signal of player 1's action. Note that
the empty history is in the center of the game. The strategic form of the game is given in the middle of
the figu陀 followed by 出 Nash 叩1 邮
In summary, the game in which player 1 moves first and her action is perfectly
observable has a unique subgame perfect equilibrium , the outcome of which is that
both players choose X , whereas the game in which player l's action is observable
with error has a pure strategy Nash equilibrium in which the outcome is that both
players choose Y, regardless of how small the error.
Thus the advantage gained by the commitment entailed in being the firstmover, as reflected in the subgame perfect equilibrium of the game with perfect
information, is completely lost in the pure strategy Nash equilibrium of the game
in which the second player's observation of the first-mover's action is even slightly
imperfec t. Why? Suppose both player 1 and player 2 chooseY, and consider the
implications of player l's switching to X. In the game with perfect information,
player 2'5 obser飞ration of X is inconsistent with the equilibrium; she interprets it
as a deviation, to which she optimally responds by choosing X, making the deviatioηworthwhile for player 1. In the game with imperfect information, player 2's
322
Chapter 1 队 Extensi￥e Games with Imperfect Information
observation of X ìs consistent with the equilibrium; she interprets it as an inaccuω
rate signal (regardìess of how unlikely such a signal is) and continues to choose Y,
making player l's deviation undesir由 le for her.
In Chapter 5 we saw that the notion of Nash equilibrium is not adequate in a11
extensive games with perfect information, and 1 developed the notion of subgarr把
perfect equilibrium to deal with the problem. HO"Vγmayweext仓nd the idea bεhind
subgame perfect 吨uilibrium to the larger dass of extensÍve games with (possibly)
imperfect information?
EXAMPLE 322 .1 (Entry game) 丁he strategic form of the entry game in Example
317.1 is shown i丑 Figure 322 .1. We see that the game has two Nash equilibria in
pure strategies, (Unready , Acquiesce) and (0风 Fight). (You may verify that it has
also mixed strategy Nash 叫uilibria in which thεchallenger uses the pure strategy
Out and the probability assigned by the ir飞α1mbent to Acquiesce is at most
Ready
Unready
Out
Acquiesce
Fight
3, 3
4, 3
2, 4
1, 1
0, 2
2, 4
Figure 322.1 The strategic fonn ot the entrγgame of Examp!e 31ï.l.
As in the version of the entry game with perfect information studied in Chapter 5, the Nash equilibrium (Out , Fight) is 口ot plausible. If in fad the challeng时
enters, the incumbent optimally acquiesces, regardless of the history that has oc静
curred (i.e. regardless of whether thεchallenger is ready for combat). In games
wÌth perfect information we eliminated such equilibria by defining thεnotion of
subgame perfect equilibri泣酌， which requires that each player's strategy b已。pti­
mal , given the other players' strategies, for every history after which she moves,
regardless of whether the history occurs if the players adhere to their strategies.
The natural extension of this idea to games with imperfect in扣rmation requires
that eaεh player's strategy be optimal at each of her information sets. 1n the entry
game we are studying , the implementation of trus idea is straìghtforward. 在le incumbent's action Fíght is unambiguously suboptimal at its information set because
the ír飞cumbent prefers A句lliesce if the challenger enters, regardless of whether the
challenger is ready.τhus the eq田libria in wruch the incumbent assigns positive
probability to Fight do not satis守在e added requiremen t. The remaining equilibrium, (Ul仰ady， Acquiesce) , does satisfy the requiremen t: the i配umbe批 chooses its
unambiguously optimal strategy at its information se t.
在le implem纪ntation of the idea in other games is less straightforward because
the optimality of an action at an information set may depend on the history that
has occurred. Consider a variant of the ent巧ì game in which the incumbent prefers
to fight than to accommodate an unprepared entrant , as iJlustrated in Fi gure 323.1.
Lìke the original game, this game has a Nash equilibrium in which the challenger
10.4 Beli告公辑 nd sequential equilibrium
323
Challer飞ger
Unreadlf
3, 2
Fight
Ac号均创ce
Fight
1, 1
4, 2
0, 3
Out
2, 4
Figure 323.1 人 variant of the entry 伊 me in Example 317.1 in which the incumbent prefers to fíght if
the challe只ger enters unprepared.τh号 cha Jlenger's payoff is listed first , the íncumben t' s second.
stays out and the incumbent fights. But given that fighting is now optimal if the
challenger enters 山lprepared ， the reasonableness of the equilibrium in thεmoφ
ified game d叩ends on the history the incumbent believes has occurred if it h部
to move. The challenger's strategy Out gives the incumbent no basis on which to
form sucha bεlief， so to pursue the argument we need to considεr what this beHef
might be. ln the next section 1 describe a sol ution for 似tensive games that ínc1 udes
the players' betiefs as part of an 叫uilibrium.
10.4
Beliefs and sequential equilibrium
ANash 叫uilibrium of a strategic game may be characterized by two requirements:
that each player暨 choose her best actìon given her belief about the other player飞 and
thatεach player's belief be correct (see page 21). The notion of equìlibrìum 1 now
define for extensive games embodies the same two requirem仍然， and , lìke the notìon of subgame perfect equilibrium for extensive games with perfect information,
insists that they hold at each point at which a player has to choose an action. When
defining a Nash equilìbrium of a strategic game precisely, we do not n硝d to consideτthe players' beliefs separately from their strategies because the re气功rement
that the beliefs be correct completely detεrmines them: each play时 's belief about
every other player's strategy is simply εqual to that strategy. For an extensive
game , the players' strategies may not completεly determine their beliefs, as we
have just seen in the game in Figure 323. 1.在lUS we are led to de在ne a notion of
equilibrium for pairs consisting of a strategy profile and a collection of beliefs.
10.4.1
Beliefs
We assume that at an information set that contains more than one history，出e
player whose turτ1 it is to move forms a belief about the history that has occu盯ed.
We model this belief as a probability distribution over the hìstorìes i旦出einforma­
tion set. (At an informatìon set containing a single history, the only possible 1予elief
assigr冶 probability 1 t。在at history.) We call a collectìor飞 of beliefs , one for each
information set of every playe乙 abeU吃fsystem.
324
Chapter 10. Extensive Games with Imperfect Information
DEFINITION 324.1 A belief system in an extensive game is a function that assigns to each information set a probability distribution over the histories in that
information se t.
Consider, for example , the entry game of Exampìe 317. 1. This game has two
information sets , one consisting of the empty history, and one consisting of the
histories Ready and Unready. Th us a belief system for the game consists of a pair
of probability distributions: one assigns probability 1 to the empty history (the
cha11enger's belief at the start of the game) , and the other assigns probabilities to
the histories Ready and Unready (the incumbent's belief after the chalienger enters)
70 .4.2 Strategies
When studying Nash equilibrium in the previous section we incorporated the possibility of randomization by a110wing players to choose mixed strategies. In the
current context , a different formulation is convenien t. Rather than alíow L."1 g a
player to choose a probability distribution over her pure strategies, we have her
assign to each of her information sets a probability distribution over the actions
available at that se t. We refer to such an assignment as a behavioral strategy.
DEFINITION 324.2 (Behavioral strategJj in extensive game) A behavioral strategy of
player i in an extensive game is a function that assigns to each of i's information
sets Ii a probability distribution over the actions in A (I i) , with the property that
each probability distribution is independent of every other distribution.
A behavioral strategy in which every probability distribution assigns probability 1 to a single action is equivalent to a pure strategy. In a11 the games we
study, mixed strategies and behavioral strategies are equivalent; 1 choose to use
behavioral strategies now only because they are easier to work with. The relation
between the two 可pes of strategy is illustrated by two examples.
In the game in Example 314.2 (BoS as an extensive game) , each player has a
single information set , so a behavioral strategy for each player is a single probability distribution over her actions. Thus in this game each player!s set of behavioral
strategies is identical to her set of mixed strategies.
In the game in Example 315.1 (the card game) , player 1 has two information
sets , so a behavioral strategy for her consists of a pair of probability distributions ,
each over the set {Raise , See}. I3 y contrast, a mixed strategy for player 1 in this
game is a single probability distribution over the set of her four pure strategies,
{(Raise , Ra 四)， (Raise , See) , (See , Ra 四)， (See , See)}. A behavioral strategy is thus
speci且ed by two numbers (the probabilities of Raise if player l's card is High and if
it is Low) , while a mixed strategy is speci且ed by three numbers. (Remember that a
probability distribution over k actions is determined by k - 1 numbers, because the
k probabilities must sum to 1.) In this sense a behavioral strategy is simpler than
a mixed strategy. However, restricting a pJayer to behavioral strategies does not
limit her options, in the sense that for every mixed strategy and eve可 list of strategies for the other players there is a behavioral stra
10.4 Beliefs and sequ黯 ntial equìlibrium
325
probabiìíty distributior飞 over outcomes. (τhis result is not limited to this gameit holds in a very wide class of games that includes all the games studied in this
d飞apter.)
10.4.3 Equilibrium
The notion of equilibrium 1 define applies to pairs consisting of a profile of beha飞，­
ioral strategi部 and a be!ief system; the following picce of terminology is convemenι
D旦FIN ITl ON
1 (Assessment) 人只 assεssment in an extensive game is a pair
consisting of a profile of beha飞lioral strategies and a belief systεm.
An assessment is an equilibrium if it satisfies the following two 把quiremeηts
(which 1 make p陀cise in the subsequent discussion).
Sequential rationalify Each player's strategy is optìmal whenever she has to
move, given her belìef and the other players' strategìes.
Consistency of beHefs with strategies Each player's belief isεonsistent with the
strategy profile.
The requirement of sequential rationality generalizes the requìrement of subgame perfect equilìbrium that each player's strategy be optimal in the part of the
game that follows each history after which she moves , given the strategy profile,
regardless of whether this history occurs if the players follow their strategies. 1n
the more general context of an extensive game, sequential ratìona !i ty
that
each player's strategy be optimal in the part of the game that follows each of her il1户nnatiol1 sets , given the strategy profile and given the player's beli<吧f about the lúsfory
in the information set that has occuïrfd , regardless of whether the information set is
reached if the players follow their strategies.
Consider, for example , the game in Figure 326. 1. Suppose that player γs strategy (indicated by the black branches) selects E at the start of the game and J after
the history (C ,F), and player 2's beHef at her information set (indicated by the
numbers in brackets) is that the hi阳y C has occurred with p时ability ~ anà
tl的i如Y D has occurred wi的robabili时 Sequential rationality 叫uires that
player 2's strategy be optimal at her inform州on s时， given the subsequent behavior specifiεd by player 1's strategy, even though this set is not reached if player 1
follows her strategy. Player 2's expected payoff ìn the part of the game starting
5 tra
刚
at her informa加 set， given h的由f， ìs ~ .0 个 l 1 = to her 挝
. l
ctt
i .1忏+~.o=j卡to 阳1 冶础r5惚t衍芷凯a舱t始e呜g白帘yG.1咄hU5
础lU5
础S 龄叫
伊
q
悄
uee阳lra
蚓
at胁
i妇
onal
叫
a剖硝lity 叫ul附 h
怡
e缸rt旬os剑
ele创
ζ. Sequεntial rationality require5 a150 that player 1's strategy be optimal at each of
given player 2's strateg予 Player 1'5 optiher two (onεel廷ment) 放飞formation
malac柱。n after the history (C, F) is J; if player 2's strategy is G, her optimal actions
at the start of the game are D and E. Thus given player 2's strategy G, playεr 1 has
two optimal strategies, D Jand EJ.
Chapter 10, Extensive Gamωwith Imperfect Information
326
1
C
1~
K
3, 0
E
D
严飞
。， 1
1, 1
2, 0
2，。
0, 3
Figure 326， 1 人 伊 me illustrating the sequenhal rationa挝、，r requirement on an assesömen t. Given
strategy, indiçated by the black branches, and player 2's belief, given by th日口umbers ìn
brackets 在 t the end of the histories C and 0 , player 2'主 expected payoff to G exceeds her expectξd
payoff to F, 50 that her optim在 i 在 ction is G
playerγs
,
Now let 伊，向 be an assessmer泣如 an arbitrary gamε(i.e. let 卢 be a profile
of behavioral
andμa belief system) , and let 1; be an ínformation set
of player i. Denote by 01; 巾，别 the probability distribution over terminal histories
that results if each history in [í occurs with the probability assigned to it by i's belief
μi (which is not necessarily the probability wíth which it occurs if the players adhere to 舟， and subseque l1 tly the players adhere to the strategy profile 卢. (Compare
this definition with that of Oi1 (s) for ar飞 extensive game with perfect information
(s机 page 165). For the inf、丁rmation set {C ， 口} and 也εstrategy of player 1 and
belief or player 2 indicated in the game in Figure 326.1 , the probabi1ity distribution
assigns p价ability ~ to the terminal history (C , F, J) and probability to (口， F) if
player 2 uses tl时t川egy F, and probability ~ to (C , G) and p时及bility to (口， G)
if she uses the strategy G.)
We can now state precisely the sequential rationality req出rement: for each
play时 i and each of her information sεts h her expected payo好 to 0 1/卢，抖 ìs at
least as large as her expected payoff to Olj ( (γ i ， ß-i) , Ji) for each of her behavioral
1
l
strategìesγ/.
The requirement that each player's belief be consistent with the strategy pro曲
file is new, though forms of it are implicit in the definitions of r如 sh equilibrium
and subgame perfect equi1ibrium. The idea is that in a steady state, each play时 's
beHef must be correct: the probabili句T it assigns to any histo巧r must be the proba伊
bility with which that history occurs if the players adhere to their strategies , The
implementation of this idea is straìghtforward at an information set reached with
positive probability if the players follo时 their strategies, but is un c1 ear at an inω
formation set not reached if the players follow their strategi时， Every history in
such an information set has probability 0 if the players follow their sh'ategies, but
ìf such an information set is reached , the play但 who moves there must believe that
some history has occurred. 飞唤 deal with this dif自culty by allowing the player who
moves at such an information set to hold any belief at that information se t.
10.4 Beliefs and seque狗ti击I equilibrium
327
τhus we formulate thεconsistency req出陀ment to restrìct the betief system
only at information sets reached wìth positive probability if every player adheres
to her strategy. Precisely, we require that t枪 probability assigned to every history
h* in such an information set by the belief of the player who moves there be equal
to the probability that h* occurs according to the strategy profile , conditional on
the informatìon se t' s being reached. Denoting the information set by Iì and the
strategy profile by 卢， by Bayes' rule (Section 17.6.5) this probability is
阳Mm·­
到一始
3
拍一吨
K
叫一m
，
…∞
了"
。一
d
不一弘
H
b-U krwv
nDS
(327.1)
Consider again the game in Figure 326.1. If player l's beha 飞!ioral strategy assigns probabìlity 1- to the action E at the start of the game, the consistency requirement places no restriction on player
beliefbecause player 2's information set ìs
not reached if player 1 adheres to her strategy. If player l's action at the start of the
game assigns posìtive probabiJity to C or 0 , however, the consìstency requìrement
bites. If player 1's strategy assigns probability p to C and probability q to 0 , so
q (where ß ìs the stratthat Pr(C acζording to ß) = p and Pr( 0 accordi吨 to ß)
egy profile) , then (327.1) ìmplies that player 2's belief assìgns probability p/ (p + q)
to C and probability q/(p 十 q) to 在 ln particula乙 ìf player 1 chooses 0 with probability 1, then player 2's belief must assign probability 0 to C and probability 1
to O.
EXA lv1 PLE 327 2 (Consistent beliefs in BoS as an extensive game) ln the game in
Examplε314.2， fcr every strategy of player 1, player 2's Ïnformation set is reached
with probability t so the consìstency condition restrícts player 2's belief ìn every
assessment. τhe condition simply r吨uires that player 2's beHef assign to B and S
the probabilities with which pl可前 1 chooses thεse actio只s. That ìs, ìn this game
consistency requìres that player 2、 belìef always be correct. τhesamεis true for
any game ìn whìch all the players move simultaneously (no player knows any
other player's action when ma挝ng her move).
‘
EXA lv1 PLE 327.3 (Consistent beHefs ìn card game) Consider the game ìn Example 315. 1. If player l's strategy selects See whether her card is High or Low, the C011sistency conditio11 does 110t restrict player 2's belief, because player 2's inform剧。n
set is not reached. For every other strategy of player 1, the conditìon determines
player 2's belief. DeI飞ote the probability that player 1 chooses Raise if her card
isH桔h by PH and the probabi挂ty she chooses Raise if her card is Low by PL. 丁h纪n
Pr( (H1gh , Raise) accordìng to 卢) = iPH and Pr( (Low , Raise) accordìng to ß) 口 jpu
so by (327.1) , the consistency requirement is that player 2's belief assign probabilìty PH/ (PH 十 p rJ to the history H and probability pd (PH 十 P rJ to the history L.
EXA lv1 PLE 327.4 (Consistent 各eliefs in entry game) In the games in Figures 317.1
and 323.1, de∞栓 by PR , pu , and po the probabilities 位e challenger assigns to
Ready, Unready , and Out. If po = 1 ，也e consistency condition does not restrict the
328
Chapter 10. E汉tensive Games with Imperfect Inform在tion
incumbent's be!ief. Otherwise, the condition requires that the incumbent assîgn
probabi1ity PR / (p R 十 pu) to Ready and probability PU / (PR 十 PU) to Unready.
In summar予 the notion of equi1ibrium 1 use for extensive games îs defined
precisely as follows.
DEFINITION 328.1 (Weak se号uential equilibríum) An assessment (ß , /l) (consisting
of a beha毛Tioral strategy profile 卢 and a belief system /l) is a weak sequential
equilibrium if it satisfies the following 队10 conditions.
Sequential rationality Each player's strategy is optimal in the part of the 伊me
that follows each of her information sets, given the strategy profile and her
beHef about the history in the information set that has occurred. Precisely,
for each player i and each information set 1i of player i, player í's expected
payoff to the probabi1 ity distribution Odß , /l) over terminal histories generated by her belief /l i at 1j and the behavior prescribεd subsequently by the
strategy profile 卢 is at least as large as her expected payoff to the probability
distribu tion 0 1,((γì， ß~i) ， μ) generated by her belief 1' i at Ii and the behavior prescribed subsequently by the strategy profile (γIf 卢 -i) ， for each of her
behavioral strategiesγ←
Weak consistency of beHefs with strategies For every information set 1ì reached
with positive probabílity given the strategy profile ß, the probability assigned
by the belief system to each history 11* in Ii is giv仅1 by (327.1).
The game in Figure 326.1 ìllustrates this notion of equilibrium. As we have
seen, in this game player 1 、 strategy E] is sequentially rational given player 2's
strategy G, and player 2's strategy G is sequentially rational given the bεliefs in叩
dicated in the fìgure and player l's stratεgy E]. Further, the be1ief is consistent
with the strategy profile (E] , G) , becausεthis profile does not lead to player 2's
ìnformation se t. Thus the game has a weak sequential 叫uilibrium în which the
strategy profilεis (EJ , G) and player 2's belief ìs the onεindicated in the figure
(or any other belief for which G is optimal). 时ow consider the strategy profile
(DJ , G). Player l's strategy is sequentially rational given player 2's s位ategy， and
player 2's strategy is sequentialI y rational given her belief and player l's strategy.
But player 2's belief is not COl飞sistent with the strategy profile. The only consister飞t
belief assigns probability 1 to D, which makes player 2's action F, rather than G,
optima l. Th us the game has no weak sequential equilibrium in which the strategy
profile is (口J ， G)
In ar飞 extensive gam合 with perfect information, only one beHef system ìs
possible-that in which ever
10.4 Beliefs and sequential equilibrium
329
In a general extensive 伊me， the requirement of sequential rationality imp1i部，
in particular, that each player's strategy is optimal at the beginning of 出e game ,
given the other players' strategies and the player's betíef about the history at each
information se t. Further, the consistency requirement implies that each player's
belief about the history is correct at any information set reached with positive probability when the players follow their strategies.τ'hus if an assessment is a weak
sequential equilibrium, then each play枝、 strategy in the assessment is optimal at
the beginning of the game , given the other pl玛rers' strategies. That is,
the strategtj pro.卢 le in any weak seque l1 tíal equilibrium 忿 a Nash equilibríum.
To find the weak sequential equilibria of a game, we may use a combination of
the techni平les fodinding 岳飞lbgamεperfect equilibIia of extensive games with perfect information and for finding 到ashεquilibria of strategic games. Consider, for
example, the game in Figure 326.1. The sequential rationality requirement implies
that in any weak sequential equilìbrìum player 1 chooses J after the hìstory (C , F)
Now we can consider two cases.
• Does the game have a weak sequential equilibrium in which player 1 chooses
E? If player 1 chooses E, player 2's belief is not restricted by coηsistency，
so we need to ask (û) whether any strategy of player 2 makes E optimal
for player 1, and, if so, (/7) whether there is a belief of player 2 that makes
any such strategy optima l. We see that E is optîmal if and only if player 2
chooses F 附hp时ability at most ~. A町 such strategy of play盯 2 is optimal if player 2 believes the history is C with probabîlity l' and the strategy
of choosing F with probabilìty 0 is opt校ir熙I叭1冶al if playe阳r 2 beli必eves the hìstory
i沁sC 飞W
川
叫i让th 拟
γ
a 1巧
盯
y proba油
bì也
l丑it咛
Y of 挝
at 沁
1e甜
飞认悦气怆々 co础
weak 肥
s equε口时t往ial equi丑li出
bri川
um飞 i迁f player l'、s strategy 总
is ε J and playe
仔r 2 either
i
位
C
ho
∞
os
使
岱s Fw
ε
川旧
t甘
t问
伊ro
P
忡
ba
抬
bi出
i且it咛
y1冬， or chooses G and believes that the history ìs C with probability
at least
!
• Does the game have a weak sequential equilibrium ìn which player 1 chooses
C ar飞d/ or D with positive probability? Denote the probability player 1 assigns to C by P and the probability she assigns to D by q. 1n such an 吨ui­
lìbrium, player 2's bεlief is constrained by consistency to assign probabilìty
p/(p 十引 to C and probability q/ (p 十引 to D.τhus G is optimal 证 p > q,
F is optimal if p < q, and ar飞y mixture is optimal if p = q. Now, if player 2
chooses G, the only optimal strategy of player 1 that assi事lS positive prob1, making G not
ability to C or D assigns probabìlity 1 to D, so that q
optimal for player 2. If player 2 chooses F, player 1's on1y optimal action
at the start of the game is C, making F not optimal for player 2. If player 2
chooses a mixture of F and G, then player 1 must assign the same probability
to both C and D , and hence must choose D with positive probability, which
is incompatìble with the fact that her expected payoff to D ìs less than her
Chapter 10. 主xtensive Games with Imperfect Informati。我
330
expected payoff to E. Thus the game has no weak sequential equilìbrium in
which player 1 chooses C andl or D with positive probability.
We conclude that in every weak sequential equilibrium of the game , player 1 、
阳 tegy is
叫 player 2 either chooses F wi问 robability 归lost i 时 believes
that the history is C with probability ~， or 伽oses G and believes 始 ttl的i纹。ry
is C with p灿 abilíty a t le拟 i. (Another method of finding 怆se 叩ilibria ìs to
find all the Nash equilibria of the game , and then check which of these eq口 ilibria
are associated wÎth weak sequential equilibria.)
EXAMPLE 330.1 (Weak sequential eq出libria of entry game) As we saw ìn Example 322 1, the entry g击 me of Example 317.1 has two pure strategy Nash equilibria ,
( UIlJ'eady， Acquies♂) and (Out ，Figlt 片. Consider the first equilibrium. Consiste:::飞cy
requires that the incumbent belie飞'e that the history is Unready at its information
set, making Acquiescc optimal徨 Thus the game has a weak sequential equilibrium 如
.vhìch the strategy pr呱坠 is (U l1 rendy , Acquiesce) and the incumbent's be !ief is that
thεhistory is Ul1 readlf. Now consider the second equilibrìum. Regardless of the ìncumben t's belief at its informatìon set , Fight is not an optimal action in the re出ain栅
de1' of the game…fore飞'e1'y be!ief, Acquiesce yields a highe 1' payoff than does Fight.
Thus no asse5sment in which the strategy profile is (Ollt ,Fight) is both sεquen叩
tially rational and consi5tent, 50 that the game has no \oveak sequential equilìbrium
lTI v由 ich the strategy profile ìs (Out ,Figlzt).
(
,
τhe games in Examples 314.2 侣。5 as an extεnsive game) and 315.1 (the card
game) havεno Nash equilibria in which some information set is not reached ,
so that for each game the set of strategy profìles that appεar in weak sequential
equilibria is equal to the set of Nash equilibria.
Why μweak" s叫时ntial equilibrium? The consisten巧 condition's limitation
to information sets reached with positive probability generates, in some games ,
a relatively large set of equilibrium assessments, some of whìch do not plausibly
correspond to steady states. Consid时， for example, the game in Figure 331.1, a
飞!ariant of the εntry game in Fig Ul‘心 323.1 in which Ready is better than Ul1 ready
for the challenger regardless of the incumber役 's action. 丁his 伊me has a weak
sequcntial equilibrium j只 which the challenger's strategy is Out , the incumben t' s
strategy is F, and the incumbent believes at its information set that the history is
U/1 ready. 1n thisεquilìbrium the incumbent belíev仍 that the challenger has chosen
Unready , although this action is dominated (by Ready) for the challenger. Thus the
ìncumben t' s be!i ef does not seem reasonable.
Notions of equílibrium that impose stronger consistency conditions narrow
down the set of 叫uilibrium outcomes 如 some games. One notion, sequential
equilibrium , ìs widelyused. However, the consistency condition ìt imposes is
not straìghtforward , and the notion does not in any case eliminate all i
10.5 Sign击 linggames
331
Challel飞ger
U
Qut
2, 4
。， 3
3, 2
1, 1
川、
卜、
4, 2
R
Figure 331.1 A variant of the entry game in Fí在 un! 323.1 in which Ready is better than UII 只如11/ for the
challenger regardless of the incumben t's actío l1 and the íncurn 如ent prefers Figll1 to Acq l/ ic兰æonly if th c:
.;hallenger is unprepared , but there is a weak seq口号公tÍal equilibr时 m in whìch the challenger stays out
and the incurnbent fights. The pJayers' strategíes in thi苦 equìlibriu沁 are indìcated by the black line;记 the
pωbabìlití的 the Íl飞 curnbe以';; belief 吕ssigns to each hístory in ìts infonn i1 tion set ilre show罚 ìn brackets
1
D
L
1A
C
2
-- -i
r
3
R
L
0, 0, 0 4, 4, 0
nu AU
F
吨'i
3, 3, 2
2
C
Figure 331.2 The garne ~n Exercise 33 1.1 (S告 Iten's horse)
set of weak sequential equilibria isηotlargεand does not include implausible assessments. For these reasons 1 work with weak seque时出 libriurr飞 and donot
consid盯吁台 finements" of this notion.
? EXERCISE 33 1.1 (Selten's hors的 Find the weak se弓闲时ial 叫uilibria of the game
iηFig双 re 33 1. 2 ir飞 which 仰ch player's strategy is pure.
(Find the pure strategy
Nash equ i1ibria , then dεtermine which is part of a weak sequential equilibrium.
τhe name of the
comes from the person who first studied it , and its shape.)
( EXERCISE 33 1. 2 (Weak sequentìal equilibrium and Nash 叫uilibrium in subgames)
Consider the variant of the game ìn Figure 33 1.1 shown iηFigure 332.1 , in which
the challenger's initial move is broken into two steps. Show that this game has a
weak sequential equilibrium in which the players' actions in the subgame following the history I I1 do not constitute a Nash equilibrium of thεsubgame.
10.5 Signaling gam制
1n many interactions , ir杖。rmation is "asymmetric": some parties are ir飞知rmed
about variables that affect everyone, and some parties are no t. 1n one interesting class of situations, the ir杖。rmed parties have the oppor切nity to take actions
Chapter 10. E宜tensive Games with Imperfect Information
332
Challenger
Challenger
R/凰\\\ u
Incumbent
A
qaA
r
咱BEA
4, 2
主
A
F
3, 2
0, 3
2, 4
Figure 332.1 A 飞'ariant of the entry gaη1日 in Fi这 ur巳 33U in which the challenger's decision ìs broken
into t飞气'0 steps.
observed by thε uninfonned parties before the 1atter take actions that affect e飞儿
eryone. ln 50me circumstances, the informed parti邸， action5 may 飞igna1" their
information.
Supp05e, for examp1e , that an emp10yer can observe whether a job applicant
has a college degree, but not her abilíty, and that a person of high ability may
obtain a dεgree at low cost , while one of 10w abi1ity may do 50 on1y 批 high C05 1.
Then the fact that a person has a degree may signa1 to an emp10yer that she has
high ability-not because college teachωan)' skills , b飞lt because on1)' high-ability
individua1s find obt吕 ining a d仓gree worth飞vhile， gi\'en the C05 1. (1 return to this
examp1e i 只 Section 10 .7.)
In a genera1 twoωp1ayer "signaling game , a se l1 der is informed about a 飞'ariable
relevant to both her and a receiver (or set of recei飞'ers)， who is uninformed. The
sender takes an action obser\'ed by the recei飞rer， \-\'ho then take5 an action that
affects them both. Depend ing on the v飞'ay in which the message and thεreceiver's
action affect the parties , the send. er may want to limit or distort the information
her signal conveys.
Such a situation may be modeled 在 s an extensive game in which the sender
has several possib1eμtypes'二 each corresponding to a value of the variable about
which she is informed. The value she observes , and thus her type , are determined
by chance. The recei飞!er does not observe the sende了 's type , b l.l t sees an action
she tak制I and then herse1f takes 汶口 action. A simple example is gi飞'en by another
variant of the entry game.
tl
EXAMPL足 332.1 币ntry as a signaJing game)
A challenger contests an incumbe时's
turf , The challenger is strO /1 g with probability p and weak with probabiJity 1 -- p,
where 0 < p < 1; it knows its type, but the incumbent does no t. The challenger
m巧 either ready itself for battle or remain unready. (It does not have the option of
staying ou t.) The incumbent observes the challenger's readiness, but not its type,
and chooses 快rhether to 卢ght or acquiesce. An unready challenger's payoff is 5 if
出εincumbent acquiesces to its entry. Preparatíons cost a stroηg challenger 1 urut
of payoff and a weak one 3 units, and fightíng entails a 1055 of 2 unìts for each
type.τ阳 incumbent prefers to fjght (payoff 1) rather than to a 叫uiesce to (payoff
10.5 Signaling games
333
2,… 1
A
F
例
口γ-FK
…
A
卜一胁
n
-oo
俨飞旷
p-m
。、…
du
一
znH
[
aL
Ir飞cumbent
C
km
ded
yry
dooo
二， 0
Incumbent
。， 1
汇A
是， 2
Rcadlf
Challenger
Ul'l ready
A
F
J生
F
5, 2
3,… 1
5, 0
3, 1
The game ín Example 332 .1. The empty hìstory is i 扫 the center of the diagram 、 τhe inactions an'λ{在 cquíesce) and F (fight)τhe cha Jl enger's payoff is Iist口 d fírst , the íncumben t' s
Figur.告 333.1
C泣泣1bent's
second
。) a v气!eak challenger (\-vho is quickly dispensed \-vithL and p陀fers to acquiesce to
(payoff2) rather than to fight (payoff -1) a strong one.
The extensi飞'e game in Figure 333.1 models this situatìon. Note that the empty
history is in thεcenter of the diagram. The first move is made by chance, which
determir飞es the challenge了 's type. Both types have two actions, Ready and Ul1 ready ,
so that the challenger has four strategies. The incumber飞t has t认10 information sets ,
at each of which it has two actions (A and F) , and thus also has four strat吨ies.
1 now find the pure weak sequential equilibria of this game. First no讼 that a
飞气leak challenger prefers ll l1 ready to Ready regardlεss of the incumbent's actions
even if the incumber飞t ac弓uiesces to a ready challenger and fights an unready one ,
the challenger prefers to be unread y. τhus in any weak sequential equilibrium
a 认reak challenger chooses Unready. 1 consider each possible action of a strong
challenger in tum.
Strong chalienger chooses Ready Both the incumber吭's information sets are
reached , so the consistency condition restricts its belief at each set. At
the top information set, the incumbent must believe that the history was
(Stro咐， Re{均1) ， and hence choose A. At thεbottom information set it must
believe that the history was (Weak , U1附ady) ， and hencεchoose F. Thus if
the challenger deviates and chooses Ullready if it 1s strong, then it is worse
off一it obtains the payoff of 3 rather than 4. We conclude that the game has
a weak sequential equilibrium in which the chal1 enger chooses Ready when it
is strong and Ul'l ready when it is weak, and the incumbent ac气功esces when it
sees Ready and fìgl飞ts when it sees Ullready.
Strongchallenger chooses Unready At its bottom information set, the incumbent
believes , by consistency, that thεhistory was (Stro l'lg, Ul'l ready) with probability P and (Weak , Ur悦ady) with probability 1 p. Thus its expected payoff to
A is 2p and to F is … p+l-p 口 1- 2p. He附 Aisop汹汹 p 三! and Fis
optimal 证 p 三 i
334
Chapter 10. Ext酷nsiveG器 mes with Imperfect Information
Suppose that p 三 i 时 theincurnbentchoo附 A in respo附 to Unready
A strong challenger that chooses Unready obtains the payoff of 5. lf it s飞Nitches
to Ready its payoff is less than 5 regardless of the incurnbent' s action. Thus if
pEjr 加抗 the garne has 附ak 叫阳
of challeng伊er c仨协00
锵$伫口mηr栩
ea
α句
d y and 也
t h飞伫 incumbent acq卢l飞uiesces to an um
飞lready
巾
dh驳圳
allen飞g
萨
但E旦. The incurnbent m
e
丑1a
叮
Y hold any beli
拉ief about the 穹
type of a ready
巾
dhall怡
eng
伊
仔r咒吼" and, depending on its belief, may fight or acquiesce.
e
Now suppose 出at p 三 1 and that the inω巾e时 chooses F in response
to Unready. A strong challeng仔 that chooses Unready obtains the payoff of 3.
If it switches to Ready its payoff is 2 if the incurnbent fìghts and 4 ìf it ac气也棚
esces.τhus for an 叫uilibrium the incurnbent rnust fight a ready challenger.
I盯fi让t beli怆
eve
岱
s 出
t ha
挝t aready cha
剖II坠
er咯盯 i岱s weak with high enough probability (at
i伫eas
滔st才j争)， 在
fii培快供
gl胁
ht
让tin
吨附
刷is ir怆材 O
g
叩P
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咐tim
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陀e时f印
向
po附臼蚓ibl沁贮U
白
ein
扫妇
nan
川e叩 ilibriu扭时?
Yes: the consistency conditiön does not restrict the incumbent' s belief upon
obs前ving Ready because this action is not taken when thξchallenger follows
its 州 tegy to choose U盯eady 吨ardle览。f its type. Thus ìf p 三 1 the game
has a weak sequεntial equilibriurn in which both types of challeng号r choose
Unready , the incurnbent fights regardless of the challenger's action, and as咆 ns prωO
伪彻
ba边b
弘斟刻
汹讪
i过l且出让
itty of 挝
a tl怡e础&叶 to th冶川
e们chal
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吨
m
n
吨
1唁
ge
啡臼r'、sb
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伦b
检
伽se附$川川
t吐h由ιa
the challenger is r已ady for battle.
Jn surnrnary, the garne has two sorts of weak sequential equilibrium.
·τh台 ζhallenger chooses Ready if it is strong and 口nready if it is wea k.
The
incurnbent believes that a ready challenger is strong and an unready one is
weak and 击 cquiesces to a ready challenger and fights an unready one.
• Th飞ech
沁
1冶a
刻
叫11险
eng
伊
er chooses Unr，
陀εad
句
f切
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伊
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es部sof 郎
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curn
主η飞b
快
er时
1让tb刊圳
eli
妇1挖
C飞ve
臼
S
t血
h旧
挝ta
a
川η
川
川1川 eady
严
f川
cha
叫
a11怡
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e
叫
m
仙
n1卢ge
町r 沁i $ 蜘
eq飞u
臼
1忍il拉ib
如ria in which the incumbent acquiesces to an unready challenger, holds
any belief about the type of a ready cha11enge毛 and takes whatever action is
optima l, given its belief, if the challe吨er is ready. If p 空 1 the game has
equilibria in whïch the incumbent fights a11 challengers and belíe飞!es a read
Separating equilibrium Each type of the sender chooses a different action (in
the first sort of equilibrium in the example, a strong challenger chooses Ready
and a weak challenger chooses Unready) , so that upon observing the sendεr's
action , tl飞e receiver knows the sender's type.
Pooling equilibrium All types of the sender choose the same action (in the sec。nd sort ofεquilibriurn in the exarnple, both types of challenger choose Unready) , 50 that thεsender's action giv础 the recei飞!er no clue to the sender' s
type.
10.5 Signaling games
335
If the sender has more than two types , mixtures of these types of equilibrium may
exist-the set of types may be divided into grcups, within each of which all types
choose thεsamεaction ar飞d between which the actions are different
? EXERCISE 335.1 (Pooling and separating equilibria i口 a signa1ing game) Consider
a game that differs from the 0附 in Fìgure 333.1 0只1y in the payoffs to the terminal
histories (Weak , Ready,..4) and (切éak， Ready, F) For what values of thεse payoffs ,
if any, does the game have a weak sequential 了'separating") 较年lÌlibrium in which
a strong challenger chooses Ready and a weak one chooses Unready? For ￥飞点at
val l1 es of these payoffs , if any, does the gamεhave a weak sequential (平。oling")
equilibrium in which both types of challenger choose U1I ready?
‘
? EXERCl SE 335.2 (Sir Philip Sydney game) Some young animals expend energy
begging for food [rom their parents-they squawk and b1eat and scream , sometimes εxtravagantly. Car飞 we expect these dεmands to signal their needs accurately? Consider the following signaling game. (The rationale for its name, due
to its originator, the distínguished evolutionary biologist John Maynard Smith, is
given in the Notes at the end of the chapter.)
to its offspring, or keep for
A hungry pare口t has a piece of food that ìt may
ìtself. It does not detect whether its offsprìng is hungry. In eíther case , the offspring
may sígna1 that it is hungry to its parent (by squawking , for examp1e). An anímal
is stronger and thus produces more offspring (i.e. has a higher "bíological fitness'丁
if it gets the food than if it does no t. Normalize the parent's strength if it keeps th巳
food to be 1, and denote its strength if it gives the food to its offspring by 5 < 1.
If the offspring does not squawk, its strength is 1 ìf it gets the food , V < 1 if it
is not hUngry ar飞d does not get the food , and 0 if it is hUngry and does not get
the food. If the offspring squawks, its strength is multiplied by the factor 1 一 t ，
wherεo < t <三 1 (i. e. squawking may be costly). Denote the degree to whìch
the parent and offspring are re1ated by r, and take each player's payoff to be its
strength plus r times thεother player's strength. Evolutionary pressure willlωd
to beha飞lior for each player that maximizes that player's payoft given the other
player's behavior. (For more on evolutionary games, seεChapter 13.) The game is
shown ìn Figure 336.1.
Find the conditions on r , in terms of 5 , V, and t , under \>vhich the game has a
separating 叫uilibrium in which the offspring squawks if and only if it is hungry
and the parent gives it the food if and only if it squawks. Show that if the offsp抵抗恙's payoff from obtaining the food when it is quiet exceeds its payoff from not
obtaining 泣， whether or not it is hungry (which means that r < (1 … V)/(1-S)) ,
then the game has such an e气uilibrium only if t > O. That is, in this case an equilibrium exists ìn which the signal is accurate on1y if 出e signal is costly. Show that if
r < (1 … 5) /(1 一口 - p)V) , then the 态ame has a pooling e电uilibrium in which the
。ffspring is always quiet and the pare时 always kεeps the food. (For some other
parameter values, the game has a pooling equilìbri也n in which the offspring is
always quiet and the pa
336
Chapter 10. Extensive Games with Imperfect Information
1
r, l
Keep \
十 r5， 5 十 r(l-t)
/
(1 … t )V 十九 1 + rV
Gíve
Keep \
1
牛 rS ， 5 + r(1 •
t)
/ Give
Parent
一制
Squawk
V'
Offspring
ae&
1
Eι
1
忖WJ
P-mb
』
帽!
『￡ t
叩门 U
j
…
N
』嗣
咱1
咱
…
d
叫
-,
ρL …
户、 m
』
萨
…忖句JV
mσo
日
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c
n 一
i1
ιγAzi
呻
nug-nH
'HM叩
'次 gd
。
位
m
俨WTQ
5
--l
Parent
Keep
Give
1 十 ïS ， S 十 r
Quiet
Keep
V 个人 1+γV
Give
1 十1' 5 ， 5 十 T
Fi晋 ure 336.1τhe Sìr Philìp Sydnev game (Exercì古e 335.2). The emp乌七 istory is in the center of the
diagram. The offspring's payoff is listed first, the parent's second.
10.6 lIIustration: conspicuous expenditure a革 a signal of quality
vVhy do some firms sponsor sporting events? Here is an explanation. The quality of scme goods cannot be detected before the goods are consumed , and some
firms sell goods of low quality while others sell goods of 1飞igh qua 1ity. A consumer
will purchase from a firm repeatedly once she detects goods of high quality, but
wi1l丑。t buy low-qualìty goods more than once. Thus a purveyor of high-quality
goods gains more from a consumer's sampling its wares thar飞 does a dealer in lowqualìty merchandise. Consumers may therefore deduce that it is worthwhile only
for sellers of high-quality goods to sponsor sporting ε飞lent告一the potential benefit
does 1飞的 exceed the cost for low-quality firms-so that conspicuous expenditure is
a sure sign of a high-quality fir凯， and lack thereof the mark of a low-quality 位m.
A high-qualíty firm is thus nudged into sports sponsors七ip.
Does this argument stand Up to careful examination? Suppose that with probability 7τa firm is one that produces goods of 号uality H > 0 and with probabilìty 1 一汀 or飞e that produces goods of quality L = O. (Note that the firm does not
choose its quality.) The cost of producÌl飞g a good of qualìty H is CH > 0 and that of
producing a good of quality 0 is CL = O.
A consumer interacts with thεfirm over two periods. ln the first period , the
firm chooses a price Pl for its output and a口 amount E 主 o to spend conspicuously,
for example, sponsoring sporting e飞咄.1Ìs. The expendi机1l'e E has no effect on 白e
quality of the good The consumer observes E and pl , butηot the quality of the
good , and decides wl飞的her to purchase thεgood. If she makes no purc阳时， the
interaction ends; her payoff is 0 and the fjrm's is - E. If she makes a purchase, she
learns the qua 1i ty of the good , the 如m chooses a price P2 for the second period ,
and she decidεs whether to purchase the good agaìn. In ar飞ypεriod in which she
‘
‘
10.6 lII ustration: co恼picUOU5 expenditure as a signal of quality
337
purchases the good her payoff is v … p, where v is the quality of the good and p is
the pτÍCe， and her total payoff is the sum of her payoffs in the two periods. (τhus
the consumer is willing to purchase the low-quality good in either period only if
its price is zer轨 }τhe firm's payoff is P1 … E 一句， where 1 is either L or H , if it
sells to the consumer only in the first period, and Pl 十 P2 E 一 2c 1, if it sells to
the consumer in both periods. (Note that thεexpenditure E occurs only in the first
period.)
Assume that H - CH > 0, so that at the highest price the consumer ìs willing
to pay for the high-quality good (namely H) , this good yields the firm a positive
profi t.
The structure of anεxtensive game that models this situation is shown in Fig耐
ure 337. 1. 1n this 自gure ∞ly one of thc possible pairs (Pl , E) of first-period actions
for the firm is indîcated , and the range of second-period prices P2 is indicated by
the shaded trìangles果
Consìder the equ i1ìbria of this game. In the second period thεplayers are fully
informed. Th us the consumer buys the good in this period only if its price is at
most 吨ual to its quali年 A price equal to H yields the high-quality fjrm a pro岳飞 so
in an equ抽血m a high-quality firm ch盯ges pf
H and the consumer buys the
good; the firm's payoff is H - CH and the consumer's payoff is O.τhepro且t of a
low-quality firm is 0 indepεndent of its price: if the price is positive the consumer
does not purchase the good , and 迁
i fi让t 怒
i s zero the 曲
fi让rrr
m
口
丑l'、s profit is zero 陀
re
咆
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盯rdles
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ioιO
付
ld P阳 p玛; 主 O
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or a low-quality 位
fi让r门miβscωomp在tible with equilibrium.
Pl - E
CH 十 pf-CHJH … Pl-pp
/
BU1f
R吃斤。 in
飞、，，j
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,
pμ
ηr
、s
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F
Pl … E
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Pl
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Ll lπ
F
R价α111
(Pl , E)
Pl … E - CL , L
及加 in
Pl
O
事民igur陀
e337.1 Anou且t往lí沁
扫
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of ar趴1 ext怆erη\Sl阿vegam
飞
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e 设汹la
放t modelsa fi如古仅rr双\ 泌
th
始&挝t 创
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必
d
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问
h
ge
阳
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en
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位如加
红.rmi总sdωen
f
阳
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时
e
d "F
i沁s shownηle firr泣's payoffs are listed firs乞 the consumer's second.
Chapter 10. Ext牵制ive Games wÎth Imperfect Information
338
1 claim 位1川 under some conditions the whole game has a (separating) weak
sequential equilibrium in which a producer of high-quality goods spends money
in the first period, while a producer of lo\^.飞回lity ones does no t. Consider the
following assessmen t.
Firm's strategy Choose (pH飞) in the first period and H in the second period
after a可 history in which chance chooses H , and choose (0 , 0) in the first
period ar飞d any nonnegative price in the second period after any history in
which chance chooses L.
Consum盯 's belief
11飞e consumer believes that the history was H if and OI飞ly if
the (price , experγiiture) pair (p , E) obsεrved in the first pεriod satisfies p 三
pH* aòd E E 气
Consumer's strategy Buy the good in the first period if and only if the (price ,
expenditure) pair (p , E) satisfies p 主 pH 本 and E = E* , and buy it in the
second period if and only if either the history starts with H and the secondperìod price is at most H , or the history starts with L and thεsecond-p时iod
price is O.
Note that the consumer's belief in this assessment is extreme: unless she observes an expenditure of exac t1 y E 民 in the first period , she concludes that the fírm
is low quality. If the firm follows its strategy, the consumer observes a first-period
action of either 怡胁 ， E*) or 币，码. The consistency condition impliεs that if she
observes (pH袭)， then she must believe that the firm is hígh ql时 ity， wher削 s if
she observes (0，的， then she must believe that it is low quality but does not restrict
her belief af犯r any other observation. The extreme belief 1 have specified makes
a profítable dε飞liatíon by a hígh-quality firm as difficult as possible , because any
deviation causes the consumer to believ巳 that the firm is low quality.
Under what condítions is this assessment a weak se气uεntial equilibrium? The
players' beHefs are consistent by construction, so we need to check only that each
player's strategy is sequentially rationa l.
Firm The firm's strategy is sequentially rational if and only if neither type of firm
can incτease its expected payoff, gi飞'en the consum侣 's strategy and belie f.
Type H If the fiτm chooses {pfh ， E 可 its profit is pf如今日- E" - 2CH , if it
- 2CH , and if it
chooses (p , P) with P < pH本 its profit is p 十 H …
chooses any other (pricε， expendìture) pair its profit is 0 (the consumer
believes it is low quality aI飞d does not purchase the good). Thus for
equilibrium we need
pH' 十日一
2(丁斤
2: O.
Type L If the firm chooses any (p , E) with E 手 E* or E = E* and p > pH飞
then the consumer believes it is low qualit乒协 that its profit is O. If it
chooses a口y (p , E*) with P ::; f严气 then the consumer believes it is high
10.6 IIlustration: conspicuous expenditure as a signal of quality
339
气uality and purchases the good in the first period , yielding it a profit of
p …E*. Thus for 吨的librium we need
。;三 pH*
Consumer If she observes (p , E 可 with P 三 pH飞 shεbelieves the firm is high
qualit实 so that her expected payoff is H pH. ìf she buys the good , and 0 if
she does 口ot. Thus for an equilibrìum in which she buys the good if and only
if the (price , expenditure) pair (p , E 可 satisfies p 三 pH飞 weneed
H … pH*;三 O.
If she observes any other (price, expenditure) pair (p , E) , she believes the
firm is low qualit予 so that her exp仅 ted payoff is 0 ~ P if she buys the good
and 0 if she does not, so that she optimally doεs not buy the good.
In summary, the assessment is a weak sequential 吨uilibrium if and only if
f 今后H~H~ P协主 min{E* ， H}.
(339.1)
If H 三 2c H, there exist pairs (卢飞 E*) vl'Ì th E~ > 0 that satisfy this condition
(see Fìgure 339 .1), and thus weak sequential equilibria in which a hìgh-quality firm
spends conspicuously (on sportingεvents， for exan飞ple汪古le firm doεs so because
if it does n仗， then the consum衍 believes it to bεlow quality, and hence does not
buy ìts good in the first period , does nol find out that it is hìgh quality, and thus
does not buy its good in the second period. EUmin在ting the expenditure saves the
firm E', but it costs it H - 2CH i汩汩st profi t. Imitating the high-quality firm is not
worthwhile for the 10以儿quality one because the cost E次 of doing so exceεds the
resulting increase in its profit of pH* (the firsi-period revenue from sεlling at the
price pH*).
T
E本
H -2CH
O
H pH大→
Figure 339.1 刀飞机haded area is the set of pairs (pH. , E*) satis乌拉g (339.1). For a咛 pair 奴役le set, the
game has a weak s叫uential 吨位ilibriUI妇 in which a firm of type H chooses (p拙，
Chapt量 r 10. 主xtensive Games with Imperfect Information
340
Wes时 that two features of the interaction are important in generating t始 re­
sult. First, a high-quality firm obtains a benefit from attracting a consumer beyond
the period in which it makes the expenditure-thεconsumer purchases the good
in the second period. If there were no repeat purchases, then a low-quality firm, by
imitating a high-quality OI恕， could inducε 出e consumer to patronize it and thus
obtain the same revenue as does a high-qua1ity firm. Gi忖飞ve创
en飞 t也
ha
挝t the low-qual且it咛
y
t业
i让rm
丑l'S un
川
川飞讨i让tcωos仗ti必s lower 由
rη
than 白
th
沁
挝to
a
时f a hi
让igh干干干.嗣-‘ q
Oα11吐
d， the consumer does not purchase the good in the first period if it obsen咄 the
(price , expenditure) pair chosen in equilibrium by a low-quality firm. If it did purchase the good in this case, a high-quality firm could dispense with its conspicuous
expenditure and still obtain the benefit of repeat businεss because the εonsumer，
having purchased its good expecting it to be of low qualit予 would discover th挝 it
is in fact of high qualit予If the consumer does not purchase from the low-quaììty
firm , why does this 自rm exist? 丁检 important point is that a high-quality firm loses
some business when it imitates a low-quality one. If we modify the model by assuming a variety of consumers, some of whom optimally patronize the low-qualìty
firm at the price it charges and some of whom do not, the qualítative feature of the
sεparating equilibria is retained.
Though the game has a separating equilibri山n if (339.1) is sa tisfìed, it also has
pooling e句uìlibrìa， whìch you are invited to study in the following exercise.
<D EXERC!SE 340.1 (Poolìng 叫uilìbria of game in which expenditure si伊als quality)
Find weak sequential 吟uilibria of the g在me in which each type of firm chooses the
same (price, expendìture) pair in the first period.
10.7
lII ustration: education as a 主ignal of ability
Why are you obtaining a collεge degree? Because you thìnk that the principles you
learn ìn your courses will prepare you for the day when you run 1 召M or preside
over Italy? Possibly-but there may be another reason. Perhaps !lothìng you learn
in college has any bearing on the job you expect to take, buryou need to get a
degree to prove to potential employers that your ability is high. 到ow does your
obtainìng a degree prove this point? Because the cost to persons of low ability
of obtaining the dεgr.εe is much higher than it is for you (they will take longer,
and 在nd the process painfu号， so that such persons cannot pr。在tably imitate you.
τhus a colJege degree signals high abìlity, even if colleges do nothìng to foster
that ability: employers know that a recipient of a c01lεdegree must have high
ability because only for such a perso丑 is it 飞Northwhile to obtaìn a degree. If the
cost of achìeving high proficiency in freestylεsηowboarding were much lower for
a person with the skills valued by IBM or the Italian citizenry than for someone
wìthout the skills , a certificate attesting to that achievεment could be your tìcket to
a rewarding job. But it is not, so you are inεol1ege.
Here is a simple model we can use to study this logìc. A worker's ability, which
is either H or L < H , is known to her but not to either of two pote时ial employers
10.7 lII ustration: education as a signal of ability
341
10 1
Worker
Workgr
/
e/H， H- '1l勺，。
"W1
广r一γ~\之)2
107
旷日 ， O， H
W2
HIπ
Chance
Firms'
/
L 11 一 π
Worker
r
Worker
Wl -
e/ L, L … 'W1 , O
Wj
\<
2
e/L, O, L - W2
Figure 341.1 An outlíne of an extensíve game that models a worker that signals her abílìty to a pair of
fírms by acquíríng educatíon. The diagram shows only ol1 cof the pu弘 ìbk education 1尔'els (' the workel
may obtain. The firms' wagξoffers ar它 made simultaneously. The worker's payoffs are lìsted first , the
firms' second and third伞
The worker chooses the amount e of education to obtain，注len 设le firms , observing
e, simultaneously offer wages wl and w2 , and finally the worker chooses one of
the wage offers. 人ssume that education is lessεxpensive to obtain for 在 v\'orker
of high ability than it is for a worker of low ability. Specifically, assume that the
cost to a worker of ability K of obtaining e units of education is e/ K, so that the
payoff of such a worker who obtains e units of education and takes a job paying
w is W … e/ K. The payoff of a firm that pays a worker of ability K the wage J:V is
K W. The extensive game that corresponds to this model is given in Figure 341.1.
1 claim that the gamεhas a weak sequential equilibri时n in which a higl:νability
worker chooses a positive amount of education. Consider the following assessment, in which e~ is a positive number.
Worker's strategy Type H chooses e = e* and type L chooses e = 0; after observ
ing the firms' wage offers , both types choose the highest offer if th咛 diff时，
and that of firm 1 if they are the same.
Firms' beHef Each firm believes that a worker is type H if she chooses e* and
type L otherwise.
Firms' strategies Each fìrm offers the wage H to a work时 who chooses e* and
the wage L to a worker who chooses any other value of e.
U飞e firms' beHefs are consistent with the worker' s strategy. (No worker chooses
an education level different from e* and 0, so the consistency co时ition imposes no
restriction on the firms' beliefs after observing such a level.)
Chapter 10. Extensive Games with Imperf，担ct Information
342
1 now fir飞d conditions on the parameters und衍 whìch the players' strategies are
sequentially ra tiona l.
Worker τhe 飞Norker's stratεgy of accepti口g the highest wage offer at the end of
tl飞e game is clearly optima l. Nm气r considεr the worker's initial action.
Type H A type H workεr obtains the payoff H - e* / H if she follows her
strategy and chooses the education level ♂， and the payoff L - e/ H if
she chooses any other education level e. The education level 0 achieves
the highest payoff, of L, for a deviar让， so for equilibrium we need H
♂ /H ~三 L， or
♂三 H(H - L).
Type L A type L worker obtair飞s the payoff L if she follows her strategy and
chooses the education 1εvel O. If she chooses anv education level other
than 俨 she obtains the same wa萨， and pays a co仗， so such a deviation
is not pro缸able. If she chooses the education level e* , then the firms
and she obtains the payoff H … t茹苦/L. Thus
believe her ability to be
for equilibrium we need
r 三 L(H … L)
Fí rms Each firm's payoff is 0, given its belief and its strategy. If it raisεs the wage
it offers in response to any value of e, its expected profit is n号gative， given its
belief, and if it 10认rers the wage its expected profit remains zero (its offer is
not acceptεd)
In summary, thεassessment is a weak sequential equilibrium if and only if
L(H - L) 三 e 主 H( 扫一 L)
Á
1 have assumed that H > L, 50 the left如hand side of thís expressíon 1S less than the
righ仁hand side , and hence va1ues of e* satisfying the expression exist. That ÌS , for
any values of H and L, the gam校 has separating equìlibria in which high-ability
workers obtaín some eduαtiol1， whereas low-ability ones do not. Education has
no effect on the workers' productivity; a high-ability worker obtains it to avoid
being labeled low in ability by potential employers.
Li ke several of the signaling games we have studied, this game has also pooli只g
equilibria , i日 whìch both types of workεr obtain the same amount of education.
That is , the model Îs c0l-1sistent both with a steady state in which only high-abilîty
workεrs obtain education and wíth one in which all workers do so.
? EXE仪CISE 3位 .1 (Pooling e气uilibria of game in which education signals ability)
Find the range of education levels e for which the gamεhas a weak sequential
equilibrium in which both types of worker choose the education level e. Compare
these leveìs with those possible in a separating equilibrium.
10.8 lII ustration: strat晤gic information transmission
343
10.8 lII ustration: strategic information transmission
You research the market for new products and submit a report to your boss , who
decides which product to develop. Your preferences differ from those of yo出
boss: you are interested in promoting the interests of your dìvision, whεreas she is
interested in promoting the interests of the whole firm. What do you tell her?
If you report the results of your research without distortion , the product she
will choose 飞机II not be best for you. Car飞 you do better by distorting or obscuring
the fruits of your rεsearch? If you systematically distort your fjnding , then your boss
wiU be able to unravel your 陀port and deduce your actua1 fir飞dings， so obfuscation
seems a morεpromising route.
To study the issues precisely, consid时 the following mode l. A sender (you) observes the state t, a number from 0 to 1 (the result of your researd才， that a rεceiver
(your boss) cannot see. Assume that the dístributíon of the state is uniform: for any
number z from 0 to 1, the probability that t is at most z is z. 丁he sender submits
arεport r , a number, to the receiver, who observes the report and takes an action
1/, a1so a number. 80th the sendeτand receiver care about the relation between the
state t and the rεceiver's action y, and neither is affected direc t1y by the valu号 of
the sender's report r. Spεcifically， assume 也at
Sender's payoff function:
一 (y … (t 十材产
Receiver's payoff function:
一 (y … tf ，
where b (the sender's 写ias") is a fixed posîtive number that reflects the divergence
betw优11 the sender's and receiv啊 's preferences.τhese functions are shown in
Figure 343. 1. Note that a reεeiver who believes that the state is t optimally chooses
y 之二 t， whereas the best actio l1 for the sender in this casεis Y = t 十主. The game is
illustrated in Figure 344. 1.
10.8.1
Perfect information transmission?
Consider the possibility of an 叫ui1ibrium in which the sender accurately reports
the st搬 she obsεrves-that her strategy is r( 叶江 t for a11 t. Given this strategy,
the εonsistency condition re气uires that the receiver believe (correct1y) that the state
ís t when the sender reports t, and hence for any report t optima11y chooses thε
t +b
y
•
Sender:
(y 一 (t 十的 )2
\
Figure 343.1 The players' payoff functions w怡 n the state is t in the game of strategic inform拉∞
transmission 刀1e number b is a positive constant
Chapter 10. Extensive Games with Imperfect Information
344
Chance 十
5ender
/吁/← (y …(t + b) 户，一 (y
t)2
(y 一 (t' + b) 户，… (y
t')2
Receiver'
r
斗----ç-
Figure 344.1 An outline of an extensi\ 巳 ganw thùt models strategìc ìnformiltion transmission. Only
two oÎ the possible states (t在 nd 1') and 011<' possible report 01 th巴比 nder arεsho \Vn. The 三ender's payoff
ìs listed Îìrst, the r己ceì飞'ετ'5 second
action t (th巳 maximizer of 一号… f )2). Is the sender's strategy a be5t response to
this strategy of the receiver? l\' ot if b > α5uppose the state is t. If the sender
f , the receiver chooses y = t , so that the sender's payoff is - (t - (t 十民 )2
. If instead she reports t 十 b， the receiver believes the state is t 十 b and chooses
y 二 f 十 b， 50 that the sender's payoff is 一 (t 十 b 一村十 b) )2 口 O. If b > 0 the
sender is thus better off reporting t + b when the state is t. 50 unless the sender's
and receiver's preferences are identical , the game has no eguilibríum in which the
sender accurately reports the state.
A similar argument shov气's that for b > 0 the game has no eguilibrium in which
the sender's strategy is any increasing function or any dεcreasing function. If the
sender's strategy is such a function r and r (t)
t' , thεconsistency condition re唰
quires that upon observing the report f' , the r巳ceìver b巳lìeve th在 t the state is 1, so
that the recei飞rer optimally takes the action t. As before , a sender 飞叶飞o observes t is
thus better off reporting t 十扛
10. 8.2
No ínformatíon transmíssio门?
l\'ow consider the possibility of a 到 equilibrium in which the send仔 's report is
constant, independent of the 飞lalue of the variable she observes-say r (t)口 c for
all t. Such a report conveys no inform é! tìon; the consistency condition requires
that if the receiver observes the report 汇I her belief must remain the same as it
vvas initial以 namely that the state is uniforrnly distributed from 0 to 1. Given
出is belieC her optimal action is Y
(This 巾 im should be plausible, giγen the
syrnrnetry of the belief and the shape of the receiver's payoff function. If you know
calculus, you rnay verify the claim precisely.)
TI飞窍 consistencv COI飞dition does not constrain the receiver's belief about the
弓 tate upon her receiving a report diff出巳1t from c, becaus已 slIch 1 report does 110t
i.
,
10.8 lII ustration: strategic information transmission
345
occur if the sender follows her strategy. Assume that the receìver completely ignores the sender's report: her belief remaìns the same as it was origina11y for every
value of the repor t. Then the receiver's optimal action is y = ~ whatever report
shεreceives. That is, the sender's report has no eff舍ct on the action chosen by
the receiver, so that the strat吨y r for which r( 斗士 c for a11 t is optimal for the
sendε主，
In summar予 for every value of b thεgame has a weak sequential equilibrium
in which the sender's report conveys no information (it is consta时， independent of
her type叶) and the 拮
r ecε
创i忖
ve
缸r i楞伊lores the 陀
repor时t (she maintains 趴e
盯r 讪
i1口1让
川iti
悦ial
叫1 beli讼
ε f about
tl盯tat怆旧，捏
reg
吨
伊a盯rdle
时硝
ssof 巾
t he 即O
仪悦创戎吟
t斗)and 瞅e创s 仙
t h飞e action
叼y=!. If b 川nalt 阳
叫e叩吨吧伊飞
ul
归1她川n is not ve眩ryat饺tra
阳C巾巳If b <
the川川ome st时es there is an action
differer川om ~ that is better for bo出 the sender and the reαiver， and the smaller
is b, the larger the set of states for which such increases in payoffs are possible. (lf
b=
for example, then for a町 t wìth 0 三 t < , b。如}时ender and the receìver
a的e阳 off if the receive内 a伽n is t 十 b than if it is ~.)
!'
t
!'
10.8.3
Somε ìnformatìon transmission?
Does the game have equìlibria in which some information is transmittεd? Suppose
that the sendεr makes one of two reports, depending on the state. Specifically,
suppose that if 0 :=; t < 杠 she reports rl and if t} ::; t ::; 1 she reports 1'2 1= rl'
Consider the recei飞'er's optímal response to this strategy. If sh仓 sees the report
1'1, she 挝10WS that the state t satisfies 0 三 t < t}; given her initial belief th挝 the
state is uniformly distributed from 0 to t the consistency condition requires that
she now believe that the state is uniforrruy distributed from 0 to t 1 . Her optimal
a伽吨iven this belief is y
~ t}. (As before , this claim should be pla 四ble gi飞ren
the symmetry and shape of the rεcei飞rer's payoff function; it may be verified using
calculus.) Similarly, if the receiver sees the report r2 she believes that the state
is uniformly dist出毗d from tl to 1, and chooses the action y = ~(tl 十 1 ).丁he
consistency condition does not restrict the recei飞rer's belief if she sεes a report other
than rl or 1"2. Assume that for each such report the receiver's belief is one of the n^l o
beHefs she holds if she sees 1'1 or 巧， so that her optimal response to every report is
either ~ /1 or ~ (t l 十 1).
Now, for εqui1ibrium we need the sender's report rl to be optimal if 0 ::; t <
tl and her report r2 to bεoptimal if t 1 <二< 1, given the receiver's strategy.
By changing her repo吮 the sender can change thε 陀ceiver's action from ~t} to
io1 十 l盯)， 仪
0 r叭刊飞
说f注icαe
V
忧们飞ver
挝归
挝佼较材f 认
a
w恼 O 三 t 三 t ρ
什
s1
1，，
~(♀机
t句1 十 1川)， and f，始
Or the report 1'2 to be optimal ìn every state t for which tl 三 f 三 1
shem瞅 i挂ik
挝e tl盯ctior口飞 ~(tl 十 1) 挝
a t le俯
ω挡
a5挝ta槌S 狡
m
I工
i如
1只飞 st恼
at怡
e h the s仪
endεer must be eχactl悖
y ìndifferent be
位twε
优
er丑1 the 肌7巧
o actior邸
lS
革， as in
Fìgure 346.1. This indifference implies that fJ叶 ìs rnidway itl and i(h +
h 叫 =iijt1 十七 tl + 1)] , so that
tIz
i … 2去
n or
(345.1)
346
Chapter 10. Extensi飞fe Games with Imperfl就t Information
o
~ t1
~(t 1+1)
t1 十 b
t1
y…
1
Figure 346.1 A value of t] such that in state t] the sender is indifferent between the receiver's actions
~ 1) and ! (t] + 1). The black curve is the sende内 payoff function in st在 te t]_ The g叫 curve on the left
is the sender's payo仔 function in a state less than t]; sh号 p时ers ~tl to! (t J 十1). Th巴 g叭.飞cu川 O时he
吨h
沁t 沁
is 臼
th
挝
es优
en
时
dεr's pa
盯
yoff
仔
ff
缸
un
川
1
We 附d t 1
0, 盼
So 出
tha
剧
a
b 三 iL
扫，由
then
啊n tl叫ame川has no 叫叩
山
u1且i毗
f陀
eports， dependin飞g on 役
t hεstate. Put differently, there is no point i Ii the recεiver's
asking the sender to submit a report if her prefereηces diverge sufficient1 y from
those of thεsender-she should simply takεthe best action for herself given her
prior bεlief.
I claim that (345.1) is not only necεssary for an equilibrìum, but also sufficien仁
That is, if t1 satisfies (345.1) , then in every state t with 0 :S; t <乙 t1 the sender
optimally rεports r1 and in every state t with t1 <二 t 三 1 she optimally reports
r2 乒1'1. This optimality follows from tl冶 shapes of the payoff functions; it is
illustrated in Figure 346.1 , in which the gray curves are the payoff functions of
senders in states less than t1 (left) and gre仗时 than t1 (right).
qu
时川心
ω1州
1
In conclusion, if b < , then the game h命a削s a w巳 ak se吨
wh嗡飞tich the se
创
en飞de
盯r trar叭1冶础$创m
丑旧
lits two 耐
d if任fe
挝古陀
创饥川
e
n1让tr陀
epo
叹rt怡s， dep岔悦1引1ding on 仙
the st怡
ati怆εε:
位 for 0 三三;
t < t1 she submits one report and the receiver takes the action ~九 and for 乌兰
t :S; 1 she submits a differe时 report and the recεiver takes the action (t1 十 1)
This equílibrium is better for both the receiver and the 使nd盯(before she observes the stâte) than the one Ìr飞 which no information is transmitted. Consider the
recεiver. (An analogous argument may be made for the sen世r.) In the equilibrium ín wl怕我o in
时
1叫forma
拭甜榈创阳阳
i阳
O川 t红汩栩主扭
rar
衍I川
1
白阳飞冶&挝t山
d
he时rpa咛
yoff
的
f打in each飞 st怡a挝t怡e t 必
iS 一
一 t片) 2 I如
nt出
hε e叫
仙
m
华
q
山1刀凶
u
ilib
缸剖
如
b
bri
川
m
泣
ium
10 认
wl飞让ich the sender tranSIηI口
n让
iits 扒A矿TO dif任ferent reports deper口飞dÌI贝19 on tl且1e state, her
payoffis --(itl - t)2 forO 三 t < t1 and
(t l +1) …斗 2 for t1 三 t < 1
i
i
(i
(i
( EXERCISE 346.1 (Comparing the receiver's expected payoff in two 叫uilibria) Plot
the receiv时 's payoff as a function of t (from 0 to 1) in each equilibrium. Given
that the distribution of the st在 te is uniforr口， the receiver's expected payo丘 in an
吨uilibrium is the negative of the area between the horizontal 拟is and the function that gives the payoff in that state. Show that the receiver's payoff is greater
in the two-report equilibrium than it is in the equilibrìum with no information
transmíssion.
10怠 lII
ustration: s在rategic information transmi莘莘ion
10.8.4
How much Înformation translγlÍssion?
347
!'
For b <
doεstr淀 game have 吨uilibria in wl临 more information is transmitted
than in the two-report equi1ibrium? Consider the possibility of an equilibri泣min
which the sender r烈的ces one of K rεports， depending on the state. Specificall予
suppose that the sender's report is 1'1 if 0 <二 t
句 I 1'2 if t1 三 t < t2 , ..., rK if
tK-l ::::: t 三 1 ， where rì 并 rj vvhenever i 乒 j. For convenience, Iet to
0 and
t K = 1.
The analysis follows the lines of that for the two-report equilibrium. If the
receiver observes the rεport rkr then the consistency condition requires that she
believe the state to 1:沱江niformly distributed from tk-1 to tkt so that she optimaIl y
takes the action i (t川十 fι If she observes a report dìfferent from any rkt the
consìstency conditìon does not restrict her belief; assume that her belief in such a
case is the belief she holds upon receiving one of the reports rk'
Now, for equi 1ibrium we need the send啊 's report I' k to be optimal when the
state is t with tk-1 ::::: t < 凡， for k = 1 ，… F 瓦 As befo陀， a sufficient condition for
optìmality is that in each state tkl k = 1" . "K , the sender be indifferent between
the reports 1'k and fk +1' a时 thus bet阴阳 the receiver's actions i (tk一 1 十 fk) and
i(九十 tk十1)' This indifference i叫1ies that tk 十 b 川qual to the ave吨e of i (tk-l 十
tk ) and i (tk 十 tk+ d:
fk+ b ::::
i [i (tk-l 十 tk) 讨(川
which Ìs equivalent to
tk+l …乌二 tk … tk-1
+ 4告.
That is , the interval of states for which the sender's report is 1}刊 is longer by 4b
than the interval for w hich the report isη. 在le length of 在e first interval , from 0
to 句， is t1 , and the sum of the lengths of all the intervals must be 1, so we r飞机d
t1 个 (t 1
+ 4的十·
个 (tl+(K … 1) 的口 1
or
Kh 令是b(l
+ 2 十.. .十 (K-1)) 口1.
Usi吨 tl时act that the sum of the 在时 n positive i附gεrs Ìs in(n + 斗， thee吨uation
lS
Ktl 十 2bK( 主… 1) = 1.
(3是7.1)
If b is small enough that 2bK 仅 -l)<l ， th盯e is a positive value of t 1 that satisfies
the equation
Suppose , for example, that 去三 b < 古Then the ìηequalìty ìs satisfied for
K <二 3， 50 that in the equilibrìum in which the most information Ìs transmitted
~ - 4b and
the sender cho05es one of three reports. From (347.n we have t1
h ke t 2 :
…ψ i坠
n 也斟is 叫u
凶ilibri
毗汩忱
bri
刘邸
'i
削
m
拟
a
nd
吐
dt扫2 (φi
妇)a缸E窍
re 吕
ili芷i耐rat剧
ed in 盹
Fi站
gu
臼r蛇
股
e3列4锦8川
肌1， 肌
and 加
th
始ee吨
伊u
q
副
d
凶
科i垃i丑泊阳刷
ib
brit
血阳
ium
日ill1 的。叼 taken by the
receive毛 as a functior飞 of the state t , i5 shown in Figure 348 .2. (1 have chosen 在i5
i
348
Chapter 10. Extensive Games with Imperfect Information
哼/伽
3
寸
1
系，.，
6φ'
，ι
,,,
.• 1
，，aE毛、
一 (y -
』J
-A-q
,-1
1A-qJ&
G
)
!(t2 十 1)
t2
1
y•
十 b) )2
/
\
Figure 348.1 The \'alues of 11 and 12 in a three-report equilibrium in the model of strategic informatìon
transmission for b 去 , 彷th出 哇们mostì趴吆
n1泛必
f吆o
(览
)r11
妇
indωìfω
阳r陀
mεR
挝tb
快
εt队
w啊川he 犯
re州优耐1附 r'、sa缸
础
M
ctí阳
1011αI
邸飞5 ~Il and ~(II + 12) , and in 归 te 12 she ìs indifferent between
the receive内在ctions !(tl 十川 and i (t l + 1),
value of b beεause it allows the diagrarn to clearly show the equilibrìurn , Note tnat
if b were a只y srnaller, a four-report equilibr妇m would exist) The values of the
reports tk do not rnatter, as long as no two are the sarne We rnay think of them as
'vords in a language; the recei飞T町 's long 以perìence playing the 伊 me teaches hεr
that rk means 飞he state is between tk 1 and lk'"
ln sumrnary, if there is a positive value of t1 that satìsfies (34ï .1), then the game
has a weak sequential 叫uilibrium 如 which the sender submits one of K different
reports , dependìng on thεstate. For any giveηvalue of b, the largest value of K
for whìch an equílibriurn exists is the largest value for which 2bK(K -1) < 1. If
2bK(K 一1) 1, then , usì吨 the quadratic formula , we have K 斗。+Jf工芳百)
Thus , in particula巳 the larger th巳 value of b, the smaller the largest value of K possible in an equilib抗议m. That is , the great时 the difference between the sender's and
recei飞F肘 's preferences, the coarser the information transmitted in the 叫uilibriurn
,
,
•
y=t 十岛
y
y 玄之 t
~ (t2 十 1)
1(t l 十 (2)
itl …--1
。
t2
f 一→
1
Figure 348.2 Tη11冶er陀εcei\飞ve臼r's action y, as 昌 f沁unctíor只1 of 仙
t he st汹
at伦
e 1, in a 秘
thr
纪
'ee
←叫r仪
epor
泛t eq江山ilíbri口 m
η ofthe
伊ameof 处
g
s tra
时f伦e
吨
均
gi比
C 口
时
m
1f伦
orm昌 阳阳
tjω
阳
on tral
of b τηh冶εda拙sh怡ed 荔ray lines show the action optímal in each state for the sender (top) and 陀ceíver
(bottom)
10.8 lII ustration: strategic inform泌io靠在 transmìssion
34争
with the largest number of steps-the "most informatìve" equilibrium. 1 claim
that the 吨uilibrium th挝 yields the receiver the highest payoff is the most ìnformative one, and this equilibrium is also best for the sendεr before she observes the
state. You should find this claim plausible, though some calculations are needed
to verify ìt.
10.8.5 Summary on strategic ínformation transmíssion
专Vhen a receiv时 takes an action based on an unverífiable report about the state
supplìed by a sender, and the sender's and 陀ceiver's preferencεs differ, no equilibrium exists in which the sender accurately reports the state. Regardless of the
difference between the sender's and receiver!s preferences, an equilibrium exists
in which the sender's report is independent of the state, so that no information
is transmitted. If the sender's ar飞d receiver's preferences are sufficiently simila巳
there are also equilibria i民 which the sender makes one of several r!巳ports， depeI飞d­
ing on the state; the rnaximal number of reports possible in an equilibrium is larger
the more similar the parties' preferεnces. The equilibrium with the maximal numω
ber of reports is better for both parties (before the state is known) than all other
equilibria.
10.8.6
Delegation
Is there a better way for the receiver to make a decision? Consider the alternative
of delegation: the receiver lets the sender choose the action. No reports, no obfuscation; the sender simply chooses the best action for herself-in each state t she
chooses the action t 十 b.
Compare this outcome with the outCOII飞.e of the best equilibri出nforthe recεiver
if she acts on the basis of a report by the sender. For b = 击， the outcome, as a
function of the state, for the three-report equilibrium (the best e伊边brium for this
value of 的 is given in Fígure 348.2, where the dashed li只台 y=t 十 b indic在 tes the
outcome of delegation.
Which outcome does the receiver prefer? Under delegation, the distance between the outcome and the receiver's favorite action is exactly b in every state.
Thus the receiver's payoff in every state is -b2 . In the three-report equi1ibrium of
the game in which the receiver solicits iruormation from the sende乙 the outcome
for most states is further than b from the receiver's favorite action; on1y 扣芷 states
within b of ~h ， !(tl 十 t2) ， or ~也十 1) is it clos臼 to the receive内 favorite action
that is the outcome of delegation. If that does not convince you that the receiver
prefers the outcome of delegation to the outcome of the three-report equilibrium,
take a look at Figure 350. 1.在加 figure plots the receiver' s payoffs in the two cases,
for each state. The horizontal line labeled
, very closεto the axis, is the re嚼
的ver' s payoff in each state under delegation.τhesca丑。pedc旧ve is her payoff in
each state in the three-report equilibrium. The state is distributed uniformly from
oto 1, so the receive玄 's expected payoff under delegation îs the negatîve of the area
Chapter 10. 在.xtensive Games with Imperfect Information
350
喝一…… r1 一…叶4
r2
, ..
r3
figure 350.1 Tl1e receiv缸 's payoff, as a function of ihe state t, in a thr佼佼port equílìbrium of the game
of information transmission 时 under delegation for b 古生 The negative of the shaded area is 他
receiver's expeζted 归 yoff in the three蜘report equilibriur口， and the negative of the area between the
horizontalline … b2 and the horÎZontal axis is her expected payoff under delegation.
of the very slim rectangle between the axis and the line …肘， while hεr expected
payoff in the thr时刊port equílibrium is the negative of the shaded area above the
scalloped lìne. Which is larger? You do not need to make an exact calculation to
be sure that the receiver prefers the outcome under delegation!
1 have analyzed 0ηly a si鸣le value of b, namely 生 For smaller values of b,
the loss from both delegation and the most informative equilibrium of the game
of information transmission is smaller, but delegation remains superior. For larger
values of b, the loss from both outcomes is larger. Delegation remains better than
the most informative equilibrium in the game of ìnformation transmission as long
asb < 占 /6 (approximately0 .29). (βCalcuαωt
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the sender sends the same report in every
state. Thus if the sender's prεferences are c1 0se enough to the rεceiver's that the
receiver benefits from consulting the sender, she is better off simply delegating the
decisìon rather than soliciting a report. T如e directive "do what you want" yields a
better outcome for the receiver than does "tell mεwhat I should do" because the
quality of the information induced by the latter is so low.
ø EXERCISE 350.1 (Variant of model with piecewisεlinear payoff functions) Con叩
sider the variant of thεgame of strategic information transmissìon in whïch the
sender's payoff function is -Iy … (t 十 b) I and the recei陀r's payoff fu阳tion is
'可以… tl (where I 工 I denotes the absolutεvalue of x). That is , in each s始 te t each
payoff function has the shape shown in Figure 71.1 , with a peak at t 十 b for the
receiver ar飞d a peak at t for thεsender. How do tl飞e k-report weak sequential
吨uilibria of this variar飞t of the model differ from those of the original model?
? EXERCIS 巨 350 .2 (Pooling equilìbrium in a general model) Consider a generaliza-
tion of the game of strategic informatiOI飞往ansmission (Figure 344.1) in which the
distribution of states is arbìtrary and each player's payoff is an arbitrary function
of 歹 and t (but, as before , is independent of r). Assume that there exists an ac挝on， say y* , that maximizes the receiver's expected payoff given the distribution
of states. Show that any such game has a weak sequential equilib
10.9 lII ustration: agenda (ontrol with imperfed information
10.9
351
lII ustratiOI言: agenda control with imperfect information
The U.s. House of Reprεsentatives assigr飞s to committees the task of formulating
modifications of the law. The bills proposed by committees are considered by the
legislature under one of several rules , rar飞ging from the "dosed rule" (or 飞ag
rule飞 under which the legislature may either accept or rεject the proposed bill
but may 口。t amend it, to the "open rule" , under which any amendment may be
made. (在le Rules Comtrjttee determines how each bill is handled.) In general , the
preferences of a committee differ from those of the entire legislature. What is the
rationale for restricting the actions thεlegisla ture ma y take?
If we treat the legislature and the committee as single players with well-defined
pref何ences， and aS!;iU me that the legislature is perfectly informed about theεnvi­
ronmen t, an ar飞alysis of the open rule is straightforward: the legislaturεsimply
chooses the b ì11 that is best according to its prεferences， ignoring the committ优 's
proposa 1. A model of the closed rule is studied in Secti∞ 6. 1. 3. In ger飞eral， the
equilíbrium outcome in this case is not the legislature's favorite bill.τhus under
perfect information the legislature has no reason to adopt the closed rule.
But of course if the legislature is perfectly ínformed , it has no reason to assi罗1
the drafting of laws to a committee to begin wíth. If it is not perfectly informed ,
then a committee has a role: it can discover the 飞tate of the wor1 d" and propose
legislation to fi t. In this environment , a committee whose preferences differ from
those of the legislature has an incentìve to report distorted information, and the
rules u口der which proposed legislation is considerεd may affect the degree of distortion. Are there circumstances lmder which the closed rule produces an outcomε
悦tter for the legislature than the open rule?
Consider the following model, which follows closely the one in the previous
section (10.8). The desirability of a bill deper飞ds on the statε t， a r飞umber from 0 to
1, which the commítt时， but not the legislature , observes. Assume that thεdistri­
bution of the statεís uniform: for any number z from 0 to 1, the probabilíty that
t :S z is z. After observing the state, the committee recommends a bill r (a number)
to the legislature. Under the open rule, the legislature may then choose any bill it
wishes. Unde主 the closed rule , it is restricted to eith仔 accept r or reject 泣， in which
case the outcome i
10.9.1
Open ru/e
Under the open rule, the model is identical to the one in the previous section (10.韵，
with the committee as sender and the legislature as receiver. Thus we 如lOW that if
b > 0, then in an equilibrium 位le committ啊's rεport obscures its information, the
Chapter 10. Extensive Game军警ithlmperf岳ct Information
352
more so the larger b. 科eals。如10W the character of thεbest (and most inform挝ive)
叩ìlibrium for the legislat附 In this equilibrìum for b = 去， for example, the
cornmìttee sends one of three different reports, as illustrated ìn Figures 348.1 and
348.2
‘
10.9.2
Cfosed rule
Under the dosed rule, the legìslature retaìns the power to veto, but not otherwise
to amend, the legisiation proposed by the committee, in which case the outcome is
the fixed status quo Yo. From Section 10.8.6 we know that if the legislature sirnply
delegates the choice of legislation to the ωmmittee， the outcomεis better for both
the legislature and the commíttee than the outcome of the best equilibríum under
the open rule, assumìng that the legislature's and committee's pr.εferences do not
díverge so much that the legislature would be b悦ter off making the decision itself.
丁he closed rule ìs dose to delegation, but the legislaturεretains the right to veto
the proposed legislature ìn favor of thεstatus 气认o. Do船航e legislature benefit
from retaìníng this vestige of control?
τhe game for thεclosed rule is illustrated in Figure 353. 1. τhe legislature optimally exerCÎses its option to choose yo if ít prefers this outcome to the committee's
proposa l. In particular, if thεcommittee proposes its favorite bi1l y 二 t + b for a11
values of t (as it doεs under delegation) , then the legislature optimally exercìses its
option whenever yo … b:二 t 三 yo + b, or equivalently whenever t … b:二 YO ~二 t +b.
(Refer to Figure 353.2.) Given this response of the legislature, the committee's proposal of t 十 b is not optimal for t with yo < t < yo 十以 the proposal t + b leads to
the outcome yo , and hence the payoff 一 (Yo … (t 十的 )2 <…肘， but if the committee
recommends the bill t 十 2b， then the legislature accepts 议， givìng the committee
-b2 . Thus in no equilib对um does the
the higher payoff - (t + 2b 一 (t+ 的户
committee propose its favorite bill in a11 states.
If the status quo yo satisfies b < Yo < 1 - 31乙 however， 1 claim that the game
has an equilìbrium in which the committee recommends its favorite bill ìn a11 states
except those between yo … b and Yo 十 b， and between yo + b and 如今给. The bill
proposed by the committee (and accepted by the legislature) in this equ注ibrium ìs
shown in Fìgure 354. 1. It is defined prεcisely as fo11ows:
t + fJ
ift <二 yo - b
íf yo - b < t ~二 yo
Yo
yo + 2b if yo < t ::二 yo + 2b
yo 十 4b íf yo + 2b < t ~二 yo 十 3b
t 卡b
if yo 十 3b < t.
For any recommendatíon of less than yo or greater than yo + 4b the legislature can
ínfer the state precisely. 丁hus the consistency conditíon re弓uires that the legislature's belie[ about the state be correct if the committee makes such a r，εcommen­
dation. For the recommendations yo , yo 十沽， and yo + 4b , the legislature cannot
ìnfer the state precisely. The committee makes each of these r.εcon飞mendations for
10.9 fII ustration: agenda ωntrol with imperfect information
r
Committee
353
一 (r 一 (t 十 b)f， 一 (r -
tf
r
:如\一 (yo … (t 十的 )2 ， -(yo ~-
Chance
tf
Legislature:
了/一。一 (t' 十 b)f， 一 (r -. t')2
Committee
r
Yo \ … (Yo … (t' 十的 )2，一 (yo - t' 严
Figure 353.1 An outline of a扪 extensíve game that models the dosed rule for the consìderation of a
committee proposal bya legìslature. 0ηIy t lVO of the possible states (t and 1') and on l' of the possíble
τeports the committee can make are shown.τhe committee's payoffs are listed first , the legislature's
second
y•
Committee:
一 (y - (t 十 b))2
事 igure 353.2 For yo betwe巳 nt 一岛在 nd 1 十 b， the legislat始 re prefers yo to t +扎在nd thus optimally exercises its option to choose y号 rather th在n a propos击 1 of 1 + b. For yo between t …占在nd t, it ís indifferent
between Yo and 21 … yo
an interval of states, so that given 出at the initial distribution of states is uniform ,
the consistency condition requires that the legislature's belief about the state after
observing one of these recommendations be uniform on the interval of sta校s that
generates the recommendation. In s田nmar予 the consistency condition requi古es
that the legislature's belief satisfy the following conditions.
State is r - b
State is uniformly distributed from yo - b to yo
State is uniformly distributed from yo t。如十 2b
State is uniformly distributed from yo 个岛 to yo 个 4b
State is r … b
ìf r < yo
if r = yo
if r = yo 十 2b
if r 工 yo 十 4b
ìf yo 十 4b < r.
The consistency condition does not restrìct the legislature's belief about the statε
when it observes a recommendation between yo and yo 十岛， or one between yo 个
2b andyo 十钻， because the committ优 does not make such a recommendation if it
follows its stra枪gy. Assume that in thesεcases it believes the state is yo. Fiηally，
Chapter 1O. Extensive G i'l1T将s with Imperfect Information
至54
1 十 b
T
y 口 t 十飞 r
y
Y
r一一-....J
YO 十 3b
Y♀十 2b
YO
占
O
t•
叫…但一'仰----
1
The bi泣山l日ly pr陀opose时d by 仙
t he committee and accept沁♀d byt协;乞挝
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W
气γr the aεction optimal in each state for the committee (top) and legíslature (b ottom)
白ig
F
伊
ur
，陀
'e3
到
54.1
st城 e t ,
the legislature's strategy is to accept a11 recommendations except those between yo
andyo + 2告， and between yo 十 2b andyo 十 4b (which are not made if the committee
adheres to its equilibrium strategy).
1 no\" argue that this assessment is a weak sequential equílíbríum. The legislature's belief satisfies the consistency condition by construction. Consider the
optimality of the comn飞ittee's strategy, given the legislature's strategy. In a11 states
less th栩如一 b or greater than yo 十弛， the outcome is the committee's favorite bill ,
so certainly no strategy yields it a higher payoff in these states. Now consider the
state 驹， in wruch its 由'ategy calls for it to reωmmend the bill yo , so that it obtains
the payoff ~b2 ， If it changes ìts recommendatioI飞 the bm passed by the legislature
either remains yo (i f it recommends between yo and yo 十 2告。rb仗weenyo 十 2band
YO 十始)， changes to less than yo (if it recommends less than yo) , or char飞ges to at
least yo 十 2b (if it recommends yo 十 2b or at least yo 十钻). None of these changεs
increases the committee's payoff (refer to Figure 355. 巧， 50 its recommendation of
yo is optima l. ln states between Yo ~ b and yo , and between yo and Yo 十 3b， símilar
analyses show that the 陀εommendation YO is a150 optimaL (Note that the committee is indifferent between the bills yo and yo 十 2b in state yo , and between the bills
yo 十 2b ar飞dyo 十 4b in state Yo 十 2扛)
Finally consider the legislature's action. A recommendation r from 出e committee of less than yo or greater than yo + 枯 rev臼Is the state to be r - b, 50 that the
legìslature prefer5 r to yo. Now consider the other possible recomrr飞毛ndationsthe
commìttee may make.
Recommendation of Yo: The outcome is the 5ame, yo， 陀gardless of the legisla啕
ture' s action
10.9 lII ustration: agenda control with imp带rfect information
YO 十岛
YO
355
YO 十 2b
y•
一 (Y 一 (YO 十 b) j2
Figure 355.1 In state Yo , the committee prefers the bill yo to aηybillleωthan y在 orgrξ在 ter than yo + 2b.
yo+2b
YO
\
\
飞、
刊叫，
(Yo … t)2
、、
气d
puu e
t 一→
一 (YO 十 2b - t)2
\
\
飞
飞
Figure 355.2τhe legislature's payoff as a functíon of the state from yo t。如十 2b if it accepts the
recomrnended bill yo 十 2b (solìd black curve) and r哇jects ít (dashed black curve)
Recommenda tion of 如今 2告: The legisl挝ure believes that the state is uniformly
distributed frorr飞 YO to Yo + 2b. lts payoft as a function of the 段时es in this
interval, is given by the solid black curve in Figure 355.2 if it accepts the committee's recommendation and by the dashed black curve if it rejects the committee' s recommendation. These curves are mirror images of each othe乙 so
the legislature is indifferent between accepting and rejecting the committee's
recommendatìon.
Recommendation of YO 十传: The legislature believes that the state is uniformly
distributed from Yo + 2b to yo + 3b. Its payoff, as a function of the states
in this interva ì, is gìven by the solid black curve in Figure 356.1 if it accepts the committee's recommendati∞ and by the dashed black curve if it
rejects the committee's recommendation. (For c1arity, the vertical scale in
this figurεis different from that in Fi gure 355.2.) 丁he solid curve lies entirely
abo飞re the dashed one, so the legislature prefers to accept the committee's
reωmmendation than to reject it.
Recommendation betwεenyo and yo 十 2b， or between YO 十 2b and YO 十 4b: The
legislature believes that thεstate is 驹， so it optimally rejects the committee's
recommendation.
10.9.3 Comparison of open ru屹 closed rule, and dεlegation
τhe legislature's action, as a function of the state, in tl飞e best equ:ilibrium for the leg-
islatu挖出der the open rule ror b = 生， is given 部 Figure 348.2. Figure 354.1 is the
corresponding figure for the equ过ibrium fOUlîd in the previous section (10旦与 for
the closed rule. Whi ch equilibrium does the legisla始re prefer? Comp盯ingthetwo
356
Chapter 10. Extensive Games with Imperfect Information
YO
YO 十 b
YO + 2b
yo 十 3各
yo 十 4b
1•
~pt
-(YO-t)2
气、、 ~eject
lYo 十是b 一 t)2
Figure 356.1 The legislature's payoff as a function of the state from Yo + 2告 to yo 十 3b if it accepts the
recommended bill yo 十 4b (soHd black c往只吧) and rejects it (dashed black CU f\吩.
figures , it should be plausiblεthat it prefers the equilibrium for the closed rule. ln
this equilibrium, the difference bεtween the outcomεand the legislatu肥 's favorite
bill is b for most states and is never more than 2b , whereas for the 叫uilibrium 旺1der the open rule the outcome 认呗nders much further afield. The legislature really
dislikes large deviations from its favorite bill (its payoff function is quadratic) , so
the 吨uìlibriur丑 undertl稳 closed rule is better for it thar飞 thee弓uilibrium under the
open rule. (Note that 1 have not argued that the equilibrium found fo川he closed
rulεyields the best outcome for the legislature , only that it is better than the best
叫u i1 ibrium under the open rul叫
For other 飞'alues of b, a comparison of the equilibria under the two rules leads
to the same conclusion: if b < yo < 1 … 3b and b is small enough that there is
an equilíbrium under the open rule in which the committee submits at least two
different reports, depending 011 the state , then there is an equilibrium 旧lder the
closεd rule that the legislature prefers to the best equilibrium for it under the open
rule.
50 in a comparison of the open rule and the closed rule , the analysis provides a
rationale for the closed rule: in limiting the legislature' s ability to take ad vantage of
the ìnformation reported by thεcommittee， the rule gives the committee an incentive to divulge that information , to the ultimate adv在ntage of both the ∞mmittee
and the legislature.
Now ωmpar较 the outcomes under the closed rule and under deJegation. Look
carefully at Figure 354. 1. In the 叫uilibrium depicted there for the closed rule, the
bill proposed by the committee differs from t 令各 in the region of yo. The devia伽
tions are symmetric about t 十 b ， as Figure 357.1 makes clear. Now, the 1吨islature's
payoff decreases at an increasing rate away from its favorite bill-its payoff to a
bill2 r5 from ìts favorite Ìs more than twice as large a negative nm到快r as its payoff
to a bill r5 from its favorite.η1US it prefers the outcome in which the bill is t 十 b
in every state f to the outcome in the equilibrium we have been studying. That is,
for this value of b, delegation is better for it than thεequilibrium we found for the
closed rule. lf it del吨ates the decisiOl飞 to a committee, tl冶 legislature loses control
of the outcome; if it retains some control it loses information, bεcause the committee' s interest leads to a distorted communication of the state. In the model we
have be
No吉普$
357
2b
i
l
l
b
O
------...斗
YO
…
b
YO
入二
如今 2b
YO + 3b t •
Figure 357.1γhe deviation of the 如ìIl propo古ed by the committee from the legisliltur哇's favorite b!ll in
an equilibríum in the model of agenda control with the c1 0sed ruJ告， as a function of the state arOl浪 d the
state yo.
τhus while the closed rule generates an equilibrium better for the legislature
than any 吨uilibrium under the open ruk delegation generates an even better
outcome. Assuming the game for the closed rule has no better e气uilibrium ihan
the one we studied, the conclusion is that if thεpreferences of the legislature and
the committee do not differ so much that the legîslature should simply choose the
legislation itself, then the legislature's best option is to relinquish a11 control over
the outcome and to cede the choice of legislation to the committee.
Notes
The notion of an extensive game is due to von Neumann and Morgenstern (1944).
Ku h..T1 (1950b , 1953) suggested the formulation described ìn 出 is chapter. The notion of sequential equilibrium, of whìch weak sequentìal equilibrium is a varia1泣， is due to Kreps and Wìlson (1982); Selten's (1975) notion of "perfect equilibrìum" is closely related. (The notion of weak sequentìal equìlìbrium is called
"assε部mente气uilibrium" by Binmore (1992); it is som巳times called "weak perfect
Bayesian 叫uilibrium" ， though the notion of perfect Bayesian equilibrium (due to
Fudenberg and Tirole 1991) is defined only for a restricted set of games.)
The models ofpoker in Example 315.1 and Exercise 316.1 arevariants ofmodels
studied by Borel (1938, 91-97) and von Neumann and Morgenstern (1944 , 19.14).
Von Neumann studi付 several other models of the game (see von Neumann and
Morgenstern 1944, footnote 2 on page 186 and Section 19) , as did some of the othεr
pioneers of game theory, ìncluding Bellman and Blackwell (1949) , Kuhn (1950a) ,
Nash and Shapley (195t时， and Karlin (1959b, Chaptεr 9).
τhe idea of the model in Section 10.6 is due to Nelson (1970 , 1974). The model
itself is a simplified version, suggested by Ariel Rubirωtein， of the one of Milgrom
and Roberts (1986). Sectìon 10 .7 is based on Spence (1 974)，因如terpreted by Cho
and Kreps (1987). Section 10.8 is based on Crawford and Sobel (1982). The model
in Sectìon 10.9 is due to Gilligan and Krehbiel (1987); the equilibrium under the
dosed rule is taken from Krishna and Morgan (2001). Banks (1991) presents many
other applicatio只s of signaling games in political science.
358
Chapter 10. Extensive Games with Imperfect Information
Example 320 .2 is based on Bagwell (1995). The game in Exercise 33 1. 1 is taken
from Selten (1975) and that in Exercise 331.2 is taken from Kohlberg and Mertens
(1986). The game in Exercise 335.2 comes fromMaynard Smith (1 991). In the Battle
of Zutphen in 1586, the your飞g British aristocrat and poet Sir Philip Sydney, having
beer飞 seriously wounded ，陀portedly gave a water bottle to a dying soldier with
the words "Thy necessity is yet greater than mine" (see Gre飞rille 1986, 77 and 215).
Maynard Smith writes that "This unusual example of altruism by a member of
the English upper classes" inspired him to formulate the game. Bergstrom and
Lachmann (1997) compare the pooling and separating equilibrìa of the
For
more discussi∞ of the game, see Godfray and Johnstone 在那时.
Variants and Extensions
11
12
13
14
15
16
5trictlyζompetitive Games and Maxminimization
Rationalizability
377
Evolutionary 主quilibrium
3臼
Repeated Games: 丁扫 e Prisoner 's Dilemma
Repeated Games: General Results 451
Bargaining 465
419
361
。
5trictly Competitive Games and
Maxminimization
11.1 Maxminimization 361
11.2 Maxminimizatioh ancì Nash equilibrium 364
11 .3 Strictlyξompetítive games 365
11.4 Maxminimization. and Nash equilibrium in strictly
competitive games 367
Prerequisite: Chaptèrs 2 and 4
HE NOTION of Nash 吨uilibrium 位rst studied in Chapter 2) models a steady
state篝It captures the idea that each player, through her experience playing the
game against v盯ious oppor虑到ts， forms a correct beHef about the actions of every
other player and choosεs her action in light of this belief.
In this chapter and the next , we study the players' behavior from a different
angl巳 We consíder the implications of each player's basing her beHef 尬。ut the
other players' actions on her introspective analysis of the game , rather than on her
expenenc已
In this chapter we focus 0口 two跚player "strictly compεtitive" games，如 which
the players' interests are diametrically opposed. In such games a simple decisionmaking procedure leads each player to choose a Nash 吨ui拉brium action.
11.1
Maxminimization
You are confronted with a game for the first time; you have no idea what actions
your opponents will take. How should you choose your action? Here is a very
conservative procedure: associate with each of your actions the worst outcorr飞e for
you , as the other players' actions vary, and then choose the action for which this
worst outcome ís best.在1Ìs procedure is knowr飞船 maxminimization
Consider, for example , a game in which player 1'8 payoffs are those given in
Figure 362. 1. (Player 2's payoffs are irrelevant to player l's choice of an action.)
Restrict attention to pure strategies (actions). 在le worst payoff the action T yields
is 0 (obtained when player 2 chooses R) and the worst payoff the action B yields
is 1 (obtained when player 2 chooses L). 百旧smaxrτúnimization among the set of
pure strategies leads player 1 to choose B.
Many of the interesting examples of maxrr1Ìnimízation involve mixed strate唰
gies , so from the beginning 1 define the concept for a strategic game wÍth vNM
preferences (Definition 106.1), though the ideas do not depend on the players' randO rr1Ì zing. Let Uí be a Bernoulli payoff function whose expected value represents
361
Chapter 11. Strictly Comp鲁吉itive Games and Maxminimization
362
player i's preferences regardíng lott衍íes over actíon profiles in a strategic game.
Denote player i's expected payoff under Ui to the mixed strategy profileαby 乌拉) .
For any given mixed strategy 纠 of player i, the lowest payoff that player i obtains ,
for any possible listα 叫"出e other players' mixed strategies, is
TmUI{筑川… i)
A maxminimizing mixed strategy for player i is a rnixed strategy that maximizes
this minimal payoff.
DE Fl NITION 362.1 (Maxminimizing mixed strateg1j in strategic game with vNM preferences) A maxminimizing mixed strategy for player i in a strategic game with
vNM preferences is a mixed strategy that súlves the problem
maxminUi 仪 ir Lt -it
ι
IJ:← 2
where Ui 识) is player i's expected payoff to tl飞e strategy profile 注:、If αi is a maxminimizing mixed strategy for player i, then minlLi Ui(叫 r 夜 i) is her maxminimized
payoff.
Note that, equiv咳 lently号在 ; is a maxmirùmizer if and OI由 if min a. i Ui (可 ， CK-i) =
max在181inι-1udal ，在一 í)
1n words, a maxminimìzing strategy for player i maximizes her expected payoff
under the (very pessimistic) assumpti∞ that whatever she does the oth仔 players
will act in a w町 that minimizes her expected paγoff.
A different way of looking at a maxminimizing strategy is usefu l. Say that a
mixed strategy 伐:í of player i guaranteεs player i the payoff ll if, when playeτì uses
the strategy 夜 i ， her expected payoff is at least lli regardless of the other players'
mixed strategies:
,
U í (泛川一 í) 主 Uí for every list CK … í of the other players' mixed s位ategies.
ln the game in Figure 362 .1, foτexample ， player 1's strategy T guarantees any
p在yoff of at most 0 and her strategy B guarar飞tees any payoff of at most 1.
A player's m拟minimizing mixed strat巳gy guara只tees her maxminimized pay梢
。ff.F盯ther， no strategy guarantees her a higher payoff. That is , if CKi is a maxmir卜
imizer of player i, then
于1飞 Uí ( α7 ，必 j) 主 I:iinlil(应 j ， fX _j) for every rr以ed strategy {X j of player i.
VSL巧311
RO2
TAnnH
Figure 362.1 Pl<lyerγs payoffs in a strategic game The action of player 1 that mòximìzes her worst
‘
payoff，在5 player 2'5 action 飞!aries， is B
11.1 Maxminimiz矗立ion
363
① Ex 乞汉CI臼 363.1 (Maxminimizers in a bargaining game) Find each play衍 's maxminimizing strategies, restricting attention to pure strategies (actions) , of the game
in Exercise 38.2.
A player may be able to guarantee a higher payoff by using a mixed strategy
than by using only pure strategies, as the follo飞Nìng example shows. The example
also illustrates a method of finding a maxminimizing mixed strategy
EXAMPLE3毛3.2 (Example of maxminimizers) Consider the game in Figure 363.1. If
player 1 chooses 丁， theI飞 the worse that can happen fo l' her is that player 2 chooses
j之; if player 1 chooses B, then the worst that can happen is that play呛 r 2 chooses L.
1n both cases player l's payoff is … 1, so that if player 1 is restricted to choose either
T or B, thea she can guara Iltee no morc than -1.
DHU''
i1
丁-A
呛ZA
τi
7 』咆 i
气f
』
ab''
誓
TB
Figure 363.1 丁hegam程加 Example 363.2
Player 1 can do better, however, if she randomizes between T and B. If, for
example, she 巾oses each action with probability ~，阳1 her expected payoff is ~
均layer 2 伽oses L and 0 if play时 2 chooses R, and thus li的etwee川 and ~ if
player 2 randomizes be t\.v een L and R. Hence by choosing each of her actions with
rr川e 阳
probability ~， player 1 g阳a附es tl即ayoff of 0, 盯
chooses either T or B.
Can she do better? Let p be the probability she assigns to T. The upwardsloping lìne in Figure 364.1 indicates her expected payoft as p varies, if pl哼但 2
chooses the actiOI飞 L; the downward-sloping line indicates her expected payoff, as
p varies, if player 2 chooses R. Her expected payoff if player 2 randomizes between
L and R lies between the two lines, and in particular lies above the lower lìne. Thus
for each value of p, thεlower of the 护wo 1ines indicates player 1's lowest payoft
as the mixed strategy of player 2 varies，证 she chooses that value of p. η1冶at 怡
i s， the
n筑句2 叫
U 1 (α
问lt α
lX 2) 扣
for忖
each 刊
v alue of p (which de
时te
凹
err
口m
挝削lÌ挝
i泣n妇飞淀陌
岱5 皮叫})i沁s ìnd
ε
出1C旧
at饺巳
ed by
value of mi如
句
tl的 lack ir附戊归
edV 丁h
挝
l们
1沱
e如ma
r 趴川
M
时11
W
认
如
咄
hich叮叮yYields her a payoff of ~
φEXERCISE 363.3 (Finding a maxminimizer) Find the m以mìnimizing mixed strat呵
egy of player 2 in the game in Figure 363. 1.
τhe maxminimizìng mixed strategy of player 1 in this example has the property
that it yields player 1 the same payoff whether player 2 chooses L or R. Note
that this indìfference is different from the ÌI飞difference in a mixed strategy 时ash
equilibriurn，也:1 which player l's mixed s红ategy yields player 2 the sam己 payo丘 to
each of her actions.
Chapter 11. 5trictly Competitive Games 喜nd Maxminimization
364
?
2
Payoff of'
player 1 :
L
R
E
1
O
-1~
-1
Figure 364.1 The expected payoff of play仅 1 in the game in Figure 363. i forωch of player 2's actions,
as a function of the probability p that player 1 assigns to T.
11.2
Maxminimization and Nash equilibrium
A simple argurr飞ent establishes that a player's payoff in a mixed strategy Nash
equilìbrium is at least her maxminimized payoff.
Fix a strategic game , and suppose that 旷 is a mixed strategy Nash 吨uilibrium
of the game. Consider player i. By de自nition of equilibrium , her expected payoff
when she chooses 可 is at least her expected payo旺 when she chooses any other
strategy, given that the remaining players choose their equilibrium strategies:
Ui( 叫，之 Uí ( 在川 =-i) 如r every mixed strategy ai of player i
No\句f， for any mixed strategy αi of player Î, we have
Ui ( α 川:1) 二> mi只 U/(皮 i ，í'cd ，
ll: ~i
so that
Ui (町， α-J 三 TirlUz(α Î， ILi) for ev台ry mixed strategy 在 i of player i
Given that this inequality holds for any mixed strategy t>.: i, it holds in particul盯 for
the strategy that maximizes the rìght-hand side, so that
Ui(α i '
>rr飞axminUi( c.: i ，cX
{t i
Cl.: _i
That is, we have the following resul t.
-LEMM 人 364.1 The expected pay唠 of each player in al1 y mixed strategy Nash equílíbriwl1
of a strategic game 必 at least equal to her m阳minimized payoff .
?二 EXERCISE 364.2 (t也sh e吨uìlibrium payoffs and maxminimized payoffs) Gi飞!e an
example of a game with a uniqu号 mixed strategy l'、Jash 吨uilibrium in which each
player'sεquilibrium payoff exceeds her maxminimized payoff. (There are exam卢
ples with two players, each of whom has t认70 actions.)
11.3 Strictly competitive games
365
L
R
T
6, 0
0, 6
B
3, 2
6，。
Figure 365.1 The strategíc game in Exercise 365.1
? EXERCIS 巨 365.1 (Nash equilibrium payoffs and maxminimized payoffs) Consider
the game in Figure 365. 1. Find player 1's maxminimizing strategy and her Nash
equilibrium strategy. Compare her payoff, as a function of player 2's strategy,
w hen she uses her 指出h equilibrium strategy and when she uses her maxminimizing strategy.
11.3
Strictly competitive games
In many games a player has no good reaso日 to believe that the other players wìll
take actions that minimize her payoff. 日ut in a two-player game in which the
players' interests are diametrically opposed , the assumption seems reasonable: in
maxim泣如g her own payoff in such a game, a player autom挝ically minimizes her
opponen t' s payoff. In this sectiOI飞 1 define such games precisely. In the next section
1 argue that 如 such games the outcome that occurs ìf each player maxminimizes is
very closely related to the Nash 吨曰librium outcome.
The players' interests in a twφplayer game are diametrically opposed if when确
ever one player prefers some outcome a to another outcome b, the other player
prefers b to a. We call such a game strictly competitive. If we restrict attention
to pure strategies, then we have the following definitio口， in which 1 assume for
conver飞ience that the players' names are "1" and "2气
DEFINITION 365.2β trictly competitive strategic game with ordinal pr学renc时 A
strategic game with ordinal preferences is strictly competitive ìf ìt has tγvoplayers
and
Ul(al ， a2) 与 Ul (b 1 ， 的) if and or句 if u2(b 1, b2) ::: u2(al , a2) ,
where (肉，向) and (b 1, b2) are pairs of actions and for i 口 1 ， 2， Uj is a payoff tunction
that represents player i's preferences.
Note that an implicatíon of this definition is that in a strictly competitive game
ul(al ， a注
Ul( 屿，告2) if and 0ηlyìfu2(al ， {/2)
= u2(b 1 ， b纣
(becauseU2(al' 的)出的 (b 1t 的 ) implies both 问(町，时泛的(屿，马) and Ul (抖 ， b2) 三
Ul ((/1 , {/2)) , and
Ul(肉 ， a2) > Ul(抖 ， b2 ) ifandonlyifu2(bl , b2) > U2(al , a2)
Notεalso that ìf Ul is a payoff ft皿ction that represents player 1's preferences in
a strictly competìtìve game, then thεpayoff function -Ul represents player 2's
366
Chapter 11. Strictly Competitive Games and Maxminimization
L R
T 2, 1 0, 5
B
1, 3
5, 0
Figure3 晶6.1 A strategíc game that is strictly competitive if attentío口 is restrícted to pure
not jf mìxed strategies are considered
bu t
preferences. That is, in any strictly competitive game there exist payoff func附
加ns Ul and U2 for the players such that 叫 a1 ， a2) 十向阳 ， a2) 口 o for every
严打{叭，何) of actìons. For this reason a strictly competitive game is sometimes
called a zerosum game.
τhe Prisoner's Dilemma (Figure 15.1) is not strictly compεtitive becàuse both
play时s prefer (Quiet， ♀ uiet) to (Fink, Fí nk). 130S (Figure 19.1) is not strictly CQmpetiti飞'e eith屹 because (for exa黯 pl 咛 both playε rs prefer ( 窍 , B) to (S , B). Match
汕
i打1抒的?
pεmη1狞1i始
t臼s (σFi培
gur陀
ε 1污
9.2) ， on the 仗
0 the
臼rh
沁飞冶
and
式， Iβs 处
s t红括玄污i仗
c tl守
yc
∞
or黠苦口lp
严
et挝it挝iv
陀
缸: pla
e
咛
严
y
仔r 1γJ、sp
e
严refe
盯f卜
臼nc
e
扭
es ov
兀.ve
吧
仔r the 扣
e
f ou
盯r outcomes are 严
p Y陀
ecis挝
el拧
yt白
h伫 T蛇伫verse of pl可er 2's. τhe game in
Figure 366.1 is also strictly competitive: player 1 prefers (在 R) to (了， L) to (B , L) to
(汇 R) ， whereas player 2's prefere配esn肌 in the opposite directíon.
Jf we consider mixed strategies , the following definition is appropriate.
峭
DEFIN lTl ON 366.1 (Strictly competitive strategzc game with vNM pr价rences) A strat，ε­
gic game with vNM preferences is strictly competitive if it has two players and
U1(且 1 ， α2) 三口1 (卢 1 ，卢2) if and only if U2( 卢 1 ， ß2) ;::日2(在 1 ，时 J
where (问， α2) and (卢 1 ，卢 2) are pairs of míxed strategies and for i = 1, 2, U j is an
expected payoff function that represents player i's preferences over lotteries.
As for games with ordinal preferences, there exist payoff functions representing
the players' preferences in a strictly competitive game with vNM preferences with
the property that the sum of the players' payoffs is zero for every action profile: if
Uj is a Bernoulli payoff function whose expected value represents player l's preferences regarding lotteries over outcomes, then -ul is a Bernoulli payoff function
whose expected value rep陀何时s play时 2's preferences.
Any game that is strictly competitive when we allow mixed strategies is clearly
strictly competitive when we restrÌct attention to pure strategies , but note that the
coηverse is fals己. Consid心r， for example, the gamεin Figure 366.1 , now interpreting
the numbers in the boxes as Bernoulli payoffs. ln this game player 1 is indifferent
behveen the outcome (1', L) and t检 lot始ry in which 泞，盯 occurs with probabiliiy ~ aI飞d (B , R) occurs with probability ~ (b ecause ~ .0+ ~ .5 = 2) , b的 player 2 is
not indifferent betwεen these two 0时comes (her payoff to (T， 王) is 1, whereâs her
expec时 payoff to the lottery is ~ . 5 十 20 工 3)
( EXERCISE 366.2 (Determining strÍct competiti\咄less) Is either the g在me in
cise 坦5 .1 or the game in Figure 367.1 strictly competitive 伊) if we restríct atte时ion
to pure strategies and (的 if we allow mixed strategies?
11 .4 Maxminimization and r也 sh equilibrium in strictlyξompetitive games
367
L
R
T 1, -1 3,…5
B 2,… 3 1, -1
Figure 367.1 One of the g在 mes in Exercise 366.2伞
11.4
Maxminimization and Nash equilibrium in strictly competitive gamω
We have seen that in any game a pl句时 's Nash 叫 uilibri出n payoff is at least her
maxminimized payoff (Lemma 364.1). 1 烈。w show 位at in a strict1y competitive
game that possess臼 a mixed strategy Nash equilibrium, the two payoffs are the
same. In fact ,
Ìl l a strictly competitive game that possesses a mixed strategy Nash equilib-
rium , a pair of mixed strategies 必 a mixed strategy Nash equilibrium if and
only if each player's strategy is a maxminimizer.
τhough the proof of this result may look complicated , the ideas ít ent在 ils are
飞Tery simple; the arguments involve no more than the manipulation of in吨ualitíes.
1 first establish a preliminary resul t. Oenote player l's expected payoff function
by U1 and let U2 = Ul (so tl飞at 口2 is an expected payoff function for player 2).
Now, foraηy function f , the minimum of f is equal to the negative of the maximum
of -f: minxf(x) = --maxx(-f(x)) (s硝 Figure 367.2).τh出 because U2 = -U1
飞与 e have
max min U2( tX 11 划 =max […n:ax U 1 (α1 ，问) 1
仪
~1
u2
飞饥
or
3sghw
守4
皮
x -4
aI
m2ma
α
"2
mι
maxminU2 (町 ， lt2)
I
(367.1)
"1
The fo l1 owing proposition is a precise version of the statement at the start of
the sεction.
f(x)
a
。
一。
工一→
气
一f(x)
Figure 367.2 The minimum of any function f is equal to the negative of the maximum of - f.
Chapter 11. Strictly Competitive Games and Maxminimization
368
• PROPOSITION 368.1 (Nash equilibrium strategies and maxminimizers of strictly
competitive games) Consider a strictly competitive strategic game with vNM pr，棋子
。lces. Let U1 be an ε勾Jected pay非functiol1 that represen公 player 1's pri收rences， and let
U2
…山 (50 that U2 represents player 2's prl咐'eηces).
a. 扩坷，咛)必 amαed strategy Nash equilibrium , 仇
t h川
r
player 1 ， α;i必s a maxminimÎzerfor player 2, and
maxmin U1 (α1 ，的) = minmax U1(11: 1 ，的) = U1 ( 叫，叫) .
a2
0: 2
b 扩 max的 mìn在2 口1 (α1 ，的
"1
min的 maxaJ U 1 (屿，我2) (a condition s拍手ied， in
particuia几扩 the game has 必 mixed strategy Nash equilibrium (sεε part a))， α;IS
a maxminímízer for player 1, and 咛 is a maxmlmmÎzer户r player 2, then ( 叫 ，11: 2 ) is
a mixed s灯'ategy Nash e号 uilíbri 11m.
Proof 1 first prove part a. Let (叫 ， tt 2) be a mixed strategy 1也sh equilibrium. By
the definitíon of a Nash equilìbríum wεhave
U2 ( lt l ，a:2)主 U2 ( 叫，在2) for 台verym旧d strategy 的 of player 2
or, gìven U2
…
U 1,
口1 (叫，叫) ，二 U 1 叫 ， (时
That is ,
U1 (叫 F
min U1(叫，夜才.
(368.2)
饨2
Now, defìne the function g by
夜Jl工lI!in U1(町，夜才 for all α1.
U2
n
Then the right-hand side of (368.2) is g( l1: , which is at most max"l g( 11:1) (by
definition of the maxim山
U 1 (tt; ， 在 2) 二 maxmin U1(在 1 ， 1t 2)
t\ jα2
Now, from Lemma 364 .1 we have the opposite inequality: a player's Nash
equilibrium payoff is at least her maxmínimized payoff. Combining the two inequalities we have U] (吨，在 :D
max 0:1 min lÀ 2 U1(lt l , tt2) , and he时e， usi鸣 (368. 2) ，
mm向民(叫，的)口 max"1 111111句 U 1 (的， lt2) ， so that 叫 ìs a maxminimizer for
player 1
人n analogous argument establishes that
is a maxminimizer for player 2 and
U2 例，可
max a2 min a1 山队 ， tt2)' From (367.1) we deduαthat U1(叫 F 吨) =
minα2 m 拟在 U 1 叭，且:2) ， completi吨 the proof of part a
To prove part b, let α咛l' bea ma阳T饥旧巾1Ì巾
ir阳
l
for player 2, and assume that max a1 mìn a2 江1 阳，叫
mÌn勺 maxaj U 1 (哟，的) .
Define
二 maxmin U 1 ( 向 F 的
minmax U1(肉，的)
在
"2
"2
0: 1
11.4 Maxminimization and Nash equilibrium in strictly competitive games
369
Frorn (367.1) we have rnax a2 rninltJ U2 (屿，的)叶飞 Because 泛咛;i必sa 泣肌rnη1Ìni
rni垃
ze
臼r for play盯 1w
附
eha
盯
Vε U l 仪(泛咛lXi ， α
向2) 兰 v. 扣
forev纪衍ry mixed 仗
s tra
州t怡
吨
e
gy 配向2 of pla
巧
严泛衍r 2,
y
andb
怡eca
趴u
山
阳1S妮
e α;引
a: Ìs a rna
以
xrn
口ünirni泣
屁
z
衍rforp
e
抖la)咄 2weh飞 ave U2( α
问1 ， 戍吨2) ~三 -v. for every
α; 如 these two ineq叫 1mixed strategy ltl of player 1. Letti吨句:在 2 and 在
ties we obtain U1(α; ，在 2) 主 v. and U2(咛， α2) 三. Given l..乌兰 U], the latter
meqωlity is U 1 伊 l'α2) 三 v飞 so that we have U1(ai , aí)
v*. Thus
口1 (叫，时兰民间， α2) for every rnixed strategy lt2 of player 2,
。乙 using U 2
一口1 again,
U2(α; ，向) s;日2(咛 ， aì) for every rnixed strategy lt2 of playεr 2,
so thatα2 is a best 陀sponse úf player 2 to 咛 Sirnìlarl予
口2(夜 1 ，乓)之 U2 (叫 ，1X 2) for every rnixed strategyα1 of player 1,
or
口1( 问， 4) 三 U 1(lt i , lt 2) for every rnixed strategy 问 of player 1,
so that IX~ is a best response of player 1 to α;.
￥Ve conclude that (呵，吨) ìs a Nash 叫uilibriurn of the garne
。
This result is of interest not only beωuse ìt shows the close relation between
Nash 叫uilibriurn strategies and rnaxminirnizers in strictly cornpetitive garnes, but
a150 because it reveals properties of the equilibria of such garnes.
First, part a of ihe result in飞plies that the rnixed strategy Nash equilibriurn
payoff of 部ch player in a strictly cornpetitive garne is unique.
部 COROLLARY 369.1 Every mixed s打ategy Nash 叫 uili切'íum
01 a stríctly competitive game
yie/ds the same pair 01 expected payo.庐，
As 飞lVe have seeI飞， in garnes that are not strìctly cornpetitive, not a11 Nash equi-
!ibria neεessarily yield the sarne pair of payoffs (consider BoS (Figure 19.1), for
exarnple).
? EXERCISE 369 .2 (Equilibriurn payoffs in syrnrnetric game) 5how that in any syrn协
rnetric (see Definition 129.1) strictly cornpεtitive garne in which U2 -U1, where
Ui is player j's expected payoff function for i
rnixed strategy Nash equilibriurn is Q.
1, 2, each player's payoff in every
Seco时， s叩 posεthat (al , lt2) and 例，马) a阳丑ixed strategy Nash equilibria of
a strictly cornpetitive garn巳 Then by part a of Proposition 368.1 the strategiesα1
and α; 附 rnaxrninimizers for player 1, and the strategiesα2 and 夜; are rnaxrnin幽
irni之ers for play町 2. But then by p盯t b of thεm由 both (肉，马) and (件的) a陀
rnixed strategy Nash equilibria of the garne. That is, we have the following resul t.
瓢 COROLLARY 369.3 1于le mixed strategy Nash equilibria
01 a stríctly C01ηpetitive game are
interchω1geable: 扩 (lt1 ， a2) and (叫， α~) are mixed strateg歹 Nash equilibria, then 50 are
(向1 ，玛 ) and (吟
Chapter 11. Strictly Competitive Games and M孟xminimization
370
The game BoS shows that the Nash equilibria of a game that is not strictly
competitive are not necessarily interchangeable.
Thìrd, d巳note player l's 吨uìlìbriull1 payoff in a strictly competìtive game by
Uí. Part a of Propositior飞 368.1 implies that any Nash equilibrium strategy of
player 1 guarantees that her payoff is at least U1, and any r、-Jash equilìbrìum strategy of player 2 g限制 tees that her payoff is at least -Uí- The 阳ond implication
means that any Nash equilibrium strategy of player 2 guarantees that player 1 、
payo任 ìs at most Ui- Th挝 is，附 have the following resul t.
• COROLLARY 370.1 Any 1气Jash equ i1i brium strategtj of player 1 in a strictly competitive
game guarantεes that her payoff is at least her equili告rium payoff, and any Nash 叫 ui即
libriU I11 strategy of player 2 guarante邵阳 t player 1's payoff is at most hεr equilibrium
payo.在
In a game that ìs not strictly competitìve the 叫uality in part a of Proposition 368 .1 does not generally hold, and a pl可衍 's equilìbrium strategy does not
ìn general have these propertìes, as the following exercise shows.
(J) EXERCISE 370.2 (Maxminìmizers ì到 BoS) For the game BoS (Figure 19 月， find the
maxrninimizer üf each player. For each ll1ixed strategy 出ash equilibriu ll1, show
that the strategy of neìther player guarantees her 叫uilíbrium payoff.
However, in any game (飞竹飞ether strictly compεtitìve or no叶， theine弓uality
maxmin U 1 ( 町，的)空 rninmax U1 (剖，问)
1% 1
a2
a2
在1
hoìds. ln 飞气!ords， the highest payoff that player 1 can guarantee herself is at most
the lowest value that player 2 can guarantee player 1's payoff does not exceed. The
argument is simplε: for a巧 mixed strategy 乓 of player 1 we have U1 (α;F 的)三
max"l U1 ( Cl: l ,1<: 2) for each strategy 的 of player 2, and Ì阳
111
泯捋i挝
闪夜们2 ma拟xaα问1 口叭l( 筑向l ， 统2
n
lt扑). This ineq础 lity holds for all 叫ues of 叫 , and thus in
particular holds for thεvalue that n飞axirnizes the left斗land side.
MAXMINI:-'1IZATION: SOME HISTORY
τh号 theory of rn拟rninirnization in general strictly competitive games was de皿
veloped by John von Neurnann in the late 1920s. The id出 of rnaxrninirnization
in the context of a specì在c garne , however, appeared two centuries earlier. In
1713 or 1714 Pierre Rémond de Montmort, a rn码。r figure in the history of probability theory (see, for exarnple, Hald 1990) publìshed Essay d'analysε sur les jeux
de hazard (Analytical essay on garnes of chanc时， in which he reports corresponden使 with Nikolaus Bernoulli (a mernber of the Swiss farnily of scientists and
mathernaticians). Montmort describes a letter dated r巾V巳mb时 13， 1713，在at he
received frorn "M. de Waldegrave" (probably Baron Waldegrave of Chewton, a
Britisl飞叭。ble born and educ没 ted in France, according to Kuhr飞 1968). Montmort ,
11.4 Maxminimization and r臼sh equilibrium in strictly comp骨titi￥告结 amω
371
Bernoulli, and 轩'aldegrave had been corresponding about the card game le Her
("the gentleman").
This two-player game usεs an ordinary deck of cards. Each play在捂住处 dealt
a single card , which she alone seεs. Each player's object is to hold a card whosε
value is higher than that of the card held by her opponent, with the ace counted
as 1 and the jack, queen, and king counted as 11 , 12, and 13, respectively. After
each player has received her card , player 1 can , if she wishes, exchange her card
with that of player 2, who must make the exchange unless she holds a king , in
which case she is automatically the winner. Then, regardless of whether player 1
exchanges her card , player 2 has the op 悦。n of exchanging hers for a card randomly
selected from the cards remaining in the deck. If the randomly selected card is
a king, she automatically loses; otherwise she makes the exchange. Finally, the
players compare their cards , and the or飞e whose card has the higher value wins; if
both cards have the same value, then player 2 wins.
1々e can model this situation as a strategic game in which an action for player 1
is a rule that says, for each possible card th批 she may receive, whεther sne keeps
or exchanges the card For example, one possible action is to εxchange ar飞y card
vdth value up to 5 and to keep any card with highεr value; another possible action is to exchange any even card and to keep any odd card. There are 13 different values of cards, so player 1 has 213 (8)92) actions. If player 1 exchanges
her card , then player 2 knows both cards being held , and she should clearly ex如
change with a random card from the deck if and only if the card she holds would
otherwise lose. If player 1 does not exchange her card, then player 2's decision
of whether to exchange is not as clear. Li ke an action of player 1 at the start of
the game , an action of player 2 is a rule that says, for each possible card that she
holds, whether she keeps or exchanges the card. Li ke player 1, player 2 has 213
actíons.
Montmort, Bernoulli, and Waldegrave argued that the only actions that could
possibly be optimal 盯衍 "exchange up to 6 and keep 7 and over" or 句xchange up to
7 and keep 8 and over" for player t and "exchaηge up to 7 and keep 8 ar飞d over" or
"exchange up to 8 and keep 9 and over" for player 2. \哨len the pl町ers are restricted
tousεon与 these actio邸， the game is eq也vale挝 to
‘
0, 0 5, -5
3，一 3
0,
号
The three scholars had corresponded about which action each player should
choose. As you can see, the best action for each player depends on t七e other
player's action, and t.l-te game has no pure strategy Nash 叫uilibrium. Waldegravemade the key conceptualleap of considering the possibility that the players randomize. He observed that player 1's mixed strategy (~! jD yields her the same
阳yoff 蛇gardless of player 2's a伽n and guaran阳 her a payoff of 号， a叫 fur­
the言~ that pl玛rer 2's mixed strategy 峰， ~) ensures that player l's payoff is no more
than
Ch挂 pter 11.
372
Strictly Competi锐￥eG矗 mes and Maxminimization
That is, Waldegrave found the maxminìmìzers for each player aJ飞d appreciated
their significance. Montmort wrote to Bemoulli that "it seems to me that [Waldegrave's letter] exhausts everything that one can say on [the players' behavior in le
Her]."
.t1~. 日 XE汉CISE 372.1 (Increasing payoffs and eliminating actions in strictly competitive
games) Let G be a strictly competitive game that has a Nash 吨uilibrium.
a.
Suppose that the strictly competitive game G1 differs from G only in that some
of player 1's payoffs in G1 are hìgher than the corresponding p玛roffs in G.
Show that U has no equilibrium in which player 1 is worse off than she is in
an equilibrium of
(Use part a of Proposition 368. 1.)
b. Suppose that the game G1 differs from G only in that one of player l's actions
in G is ab段时 from her set of actions in G1 . Show that G' has no equilìbrium
in which player l' s payoff is higher thar飞 it is 如 any 叫uilìbrium of
ε.
Give examples to show that neither of the above properties necessarily holds
如r a game that is not strictly competitive.
ø EXERCl SE 372.2 (Equilibrium in strictly competitive games) 日出er prove or glVe a
counterexample to the claim that if the equilibrium payoff of player 1 in a strictly
competitive gamεìs v, then any strategy pair that gives player 1 a pay(￥ff of v is an
equilibrìum.
:]去CISE 372.3 (Morra)
In the two-player game "Morra 气 the players simultaneω
ously hold up some fü飞gers and each guesses the total numbεr of fingers held up.
If exactly one player guesses correctly, then the other player pays her the amount
of her guess (in dollars , say) If either both players guess correctly or neither does
so, then no payments are made. (The game was played in ancìent 1之ome， where
Ít was known as "micatio" (Carcopìno 1940, 251~252).) Consider a version of the
game in which the number of fingers 制ch player may hold up is restricted to be
either onεor two.
a.
Model this sìtuation as a (strictly competitive) strategic game.
b. Given the symmetry of the 伊me， each player's equilibrium payoff is 0 by the
result in ExerCÌse 369.2. Find the mixed strategies of player 1 that guarantee
that her payoff is at least 0 (i.e. the strategies such that her payoff is at least
o for each pure strategy of player 匀f and hence 负nd a11 the mixed strategy
equilibria of the game.
( EXERCISE 372 .4 (O'Neill's 伊me) Consider the game in Figure 373. 1.
a. Find a mixed strategy l'也sh equilibrium in which each player assigns positive probability to each of her actions and assi萨s the same probab i1ity to the
actions 1 ，立， and 3.
11 .4 Maxminimization and Nash equilibrium in strktly competitive games
1
2
3
1
2
1
1, -1
1, -1
-1 , 1
1 ，一 1
一 1，
一 1，
1
1, -1
1, 1
3
1, -1
1, -1
1, 1
-1 , 1
373
1
-1 , 1
一1， 1
1, -1
一1，
Figure 373.1 The game in Exercise 372.4
b. Use Corollaries 369 .1 and 370.1 to show that the equilibrium you found in
part a is the on1y equilibrium of the game
EMPIRICAL TESTS: EXPERIMENT 轧 TENNIS ， AND SOCCER
The theory of maxminimization makes a sharp prediction about the p1ayers'
strategies in a strict1y competitive game. Does human behavior bear out this
prediction?
Experiments
To test whether experimental subjects choose maxminimizers (or Nash equilibrium
strategies) in a specific strict1y competitive game, we need to know their prefer
ences given the monetary payoffs with which we confront them. When working
with mixed strategies，口ot only do we have to be sure that the subjects care on1y
about their own payoffs (see the discussion on page 24) , but a1so we need , in genera l, to know their attitudes to risk (see Section 4. 1. 3). However, in games with
on1y tvvo outcomes , risk attitudes p1ay no r01e. If each p1ayer in such a game cares
only about the amount of money she receives, her preferences are represented
by the expected va1ue of the Bernoulli payoff function that assigns to each outcome this amount of money. This fact makes two-outcome games attractive for
expenments.
One experiment used the game in Exercise 372 .4 (O'Neill1987). Each p1ayer's
unique maxminimizing (and hence Nash equilibrium) mixed strategy in this game
is (0.2 , 0.2 , 0.2 , 0 .4) (with the actions ordered 1, 2, 3, J, as in Figure 373.1). 1n the
experiment, each of 25 pairs of peop1e p1ayed the game 105 times. Thi s design
raises two issues. First, when the same two p1ayers confront each other repeated1y, strategic possibilities absent from a one-shot game emerge: each p1ayer may
condition her current action on her opponen t's past actions. However, an ana1ysis that takes into account these strategic options 1eads to the conclusion that,
for the game used in the experiment, the p1ayers will eschew them. Second, because of the repeated p1ay in the experiment, each subject faced more than two
possib1e outcomes. However, under the hypothesis that each p1ayer's preferences
are separab1e between the different trials, each p1ayer's preferences in the whole
experiment may be represented by her expected total monetary payoff.
374
Chapter 11. Strictly Competitive Games 矗nd Maxminimization
Each subject was gîven $2 .5 0 in cash at the start of the game , was paid $0.05 for
every win, and paîd her opponent $0.05 for every 10ss. on avεrage， subjects in the
role of player 1 d附e the actions with probabìlìties (0.221 ， 0且 5， 0.203川0.362) and
subjects in the role of player 2 chosεthem with probabllities (0.226 , 0.179 , 0.1 69 ,
G役的. These observεd frequencies 附那 close to those predicted by the theory of
maxminimization. Are they close enough to be consistent with the theory? A staI子
dard statistical test (X 2 ) asks the 伊estion: If each player uses her maxminimizer,
what is the probability that the observed frequencies deviate at least as much from
the predicted ones? Applying this test to th巳 aggregate data on the frequencies of
the 16 possible outcomes of the game leads to the decisive rejectioηof the hypothesìs that the Sl均ects were using their m拟minimizing strategies. (The probabìlíty
of a devìation from the predìction at least as large as th拭 observed is
than 1
ìn 1,000.) However, this statistical test is severe: for a game in whìch a player has
a pure strategy maxminìmizer, for example, it would reject the hypothesis that the
player is usìng the maxminìmÌZer if there were a single ìnstance in which she used
another strategy. Ne飞rerthe1ess ， a variety of other tests suggest a1so that the obser飞，ations are 机。t closely consistent with the theory of maxminimizatìon (Brown
and Rosenthal199时， so that at best the experìment appears to offer only limìted
support for the theory.
Tennis al1 d soccer
One explanatìon for the relatively poor performance of the theory of maxminimìzation in the experiment using thεgame in Fìgure 373 .1 is that because the
su问 ects had ìnsufficient time to gain experience playìng the gam仑， their p1ay had
not reached a steady state. This exp1anation suggεsts that the behavior of professìona1 game players might yìe1d better 陀sults. Professional athletes , for example,
spend their lives gainìng experience and have huge monetary incentives to optimize. Most 兹在mes played professionally are too complex to test straightforwardly,
but parts of tennis and soccer have been successfully isolated for scrutin予
百1εserver in tennis would like to hit the ball as close as possible to onεof the
sides of tl飞e receiv啊 's courιThe receiver must decide whether to prepare for a
serve to the 1号ft or to the righ t. Assuming that the receiv舍r must make her decision
before the ball is served , we may model the play of a poìnt as a strategic game
in which the server's actìons are "play to the 1eft" and "play to the right" , and the
receiver's actions are "prepar.εfor a serve to the left" and "prepare for a serve to the
right气 The serve does not fully d的ermine the outcomεof a poi时， but SigI飞ificantly
influences it: the probabilìty of the server's ultimately winning the point depends
on the initial actions of the server and t1双陀ceiver.
The server's action on each point is easily observable. The receiver's action
and the probabìlìties that the server wins the point for each of the four action
pairs are , however, not observable, so we cannot directly test whether the server is
playing an 吨ui1ibrium strategy We can, howeveζcheck two implications of the
theory of maxmìnimization (and 到ash equilibrium): the equalíty of the server's
‘
撼。tes
375
payoff (probability of winning) to her two actions, and the statistical independence of her actions over a serìes of points. Walker and Wooders (2000) use data
from ten matches 如 major tournaments between top players who had played each
other many times They find that the probabìlities of the server's winning a point
played to the left and to the right are close enough to be consistent with their beir飞g
equat as the theory imp挂 es， but the server's choices frorr飞 point to point arεnot
statistically independent.丁hese results thus support the theory, but not without
qualification.
Penalty kicks in soccer matches provide another set of àata to test the theo巧巳 A
penalty kick may be modeled as a two-player strictly competitive game betw时n
a kicker and a goalkeeper. To have a good chance of scoring a goal , the kicker
must propel the ball to one of the two sides of the goal; to prevent a goal's being
scored, the goalkeeþer must move to the side the kicker chooses. Given the spεed
of the ball and human reaction times , the goalkeeper must act simultaneously wìth
the kicker; waiting to see at which side the kicker is aiming is not practica l. Both
players' actions-the side to which the baU is kicked and the direction in which
the goalkeeper jumps-are observable. 盯si吨 data on highly skilled players who
often faced each otheζPalacios-Huerta (2003) performs tests like those used on
the tennis data. He finds that the data are consistent, for almost a11 kickers, with
the probability of scoring a goal 'v\'hen the baH ìs kickeà to the left being equal
to the probabilìty of scoring a goal when the ball is kicked to the right, and , for
almost a11 goalkeepers, with the probability of pre\咽1ting a goal by jumping to the
left being equal to the probability of pre飞'entìng a goal by jumping to thεright.
Further, the players' actions from one penalty kick to the next are ∞nsistent with
their choices being statistically independent. τhus these data are consistent with
both implications of the theory.
ln conclusion, the theory of maxminimization appears to perforrr飞 well in situations in which players are experienced; it perfonτ1.S much less 斗vel1， as we would
expect, in games between inexperienced players.
‘
Notes
τhe main ideas and results in thís chaptεr are due to von Neumam1 (1928); the
theory was de飞reloped by von l'\eumalli飞 and Morgens挖出 (1944).
丁he material in the box on page 370 is based on Todhunter (1865) and Kuhn
(1968). Guílbaud (1 961) rediscovered Montmor t' s report of Waldegrave's work.
The box on page 373 is based on 0'1'运εill (1987), Brown and Rosenthal (199时，
Walker and Wooders (2000) , and Palacios-Huerta (2003).
The game in Exercise 365.1 is taken from Aumann (1989 , Example 2.21) , and
the one in ExerCÎse 372 .4 is taken from O' l'\eill (1 987).
飞
Rationalizability
12.1
12.2
Rationalizabìlity 377
Iterated élimination of strictly dominated actions 385
1 二3 Iterated elimination of weakly dominated aζtions 388
12.4 Dominance solvabîlity 391
Pre(equisite: ζhapters 2 and 4; Chapter 5 for Seζtion 12.3
HAT outωmes in a strategic game are consistent with the players' introspec刷
出'e analyses of each others' rational behavior?
The main solution notion
we have studied so far, Nash 叫uilibrium， is not designed to address this question;
rather, it models a st，εad)' state 出 which each playεr has learned the other players'
actions from hεr long experience playing the game. In this chapter 1 discuss an 呼"
proach to the question that considers rational players who attempt to deduce their
opponents' ratíonal actions from their opponents' prefereηces and from analyses
of their opponents' reasoning about their own rational actions.
12.1
民a主ionalizability
12. 1.1
Introductoryexamples
We say that a player's action is rational if it maximizes h衍 payoff， giveη some belief
about the other players' actions. (Note that we place no restriction on the play时 's
belief.)
In some games, the assumption of rationality sígnificantly restricts the players'
actíons. Consider, for example , the Prisoner's Dile111ma (Example 14.1). F(J r any
beHef about thε other player's action, Fùη1k yields a payoff higher tha1贝飞 does Quiet
(Fù仅1汝支 i玲s bet仗ter than QtμfÍ，岔
et
gω11η♂l'd
泛
only rat挝10r抗1al aζt注io
仪n for 础
e ach pl弘
句yε
a
衍f
汇位. (στf豆1让
挝is action 必
i s a1弘
仰
S
o of course the ofÙ y Na
部sh飞
叫uωi1胁
i岛
bríum action.)
In other games, the assumption of rationality ís less restrictíve. CO王飞sider the
varíant of the Prisoner's Dilemma in Figure 378.1. If player 1 believes that player 2
飞Nill choose Q, then she should choose Q, whereas if she believes that player 2
will choose F, then she should choose F. Thus both Q and F are rational for
player 1. Are the belìefs th拭 support these actions reasonable? Suppose we restrict playerγs belief to be consistent with player 2's rationali町，百1e ofÙ y ratioηal
action of playεr 2 is F (as in thε Prisonel乍 Dile疗1m啡， and player l's ofÙ y rational
action given the beHef that player 2 chooses F is F. 币lUS F is the or飞ly a~tion of
::1.
77
Chapter 12. Rationali芷矗 bility
378
吁'』
nHU 唯i
i--M
1
;FF
)'20
•,,
QF
31
Figure 378.1 人 variant of the Pr;sùl的 '5 Dilel111ì1β
pJayer 1 that is both ratior飞al and supported by a belief consistent with play时 2's
action being rationa l.
In yet other games , the outcome is not signìfìca时ly restricted by the assumptions that each player's action ìs rational and that her belief is consistent with her
opponents' ratior飞ality but is restricted by further steps in the players' reasoning
process. Consider the game in Fìgure 378.2.币le action ♀ of player 1 is rational
(supported by the belieÍ that player 2 chooses Q) and the action F is also ratíonal
(supported by the be!ief that player 2 chooses F). Further, both Q and F are ratìor飞al for player 2 (support泛d respectively by the beliefs that player 1 chooses X
and either Q or F). Thus player 1's assuming that player 2's action ís rational does
actions , so that piayerγs actions Q and F are both
not rule out either of player
consiste时 with her rationalìty and her assuming that player 2's action is rationa l.
Now for an additional step: suppose that player 1 assumes that player 2 assumes that playerγs action is rationa l. Player 2's assumption that 抖在yer 1's action is rationalleads her to conc1 ude that player 1 'will not choosεX (b ecause this
action is not a best response to a只y belief of player 1 about player 2). But then
playeτ2's rationality leads her to choose F, because Q is not a best response to any
belief of player 2 that assign主 probab过 ity zero to X. Fina 1l y, if player 1 ass江mes that
player 2 assumes that hεr action is ratíonal, she knows that player 2 will chooses
F， ωal叫 thus γ臼at阳
州
of actions ∞
c ons
飞陈si总st怡阳
刽1川
e
1甘t 认
w罗 i让th飞 (仙
i) each飞 P
抖la
盯y罗e
世r's assumηII凶
l只飞gt白
ha
放t the other playe
盯r assumes
t白
ha
挝t her 缸
a ctior口1 怡
i s ra
挝t往ion旧al ， (i í) each player's assuming that the other player's action
is rationa l, and (Îii) each player's action bεing rationa l.
For each of these three games 1 ha 飞!e argued that tl1εonly action for each player
consistεnt with the playε凹， rationality and up to two levels of introspective reasoning about each other's rationalìty is F. 1 further claim tha 1, in each game , the
action F for each player survives a 1l higher level reasoning of the same type.τ1at
is, for each player i in each gam 己的 there is a belief about the other player's action
(she w il\ choose F) under 飞vhich F is optima l, (íi) there is a belief about i's action
(sbe 飞i\' ill choose
under whích the action hypothesized for the other player in (i)
is optimat (iii) there is a belie
、
、1'7 』 1Anυ
;'r?,,,
,..
呵咆
JV
,
1JVAHV 叮4
斗A43A
a
QXF 012
Figure 378.2 A further vari牵引 t of the Príso l1 cr's Dìlem111日
12.1 究ationalizability
379
under which the action hypothesized for the other player in (i i) is optimal, and so
on indefinitely. At each stage , we can find a belief that justi白es the action assumed
in the previous stage. We say that an action that can be supported ín this way is
"rationalizable" .
I now formulate these ideas precisely and describe their implications. We sha11
see that in some games (like the three examples) , the players' introspective reaso仆
ing about each other's actions significantly restrìcts the outcom仑， while in others it
does 口ot.
12.1 .2
De自niîions
A "rational" player forms a probabilistic belief about the othεr players' actions and
chooses her actìon '(or mixed strategy) to maximize her expected payoff given this
belief. (1broughout this chapter I work wìth strategic games with vNM preferences.)
DE Fl NITION 379.1 (Player's beliefalld rationality in strategic game) A beHefof player i
about the other players' actions in a strategic game with vNM preferences is a prob
ability distribution over the set A 叫 of lists of the other players' actions. Player í's
mixed strategy (X i is rational if there exists a belief 11j of player i about the other
players' actions under whichαi maxìmizes player i's expected payo任
汇
f1ì(a-i)U i (α i， a-i)'
。… jEA 呼 j
where Ui( iX ú a-i) is player i's expected payoff when she uses the rnìxed strategy 夜i
and the other players' actìons are a… i. In this case we say that 向 is rational for the
belief ri μor a best response to f1 i'
(J) EXERCISE 379 .2 (Best responses to beliefs) Give an example of an action that is
a best response to a belief but is not a best response to any beHef that assi伊s
probabilìty 1 to a single action.
Note that thεdefinition a110ws a player to hold a beHef that Ìs not derived from
ìndependent mixed strategies of 自己 oth己r players: a beHef of player i may be any
pro忌abi1ity distrìbution over A-ù regardless of whether the probabilities this distribution assigns to lìsts of actions are products of the probabi1ities assìgned by
mixed strategies of the other playεrs. That is, the definition allows a player to believe that 出eothεr players' actions are "correlated". For example, in a three-player
gamεin which the actions of player 2 are A and B, and those of player 3 are X and
Y, player 1 may hold the beHef that the actions of players 2 and 3 will be 忡， X)
with probabili可 i and (B , Y) wi由 p协abili时 iF a p时ab让i时 distribution that is
not possìble if players 2 and 3 independently randornìze.
In the terminology of the definìtion, the notion of Nash 叫uílìbrium requires
not only that each player's strategy be rational but also that each player's beHef
be correct. Precisely, a mixed strategy prof过e (X* is a Nash equilibrìum if, for each
player i ,
ζhapt告 r 12.
380
Rationalizability
31
HFe
g*
:
Plaver l's actions
Pl在Vεr 2' s actions
Fig淤e 380.1 An ilJ ustratíon of t]-飞\e
盯
co州 ition tl硝 player 1γF、sa
趴巾
c:
叹附S叩
s归
po
on
时s淤
附e 如 the be !ief 111; the set of actions of player 2 ，tüW警
怡 h始 t怡
hi比5山b咒甘硝
elí
口i淤
们
e
f归击 55旦1鄂
♂I1S püs缸
g
Slt
共t川 Pω
协
ba必b
汹削
圳
1且l 川
i沁 X~.
·可 is rational for the belief 111
• for every list a… i of the other players' actions , the probabi 1i ty 圳 a_~I) that i's
belief 灼 assigns to a-i is equal to the product of 可(的) over all j 乒 Î， the
probability under 旷 that the list of the other players' actions is a-í.
The rεquirement of Nash equilibrium that each player's belief about the other
players be correct is not appealing for a player who confronts a game in which she
has little or no experience. 1n the previous section (12. 1.月 we considered the impli
cations in some examples of the alternati飞沧 requirement that each player knows (or
at least assumes) that the other players' actions are rational , that the other players
assume that her action is rational , that the other players assume that she assumes
that their actions are rationa l, and so on.
1 now formulate these requirements precisely, initially restricting attention to a
two-player game
Suppose that the actìon aT of player 1 is rationa l. 刊T11.削
e
to some belief , say fl} , of player 1 巾ut player 2's actions. Tl泣s co叫恼1 is illustrated in Figure 380.1 , where the rectangles repr，巳sent the players' action sets
and the sl回ed set
on the 句ht is the set of player 2's川袅ω川
a
assig时d
句
pos白i挝圳
tiv
划
pr∞蚓
ob双灿
abili让tyby
fll
Xi
Now suppose that ∞t only is the ac校也1 aî of player 1 rational , but
11
there is a beHef !l1 of player 1 to which aí is a best response such that every
action of play前 2 assigned positive p r:obability by fli is rational for player 2
τhis condition is illustrated in Fi gure 38 1.1, which sho、气IS 队'0 representative ac-
tions a~ and a~ to whÍch 时 assigns positive probabilit予 together with beliefs of
player 2，!缸a~) and fl~(吁)， to which these actions a把 best responses.τhe 臼ts of
actions of pl吧Ter 1 to which thεse two beHefs assign positive probability are the
sεts inside the black ellipses on the lef t. (These sets may or m咛 not intersect; ìn
381
12.1 民ationalízability
(nuH ll
fe 句L
付ζ 句L
i
'jyt
H尸
il
f1~(a~)
Player 1'8 action8
Player 2's actions
Fìgure 381.1 An íllust础ion of al飞 actíon aj of player 1 th在 ti内付1 础íonal and s叩阴阳d by a be!ief
111 with the prope巾 ωt every action of player 2 assìg只ed posítìve prob击bílíty by /11 ís rational for
player 2
the diagram , they do.)τhe shaded set of player l's actior飞s consists of all aεtions
削吗时 po协叩拍拍ility by a belief 阳ts叩ports some me时机 of
11附 s，
the shaded 制 is thεunion over all mer地ers a2 of X~ of the 使ts of actio邸 assigned
positive probabi 1ity by JI~ (a2).
Taking the argument one step further, suppose that not only is the action ar of
player 1 rationa l, but
Xi.
i
• there is a belief Jl of player 1 to which aî is a best 附印。只优 such that
ction
川
a20f印
play川削删ig时 po协
o foτeverya缸削削
is a be !ief Jl~ (的) to which a2 is a best 附ponse such tl时 every action of
player 1 as吨ned posi抗ve probability by Ji~ (向) is ratìonal fo 1' player 1.
This condition is 山lstrated in Fi gure 382. 1. The beHefs 川der which a~ and a{ are
rational are 严i (ll~) and Jl (a{) respectively, and the actions of player 2 to which
these beliefs assign positive probability are enclosed in the black ellipses on the
righ t. 11时arkg时 set on the 咔ht，坞， is the set of all actions of player 2 a5signed
positive proba bil句 by a be!ief that s叩ports 州州州巾er of Xî. (The sets X~ and
X~ may or may not intersec t.)
叭'e may be able to contím法出e process further, a t each stage t finding , fo
臼reach
act挝io丑
na丐j 垃
1 nX
乓: 1r ib7沱eli必G咄
fs川
时fF抖4臼a玛
O
均ye盯r忖
叫j 飞md臼 w
灿
hich刊必j
a川is川ra挝t刽iona剖1. lf we ca川on
归
tm
川
1
1
the proζe55 妇
indef主ini投tel 予
a 捻sm
汉斜必
μti切
仰
O
阳
η
ali泣za
♂忡
bl扣E巳. τh飞a
时t is, an action ai of
J斗~ we 5ay that 咛
player i i5 ratìonali之able if player i can 5ay to hers附el旺f"气1 命
sh阳
O
〈怜
i斗) 技
it 怒
i5 rat挝ior口1冶
挝1， (ii) there is a be如f that makes it rational that assigns positìve
a
probability only to rational actions of play校 j， (iii) there are beliefs of player j that
make each of these actions rational that 部sign positive probability only to rational
actions of mìne'二 and 50 on inde在nit，εly.
ln the game in Figure 378 .2, for exarr飞ple， the action Q of play缸 1 is not ratìonalizable. It is rational, and there is a be!ief 11} that suppo出 ít that a琐事lS positive
prob抽出tyo均 to ratìonal actions of player 2 (f1 assigns probability 1 to 窍， which
f
i
382
Chapter 12. Ratior时izability
严l(aD
玛 (a~)
pi
‘、
iιa
-
/，
2
句4
m
?
伽，
叫/4
、z
、、卢，
i
t.
嘻a
βMHnMH
μaμ'
，
f
，抖
句3
、d习 A
Player 1's actions
))
HHIH2
Player 2's actions
Figure 382.1 An illustration of an action aj of player 1 that 但) ìs rational and (b) is supported by a beliξf
l
.u} is (i) mtωnalfor
pl在yer 2 and (í i) supported 七ya be!i ef with the property that e\'ery action of player 1 assigned positive
probability by this be !ief is rational for player 1
Jl w抽 the prope咛 that every action 0年以yer 2 assigned positive probabili句I by
ís rational for player 2 if she believes that player 1 chooses X with probabílíty 1).
But no belief that supports it assigns positive probability to actíons of player 2 that
are themselves supported by beHefs that assign positive probability only to rational actions of player 1 The point is that play，εr 1 rationally chooses Q only if sh已
believes that player 2 is sufficìer飞tly likely to choosεQ， an action that is optimal
for player 2 only if she believes that player 1 is sufficiently likely to chooses X, an
action that is not rational for player 1.
ln this game, the action F of player 1, however, is rationalizable, supported by
beliefs for each player that assign probability 1 to F. That
player 1 thinks that
player 2 will choose F, and thi放s that player 2 thinks that she will choose F, and
thinks that player 2 thinks that she thinks that player 2 will choose F, and so on
inde在nitel予
We could now define rationalizability precisely by formulating the ìnfinite sequence of conditions. But alternatively, the informal discussÎQn suggests a much
simpler definition that circumvents the in位üte sequence. Suppose that there are
sets Z} and Z2 of actíons of players 1 and 2, respectively, such that Z1 contains 叫，
every action in Z1 is a best 陀sponse to a belieÍ of player 1 that assigns positive
probability or飞ly to actìons in Z2 , and every action in Z2 is a best response to a
belief of player 2 that assigns positive probability only to actions in . (丑ef，每e轩r to
Figur陀
e383袋.1.) Then we car趴1 construct 使
s eque
衍nc
使
εs sat挝ìsfy
抖m屯
g the conditìons discussed
站ov
四e陀
ε: 刽 mpl污
yse
供tX
乓; z Z
岛1 阳
fO
时r a11 eve
钮n
钮 vah阳 oftι， and
飞吐
dX
乓; 口 Z
马2 ，
fO
时
阳ra
拮
1拄1 odd 叫
vaalues
of t. τhus aj is rationali之able.
Conversely, suppose that we can find an i盯住ütεsequence of beliεfs and sets of
actions as described above. Let Z1 be the union of {哨 with the union of the sets
Xi for a11 eve川alues of ( and let
be the union of the sets X~ for a11 odd values
of t. ηìen every action in Zi is a best response to a belief that assigns positive
probability only to actions in Zj for í = 1 and j 口 2， and for i = 2 and j = 1.
12.1 院矗tion昌 lizability
383
泣'
，，
鸡-、
vaa
t J! )
Player l's actions
，
nι
?& Hum
、‘ι，，J
i
飞
tl
'Lr·
Player 2's actions
Figure 383.1 An ilIustratìon of the definítion of a rationalízable actìon aj _Each actìon ìn Zl ìs a besl
response t。在 beIÌt，f thal assìgns positive probabilí 、f only to members of Z2f and each actìon in Z2 is a
best response 10 a be!ief thal assigns positive proba 七 ìlity only 10 members of Z1- The fígure shows an
example of a rr飞机nber n1 of Z1 and the set of actions of player 2 to which the be !ief /1 1(n1) that supports
it assìgns posit 孔'e probabìlìty, together \'vith an example of a member 112 of Z2 and the set of actìons of
player 2 to whìch the belief
tl回$叩po白 it assigns posit州 probab汹y
Thus we define an actíon to bξrationalizable as follows
where Z一; denotes the
\
set of a11 collections a… i of actions for the playεrs other than í for w hich aJ E Zj for
all j.
J
DEFINlτION 383.1 (Rationalizabie actioll Í l1 strategic game) The actíon ai of pJayer i
in a strategic game with vNM priε ferences is rationalizable if for each player j there
exists a set 马 of actions such that
• Zi CO时 ains aj
• for every player j , every action aj in Zj is a best response to a belief of player j
that assigns positive probability only to lists of actions in Z…112. 1.3 Properties of ratio门。lizable actÎons
Suppose that 矿 is a pure strategy Nash equilibrìum. Then for each player i the
action aj is a best response to a belief that assigns probability one to a-=-i- Setting
Zi {aj} for each Í , we see that a* 妇
i sr盹
挝t纣i∞ali泣
a
za
抽
ble
巳. 1n fa
前甜
仕t， 附
C
w eh驳
a刊 the 如
f硝
01
削
i丑10
队毛W吟γ叩咄
i扫
in
挝troτn飞ger resul挝t， which follows from the fact that ín a mixεd strategy Nash equílìb刷
S
rium , any action a player uses with positive probability is a best response to the
list of the other players' strategies.
疆 PROPOSITION 383.2 Every actìo l1 used with positìve p/'O bability in so附 míxed strategy
Nash equili如ium is ratiol1alíz始le.
Proof Let 旷 be a míxed strategy Nash equilibrium, and for each playεr i let Zi
be the set of actions to which player í's strategy ai assigns positive probability.
By Proposition 116.2, every action ín Zí is a best response to the belief of player i
generated by the other players' strategies (i.e. the belief that assigns to the list of
Chapt坦 r 12.
384
L
C
R
了
。， 7
M
5
5, 2
2, 5
3, 3
2, 5
5, 2
7，。
Rationalizability
7，。
。， 7
Figure 384.1 A g在me in which the actions T and B of player 1 and L and R of player 2 are not lIsed
with positive probability in any Nash equilibrium, but are rationalizable
actions a-i a probability equal to the product of iX j (的) over all j 并 i) ， and this belief
by definitior飞 assigns positi飞'e probability only to lists of actions in 2-i' Her飞ce
everyac延on in
is rationali之able.
口
Actions not used with positîve probability in any Nash equilibrium may also
be rationalizable. First consider 'a game in which each action of every player is a
best re叩onse to S0111e list of strategies of the other players. For such a game, the
conditions in Definition 383 .1 are satisfied when, for every player Î , 2i is equal to
the set of all player Î's actions. Th us eVClγaction of every player in such a game is
rationalizable
1 claim that the game in Figure 384.1 (a) is such a game and (b) has a unique
Nash 叫uilibrium， (M , C).
,
( EXERCISE 384.1 (Mixed strategy 叫uilibrium of game in Figure 384.1) Show th拭
the game in Fígure 384.1 has no mixed strategy equilibrium in which any pl可衍 's
strategy assigns positive probability to more than one action.
Each action of each player is a best response to some action of the other player (for
example, T is a best response of player 1 to R, M is a best response to C, and B
is a 怡然陀sponsεto L) , so by the argument of the previous paragraph, a11 actions
of both players are rationa吕立abl巳 1n particula乙 the actions T and B of player 1
are rationalízable , though they are not used wíth positi飞'e probability in any r让我sh
equilibrium. For example , the action T is rational for player 1 if she believes that
player 2 will choose R, which is rational for player 2 if she believes that player 1
1气rill Ch005心 B ， which is rational for player 1 if she thinks that player 2 will choose
L, whìch in turn is rational for player 2 if she thinks that player 1 will choose T , at
which point wεare back to the beginning of the cycle.
苔~ven in gamεs in "vhich every rationa吕立able action is used with positi飞re probabilìty in some Nash equilibriuffi, the implications of rationalizability differ from
those of Nash equilibrium. The reason is that the notion of Nash equ i1ibrium isolates a set of strategy pr哨的， while the notion of rationalizability isolates a set of
配 tions for each player. C'imside巳 for example , the game in Figure 385. 1. This game
has exactly two Nash equilìbria , both in p 旧e strategies: (T , T) and (B , B). The 叫一
tion of rationalizability, hO\电rever， does not restrict the outcome at all: both T and
B are rationalizable for both players, so that a11 four pure outcomes are consistent
with each player's action being rationalizable.
Finding a collection of sets 2i of actions that satisfíes Definition 383碰 1 can be
difficult. ln the next section 1 describe an alternative concept that is easier to work
with and
12.2 Iterated elimination of strictly dominated actions
T
B
Figur骨 385.1 Agame 钊 íth
12.2
385
了
B
2, 2
0, 1
1, 0
1, 1
two Nash eq口 iJíbria ， (T , T) and (B , Bl.
Iterated elimination of strictly dominated actions
τhe notion of rationalizabilit)飞 in requiring that a player act rationall予 eliminates
from consideration actions that are not best responses to any belief. Such actions
are called never阳best responses.
OEFINITION 385.1 (Nevel 位st response in strategic game) The aclion a; of player i in
a strategìc game with vNM preferences is a never-best response if for every belief
flj of player i about the other players' actions there exists a mixed strategyα j of
player i such that player i's expected payoff toαi exceeds her expected payoff to a;:
ζ
}严扣川幻川
j快
( ρ 一JiJ}川叫叫口叫叫川
j .
a_i EA 叫
a_i 在 A
where Ui 怡 ， a -i) is player i's expected payoff v巾nsh川ses the IIÚXεd strategy lX i
and the other players' actions are a_ Ít Ui is her Bernoullí payoff functio日， and A … 1
is the set of lists of the other players' actions.
Another criterion that we might use to eliminate an action from consideration
is domÎnation. 1讼call that an action aj of player i in a strategic gamεwith vf吗?叫
preferences is stríctly dominated if player i has a mixed strategy that yields her a
higher payoff than does ai regardless of the other players' actions (Oe位让tio口 120.1).
An argurnent on page 121 shows that a strictly dominated action is a neverbest respoI飞船 (a result you should find unsurprising). Here is an argurnent usi吨
the current terminology: if is strictly dominated by the mixed strategyαi ， 白
th阳
凹I
e
口i(α
内j ， a 一 i) >叫d4:， a 一 i) for a11 αL
一 iρ， so that (385.2) is S3挝ti日sfi陀ed for any belief Jl i of
player í abo时 the other players' actio盹 and hence is a never-best response
1n a game in which each player has finitely many actions, thεconverse of this
result is also true: every r飞机re卜best response is strictly don飞inated. Although you
may be able to convince yourself that tl让s result is plausible, the proof (which 1
omit) is 口。t trivial. In summary, we have the followìng resuìt.
a;
a;
磁 LEMMA 385.3
A pl叼er's action in a strategic gûme with vNM pr价rences Íil which
each player has 卢nite与 many actions is α neve冲岱 t respo l1 se if and only 扩 it is strictly
dorη inated
Now recor飞sider the argument behind the notion of rationalizability. First wε
argue that a player will not use a nεver-best respor飞前， or equivalentl予 a strictly
dominated action. Next we argue that if she \母íorks 山飞der the assumption that
her opponents are rational, then her belief should not assign positive probability
to any of her opponents' actions that arεnev啊-best respor飞ses. PU役ing thesεtwo
argurnents together, a player should not choose an action that is strictly dorninated
Chapter 12. 民ationalizability
386
in thεgame that results when we eliminate a11 her opponents' strictly dominated
actions. We argue further that a player who works under the assumption that her
opponents assume that she is rational will assume that her opponents' actions are
best responses to beliefs that assign positive probability only to actions of hers that
are best responses to some belief. τhat is, shεwill assume that her opponents'
actiop.s are not strictly dominated in the game that results when a11 of her strictly
dominated actions arεeliminated. Thus she wíll choose an action that is not strictly
domina ted in the game tha t results w hen , first , a11 of her strictl y domina ted actions
are eliminated, and then a11 of the other players' strictly dominated actions are
eliminated.
We see that given Lemma 385 .3，制ch step in the argument for rationalizabìlity
is equivalent to an additional round of elimination of strictly dominated strategies,
so that tlle rationalizable actions are those that remain no mattεr how many rounds
of elimination we perform. 丁hat is, an action prof闭过e is rat挝iona Ji泛able 证
i f ar衍ld orη11y if
su
盯rv
时i忖
陀
V
岱S 价
e
1 t衍era
α tε
ed εelim
价州
1斤
m
η1巾in
巾
η1at
让ti，纱
Ot门1 of sfrÎ亢
Uctly
ε 正d
由i沁
1ωf
01
阳仰?
it 阳
Pr陀
ecis时e已l衫
予
yf二} 飞w
币re def自ine a set of action profiles to survi飞re iterated elimination of
strictiydOITUnated actions as foilo飞车队
DEFIN ITl ON 386.1 (I terated elimÎnation of strictly domi l1 ated actÎon5) Suppose that for
each player i in a strategic game and each t = 1,..., T there is a set X; of 配tions
of player i (the set of actions remaining at the start of round 1 of elimination) such
that
• XJ = Ai (we start with the set of a11 possible actions)
• X:?
十 1 is a subsε挝tωOdf
们 X! f包
挝f刊
O
盯ead
川川
ch t
1,..., γ?民
T. 1 (怡at怆汹
e臼ach st怡age们叭
E认附飞
ac
位t悦
柑ions)
• for each t
1,. . ., T - 1, every action of play衍 i in X; but not in Xf+l is
strictly dominated in
丑1 山
t he game in 飞认
W
句凉r括lÌch
h
the set of actions of ε臼ach 抖
p lay
严心盯r j is
巧 {队认附
7哈e川川刊
叫
硝归
ell且
limi如蚓叭
附a批叫
m
t怡
翻eo
∞
肉i与
n
向
ys付如
t红r剖
挝i呻 do
臼佣棚删盯
mi
盯1让inat
• ∞ a削ctωt削创io丑Miid7 i岭ss纣弛位阳
t扫仪
nn
红挝冗
1汇ct
ct
川句 do
倪3双侃
I红
m
况飞11m
each pla叮yer j 怒
i sX丁 (atthe e叫 of the proαssnoaεtion of ahy player is strictly
dominated).
Then the set of a巾n profiles a 以h that aì 巳 Xr for every player i survives
iterated elimination of slrictly dominated actions.
This definition admits the possibílity that more than one set of action profiles
survives iteratεd eJi mination of strictly dominated actio贝尔 that different orders
and rates of elimination lead to different sets of surviving adion profiles. But in
fact we can show that the set of survivors Îs independent of the order and rate of
eliminatioll, and, as 1 have argued informally, coìncides with the set of profiles of
ratìonalizable actions (1 omit the proof)
阳 PROPOSITION 386.2 For any strategic game with vNM pl咕rences il1 which each player
has 卢nitely ma呼 actio毗 a ul1 ique set of actioll pro.卢les su 叩ives iterated elimination 旷
的'íctly dominûted actior吼。 nd this set is equal to the set of p功。les of rationalizable actio部
12.2 Iterated elimination of strictly domìnated actions
B
B
S
3, 1
0, 0
S
0, 0
1, 3
387
X
B
2
0, 2
4, 3
0, 0
B
5
S
0, 0
3, 4
X
2, 1
1, 1
Figure 387.1 Two variants of 1305
EXA fv1 PLE 387.1 (Rationalizable actions in two 飞!aríants of BoS) Consider the game
in the left panel of Figure 387.1. Neither actior飞 of playεr 1 is strictly dominated,
but the actior飞 B of player 2 is strictly domìnated by X. 1n the game obtained by
eliminating B for player 2, the action B of player 1 is 日 trictly dorni且ated. Finall予
in the game obtained by eliminating B for player 2 and B for player 1 the action X
for player 2 is strictly dominated. Wεconεlude that the only rationalizable action
of eac~1 player is S.
Now consìder the game ir飞 the right panel of Figure 387.1. 1 claim that the action
X of player 2 is strìctly dominated by mixed strategies that assign posìtive proba 辄
bilities to B and S. To establish this claim, dε∞teby 吨 the strategy of player 2 that
assigns probability p to B and probability 1 p to S. For 唁 to strictly dominate the
actíon X, we need 3p > 1 (the expected payoff to α2. exceeds the expected payoff
to X when player 1 chooses B) and 4(1 - p) > 1 (the expected payoff to IX :i exceeds
the expected payoff to X when player 1 chooses 句" τhese two cor趴飞dí垠
tions are sa
挝t阳
i 巾di证f ~ < P < ~. Now ∞
cO邵邸i挝
白e机
d
制
M
栩川r叫.
白
the 伊
gan时 t出
h飞a挝E打
M
陀阳
e阳叫lts
怡sw
.气
.vhen
臼nX 沁í $川叫
elim
mín
盯
t山
hi沁sg
伊
ame
巳， for each pla
吗咛
ye
旺r the ac
时
cti
创ion B is a best response to th ε beHef that assigns
probabìlity 1 to B, and tl飞e action 5 ìs a best response to the belief that assigns probabilìty 1 to 5, so that neìther action of either player is strictly dominated. Hence
both B and 5 are rationalizable for each player in 位le whole game.
(1) EXERCISE 387.2 (Fínding rationalizable ac柱。只吟Fi时 the set of rationalizable actionsof εach player in the game in Figure 387.2.
? EXERCISE 387.3 (Morra) Fi nd the rationalizable actions of each player in the game
Morra (Exercise 372.3).
? EXERCISE 387.4 (Guessing two-thirds of the average) Find the rationalizable ac楠
tions of each player in the game in Exercise 34. 1.
( EXERCISE 387.5 (Hotelling's model of electoral competition) Consider a variant of
Hotel1il鸣 's model of electoral competiti∞ (Section 3.3) in which there are two candidates and the set of possible positions is {O, 1, 2尸. , e}, where f is even. Assume
that there is only one position m with the prop仅ty that exactly half of the voters'
favorite positions are at most nl , and half of the voters' favori校 positions are at
least 1n. Show that this position Ìs the urtiquεrationaJizable action.of each playe汇
•
L
C
T . 2, 1 1, 4
B 1, 8 0, 2
R
0, 3
1, 3
Figure 到 7.2 The game in Exercìse 387.2
Chapter 12. Rationalizability
388
In the games on which the following exercise and exa皿ple are based, each
player has infinitely many actions. ln each case 1 describεa finite variant, so tha t
Lemma 385.3 and Proposition 386.2 can be applied.
主 EXERCISE 388 .1 (Contributing to a public good) Show the following results for the
variant of the game in Exercise 44.1 in which the number of players is an arbitrary
positive integer n and the contribution of each player i is restricted to be a nonnegative multiple of some small amount 0 > 0 that is at most 1O i. Assume that 0 is such
that a11 the specific contribution levels you need in your argument are multiples of
(j (so that you do not need to worry about the finiteness of the action sets).
a. Any contribution of more than wd2 is strictly dominated for pl咛er i by a
contribution of wd2.
3 and Wl
W2
W3 口 w， then every contribution of at most w /2
is rationalizable. [Show that every such contribution is a best response to a
belief th挝 assigns probability 1 to each of the other players' contributing some
amountatmo日t 叫ual to w/2.]
岛.
If 11
C.
If 11
3 and ω-- 的<抖，加1 the unique rationalizable contribution
of players 1 and 2 is 0 and the unique rationalizable contribution of p1ayer 3
is W3. [E1iminate strictly dominated actions iterative1y. After eliminating a
contribution of more than Wi /2 for each p1ayer i (b y part a) , you can eliminate
small contributions by player 3. Subsequently you can eliminate any posi出'e
εontribution by players 1 and 2.1
(j) EXERCISE 388.2 (Courno t' s duopo1y game) Consider the example of Courn仗's
duopoly game with constant unit cost and linear in飞rerse demand studied in Sec翩
tion 3. 1.3. Restrict each firm's output to be a nonnegative mu1tiple of a small
amount ð > 0 and to be at most 此; assume that l 归一 c) is a multiple of o. Use
Fígure 58 .1 to argue that every output grωter than ~(统一 ε) is strictly dominated
by the output !何一 ε). Then use a simílar a电ume时 to show that when a11 such
outputs are eliminated from the game , every output less than i(a - c) is strictly
dominated by the output i 但- c). Continue a10鸣 similar Ìines, using Figu祀 59.1
to obtain the cutoff outputs, to reach the conclusion that the only rationalizable
outp 时 for each firm is the Nash equilibrium output l 仪- c)
12.3 Iterated elimination of weakly dominated actions
Recall that we say that an action ai of player i is weakly dominated 证 player i has a
mixed stratεgy that yields her at least as high a payoff as does ai whatever the other
players' actions and a higher payoff than does aj for some actions of the other
players (Definition 121.1).
We may define a set of action profiles to survive iterated elimination of weakly
dominated actions by replacing "strictly" with 气N"eakly" in Definition 386. 1. τhat
is, we obtain a set of action profiles that survives iterated elimination of weakly
dominated actions as follows. First we mark actions of each player that are weakly
12 .3 Iterated elimination of weakly dominated actions
L
C
38吉
R
T . 1, 1 1, 1 0, 0
B
0, 0 , 1, 2
1, 2
Figure 389.1 A game in which the set of actions that survive iterated elimination of weakly domìnated
actions depends on the order in which actions are eliminated.
dominated. Then we remove a11 the marked actions, and again mark weakly dominated actions for every play旺Once again, we remove a11 the marked actions. We
repeat the process until no more actions can 七e eliminated for any player.
This procedure has 1εss appeal than that of iteratively eliIτùnating strìctly domìnated actions. First, in some games the set of actions that survives iterated elim自
ination of weakly dominated actions depends on the order in which actions are
eliminated. Consider the game ìn Figure 389. 1. If we first elimìnate L (weakly
dominated by C) and then T (weakly dominated by B) , tl飞台 outcomes that survi飞re
a陀侈， C) and 悍， R) ， yielding the payo旺 profile 口， 2). But if we first eliminate
R (weakly domìnated by C) a双d then B (weakly dominated by 盯， the outcomes
that survive are (T , L) and (T , C) , yielding the payoff profile (1 ，刊 f and if we first
eliminate both L and R, the outcomes that survive are (汇。 and (B ， 白， yielding
the payoff profiles (1 , 1) and (1, 2) , respectively
Second, a weakly dominated action that is not strictly dominat付， unlike a
strictly dominated action, is not an unambiguously poor choice: by Lemma 385.3
such an action is a best respon使 to some belief 古lUS， whereas a strictly dominated
action remains strictly dominated in any game obtained by deleting any actions of
the other players, a weakly (but not strictly) dominated 在ction does not necessar懈
ily remain weakly dominated in a game obtained by deleting some actions of the
other players. In the game in Figure 389.2 , for example, the action B of player 1
is weakly dorninated by T but is not weakly dominated in the game obtained by
dele也19 the action R of player 2.
在lÌs feature of a weakly (b ut not strictly) dominated action means that the removal of an action at some stage in the iterahve procedure may eliminate the rationale for the removal of some other action at an earlier stage. Suppose that in the
game in Figure 389.2, for example, we 如st elìminate B and then eliminate R. In
the absence of R, player 1 no longεr has a reason not to choose 旦
Despite these difficulties with the iterated elimination of weakly dominated
actìons, the procedure retains some appeaL One reason Ìs a connection between
the subgame perfect 吨uilibrium outcomes of a c1 ass of extensive games with per-
L
鸟在
R
1, 0
0, 2
Fi惧ue 389.2 A game 过lustratìng the iterated elimination of weakly domìnated actions.ηle action B of
player 1 is wcakly domìnated by 丁 but is not weakly domìnated ìn 也eg豁口e obtaìned by deletìng the
action R of player 2
‘
390
Chapter 12. 民ati础al泣ability
fect information and the action profiles that survive it时 ated elimin剖ion of weakly
dominated actions in the strategic form of such games. Specifically, let f be a 如lÏte
εxtensive game with perfect information 妇 which no player is indifferer飞tbetween
any two terminal histories (50 that, in particula乙 f has a un坷ue subgame perfect
叫uilibrium). We can argue straightforwardly that there is an order of eliminatior飞
of weakly domÌ1飞ated actions in the strategic form of f for which every member
ofthε sur飞如ing set of aεtion pr。在les generates the subgame perfect eψilibrium
outcom号 of f. (Note that 1 do not assert that the surviving set of actior飞 profiles
contains on1y subgame perfect equilibria: every surviving action profile generates
the unique subgame perfect 叫飞tilibrium outcome, but may not itself be a subgame
p衍fect equilibrium.)
The order of elimìnation that yields the result corresponds to the steps in the
procedure of backward induction. Fì rst consíder subgames of length 1. Take one
t ha
挝t pr陀
ec
优
ed扫
岱s the subg
e
伊
am
引le， by i the player
such subgame. Denote by 11 the history 出
认
w巾
F
every s位ategy of pla叮yer i that (α叶) chooses act挝i(ζonscωonsiste衍n飞t 飞wγi让th h and (b) chooses
ηac时ti扫
on飞 di迁ffere口t from ai in the subgame. Every such stratεgy is weakly domian
nated by the strategy that differs only in that in the subgame it chooses aj. After
eliminating these strategies, we replace each subgame of length 1 wíth the outcome that results from the subgame perfect 叫uilibrium action in the subgame.
Then , in the second stage, we repeat the exercise with the subgames of length 1 in
the remaining game. We cor飞tinue until we 在rrive at the start of the game.
τhe game in Figure 391.1 illustrates the procedure. Assume that player 1 prefers
d to a to b and player 2 prefers a to c ar飞d d to e, so that the subgame perf，εεteq飞lÍ嗣
librium actions are those indicated by black line segments. Denote the strategic
form of the game , sho马代rn in the right panel of the figure , by G. We first look at
the subgames of length 1, namely those following the histories (A., C) and B. In
the subgame following (A , C) , player 1 moves , and the fact that she prεfers a to
b means that in the strategic form her strategy AH weakly dominates her strat棚
egy A l. (在仅 strategies BH and B1 are inconsiste时 with the history (A , C), so we
do not consider them.) In the other subgame of length 1, follo.wÎI飞g the his拍
12 .4 Domin矗院cesolv在 bílity
391
1
A
2
严
a
D
C
\乡 2F
AH
A1
BH
B1
e
d
cε
CF
DE
DF
a
b
d
d
G
C
C
b
C
C
t
d
d
E
t
b
Figure 391.1 An extensíve game with perfect infoτ飞机a tion (left) and its strategic fOI狈 (right). 丁he
actions taken ín the unique subgame perfect 告quilibrium are shown in black.
12.4 Dominance solvabili句
A version of the procedure of iterated elimination of weakly dominated actions of
particular interest is that in which all weakly dominated actions of each player are
eliminated at each stage. If all the players are indifferent between all actìon profìles
白atsur飞巾 e when we perform such iterated elimination, we say that the game is
domina l1 ce solvable.
τhe game in Figure 389.1 ìs not dorniI飞ance solvable, because at the fìrst stage
the weakly dominated actions are L and R for player 2 (neither action of player 1 is
weakly domin挝ed)， and once these are eliminated, no actíon is weakly dominated ,
and play，εr 2 is not indifferent betweeI飞 the two remaining actìon pairs. 百le game
in Figure 389.2, howeve瓦拉 dominance solvable: B are R are the weakly dominated
actions at the first stage, and M is the weakly dominated action at the second stage,
leaving the single outcome (了，斗.
In the following ex校cises you need to find , at each round, the set of all weakly
dominated actions. Note that a player's action is not weakly dominated ìf it is a
strict best response to some list of the other players' actions.
( EXERCISE 391.1 (Example of dominance-soh咄 le game) Find the set of Nash e气时伽
libria (mixed as wel1 as pure) of the game in Figure 391.2. Show that the game is
dominance solvable , and find the pair of payoffs that survives. Find an order of
elimination such that more than one outcome survives.
Q) EXEF之CISE 39 1.2 (Oi飞!iding money) Show that the game in Exercise 38.2 is domi-
nance solvable and find the set of sm飞riving actìon pairs
EXERCISE 391 .3 (Voting) Consid盯 the model of voting in Section 2旦3. Suppose
there are three candidates and assume that the conditions in Exercise 49.1 are satisfied. Show that if more than two-thirds of thεcitizens rank the same candìdate
last, then the game is dominance solvabl巳
L
TMB
Fig始 re 391.2
C
R
The game for Exercise 39 1.1
392
Chap重er 12. Rationalizabili勾
.t EXERCISE 392.1 (Bertrand's duopoly game) Consider thεvariant of Bertrand' s
duopoly game in Exercise 67.2, in which each firm is restricted to choose a price
that is an integral nurr飞快r of cents. Assume that the profit function (p 一叶。 (p)
has a single maximizer and is increasing up to this maximizer. Show that the game
is dominance solvable and find the set of surviving outcomes.
(J) EXERCIS 主 392.2 (Strictly competitive extensive games with perfect information)
Show that a finite strictly competiti飞'e extensi飞re game with perfect information in
which there are two possible outcomes is dominance solvable in (at most) a single
step.
Notes
The notion of rationalizabilìty is due to Bernheim (1984) and Pearce (1984); Spohn
(1982) is a precurso立 Gale (1953) and Luce and Raiffa (1957, 108… 109, 173) first
studied Ín detail versions of the proced 议re of ìterated elimination of dominated
strategies. The formulation here is due to Moulin (1979). Lemma 385.3 is due to
Pearce (1 984); it is c1 0sely related to Lemma 3.2 .1 of van Damme (1983). Proposition 386.2 is due to Pearce (1984 , 1035). Aumaru飞 and Brandenburger (1995) make
prεcise the idea that rationalizability is equivalent to each player's being rational ,
assuming the other players are ratior飞at assuming that the other players assume
th ü the first player is rationa l, and so on. The notion of dominance solvability
is d ue to Moulin (1979; it is d部ely related to the notion of "so1\咄 ility in the
complete weak sense" of Luce and Raiffa (1957, 109).
The game in Exercise 387.4 总 taker飞 from Moulin (1986b, 72) , Exercise 388 l
was suggested by Stuart MesteÌman, the result in Exercise 388 .2 is due to Gabay
and Moulin (198时， Bernheim (1984" and Moulin (1984), and the rεsult in Exer附
cise 391. 3 Ís due to Dhillon and Lockwood (2004). Ewerhart (2000) shows that
every finite strictly competitive extensive game with perfect information th肘 has
thrεe possible outcomes (l ike chess, in which the outcome may be a win for Black ,
a win for White, or a tie) is dominance solvable in at most two'steps, extending the
result ín Exercise 392.2.
,
‘
Evolutionary Equilibrium
13.1
13.2
13 .3
13 .4
13 .5
13.6
Monomorphic pure strategy equilibrium 394
Mixed strategies and polymorphic equilibrium 400
Asymmetric contests 406
Variation on a theme: sibling behavior 411
Vari号tion on a theme: the nesting behaviorofwasps 414
Variation on a theme: the evolution of the sex ratio 416
Prerequisite: Chapters 2 and 4
cco 民 DING
to the Darwinian theory of evolution, a mode of behavior sur1n
anεnvironment in which organisms interact, the reproductive success of a mode
of behavior may depend on thεbehavior of other organisms. If a11 organisms act
aggressively, for example , then an organism may be able to survive only if it is
aggressi飞吃 whereas if all organisms are passi飞电 then an organism's reproductive
success may bεgreatεr if it acts passively than if it acts aggressively. Gamεtheory
provides tools with which to study evolution in such an en飞rironment.
1n this chapter 1 describe models based on the notion of a strategic game. 在le
players are interpreted to be members of a到 evolving population of organisms
(humans, animals, pla抗ts， bacteria,…), pairs of whom interac t. Each player's set
of actions consists of the modes of behavior an organism could ac伊ire through
mutation, and its payoffs measure its biological 卢tlless， or reproductive success
(expected number of healthy offspring).
We do not interpret the players' payoffs as reflections of their subjective feel严
ìngs , nor their actions as conscious choices. Rather, each player's payoff is a concrete measure of its fitness , and we assume that each player is programmed to
follow a certain mode of behavior, which comes from one of two sources: with
high probability it is inherited from the player's parent (臼 paren时I and with low
(but positi飞re) probability it is assigr飞时 to the player as the result of a mutation.
For most of the models in this chapter, inheritance is conceived very sirnply: each
player has a single parent, and, unless it is a mutant, simply takes the same action
as does its par乞nt. This model of inheritance captures the essential features of both
genetic inheritance and social inheritance: players either follow the programs e仆
coded in their genes , which come from their parents , or learn how to behave by
imitating their parents.
1f, given the modes ofbehavior of a11 other organisms, an orgar吐sm's biological
fitness associated with the action a exceeds its fitness associated with the action a' ,
飞rives only if no other mode is more successful at producing offspring.
:(Q 丁2
Chapter 13. Evolutionary Equilibrium
394
then adher创lts of a reproduce faster than ad始rents of a' , causi抗g the composition
of actions in the population to change. Thus , roughly, a configuration of actions is
stable only if each orgar飞ism's action is a best rεsponse to the environme拙， which
suggests tl1 at evolutionarily stable outcomes are related to Nash equilibria.
13.1
Monomorphic pure strategy equilibrium
73.1.1
lntroduction
认是= consider a pop飞llation of organisms whose members are repeatedly randomly
matched in pairs. The population is largεεnough that the probability of an organism's being matched with the sam己 opponent more than once is negligible. The set
of possible modes of behavior of every organism is the same , and the consequence
of an interaction for an organism depends only on its action and that of its opponent not on its name. Each organism regularly produces offspring (r叩roduction
is asexual). An organism's mode of behavior is, with high probability, that of its
pare时， Jnd with low probability a randomly chosen 气nutant" mode of behavior.
As an example, think of a populatio日 of identica\ animals , pairs of which are
periodically engaged in conflicts over pre予丁he actions available to each al飞 imal
could correspond to various degrees of aggression, and 飞Ne could reasonably assume that the outcome for each animal depends only on its degree of aggression
and that of its opponen t.
We may model the interaction bet飞lVeen each p在 ir of organisms as a two-player
symmetric strategic game. 1n this context, suc妇 a game consists of a cümmon set A
of actions and a pay。在 fun础。n 11 , where for any pair of actions (矶的， the payoff
u( 斤，的 is interpreted as the fitness (expected number of offspring) of an organism
that takes thεaction a when its opponent takes the action af • (Note that each fun伫
tÌon II gives rise t。在 different game; no t飞\'0 payoff fur飞ctions give rlse to the same
game, as they do when the game is specified by the players' preferences.) We assume that the adherents of 拍拍 mode of behavior multiply at a rate proportional
to their payoff and \ook for a configuration of n飞odes of behaviòr in thεpopulation
that is stable in the sense it cannot be "invaded" by any other mode: if a small
fraction of the populatio日 consists of idεntical mutants , e飞'ery mutar飞t obtains an
expected payoff lower than that of any nonmutant (We ig只ore the case in which
mutants taking different actions are present in the population at the same time.)
1n this section 1 restrict attentìon to situations in which all organisms , except
those resulting from mutation , foliow the same mode of behavior, which has no
randorr飞 componer飞仁在1at is , 1 consid仔 only mOI I0 11LOrp11íc pure sfrategy 吨uilibria
("monomorphic" "one form").
‘
13.1 .2
Examples
To get an idea of the implications of evolutionary stabilit予 consider two exam
ples.
suppose that thεgame between each pair of organisms is the one in
13凡在走。往omorphic
pure strategy equilibrium
X
Y
395
x
Y
2, 2
0, 0
0, 0
1, 1
Figure 395.1 A strategic game in which both actions of each player are evolutionarily stable
,
Figure 395. 1. Suppose that every organism normally takes the action X. Suppose
the population contaìns the small fraction εof mutants who take the action Y.
The population is large, so the probability of any given normal organism's being
matched with a mutant is approximately ê; 1 take it to be exactly ε. Th us a normal organìsm's opponent is another normal organism with probability 1 一 εanda
mutant with probability f , so that a normal 0丐anism's expected payoff is
2.(1
f) 十 O.ε= 2(1ε) .
Similarl予 a muta时's expected payoff is
O. (1…玛丽← 1.ε±ε.
If εis small (less than
then the 在古st payoff exceεds the second, so that the entry of a small fraction of mutants leads to a situation ìn which the expected payoff
(fitness) of every mutant is lower than the expected payoff of every normal organism. Thus normal organìsms reproduce f部ter than do mutants, ar飞d the action X
is evolutionarily stable.
Nowsuppos台 that every organìsm normally takes the action Y. Mutants taking
the action X who meet each other obtaìn a payoff higher than that of normal or幽
ganisms who meet each otl飞机'. But when f is smal1, a mutant is lìkely to be paired
with a normal organism, in which ca优 its payoff is O. Making a calculation like
that in the previous case, a normal organism's expected payoff is 1 - f , whereas
a ml阳1t's expected payoff is 2ε， so tha t íf f is small (less 阳n ~ in thisω时， the
first payoff exceeds the second , and the action γis evolutionarily stable. The value
of f for which a normal organism does better than a mutant is smaller 国 this case
than it is in the case that the normal action is X. 泣。毛刊￥rer， the sign证icant point is
that in both cases there is some 飞ralue of εsuch that for smaller 飞ralues no mutant
can invade: we 飞，飞'Ìsh to capture the idea that mutation is extremely rare, relativε
to normal behavior, so we are satisfied with the existence of some value of ε 也at
prevents invasion by mutants.
Now consider the game in Figure 396.1. By an argument like those abovε， the
action X is evolutionarily stable. Is the action Y also evolutionarily stable? In a
population containing the fractionεof mutants choosing X, the expected payoff of
a normal organism choosing Y is 0 (it obtains 0 whether its opponent is normal or a
mutant), wherεas the expected payoff of a mutant is 2f (it obtains 2 against another
mutant and 0 against a normal organism). Thus the action γis not evolu悦。narily
stable: for any valuεof εthe expected payoff of a mutant exceeds that of a normal
orgamsm.
Chapter13. 主valution在ry
396
x
X
Y
Equilibrium
Y
2, 2 0, 0
0, 0 0, 0
Figure 3吉岳 .1 A strategíc game ín 飞飞'hich only the actíon X of each pl在yer ìs evolutíonaríly stâble
1n both garnes , both (X , X) and (Y , Y) are Nash equilibria, but whereas X is
an evolutionarily stable action in both garnes， γis evolutionarily stable only in the
first garne. What is the essential difference between the garnes? If the norrnal action
is Y, then in the fìrst garne a rnutant that chooses X is worse off than a normal organism in encounters with norrnal organisms, wh巳reas in the secor飞d garne such a
mutant obtains the same expected payoff as does a norrnal orgarusrn in encounters
with normal organisrns. Thus in the first garne , for εsmall enough, the gain (relativεto the payoff of a norrnal organisrn) that a rnutant obtains with probabilityε
when it faces another rnutant doεs not canoε1 out the loss it incurs with probabîlity 1 … εwhen it faces a norrnal organisrn. 1n the second garne , however, a mutant
loses nothing relati飞'e to a norrnal organism , so no rnatter how srnall f is, a mutant
is better off than a norrnal orgar飞isrnτhat is, the essential differen但 between the
games is that 叫 X， Y) < u(Y, Y) in the game in F耶lre 3纯 1 ， butu(X， Y) =叫 γ， γ)
in the game iηFigure 396. 1.
13.1 .3
General dennitions
Consid缸 now an arbitrary syrnrnεtric twoωplay盯 strategic game in which each
playεr has finitely many actions. Under what circumstances is the action a* evolu-
tionarilv stable?
Suppose that a small group of mutants choosing an action different from 旷
enters the population. The notion of evolutìonary stabilìty requires that εach such
muta时 obtain an expected payoff less than that of each normal or吝anisrn， so that
the rnutants die ou t. (If the mutants obtained a payoff higher than that of the
norrnal organisms, then they would e\咽ltually cornεto domi叫te the population;
if they obtained the sarne pa yoff as that of the normal organisms , then they would
neither multiply nor declin巳 Our notior飞 of stability excludes the latler case: it
requires that mutants 快 driver飞 out of the population.)
Denote the fraction of mutants in the popuJation by 乙 First consider a mutant
that adopts the action 扛 ln a randorn encounter, the probability of its facing an
organism that adopts the action ♂ is approximately 1 一 ιand the probability of
Ìts facing a mutant, which adopts b, is approximately f. Thus its expected payoff
1S
(1 一时 u(b， 。可个印 (b)丰
Sirnilarly, the expected payoff of an orgarusrn that adopts the action a* is
(1 … ε) 叫矿 ， a*) + ε u( 。本 ， b).
13.1 Monomorphic pure strategy equilibrium
397
For any mutation to be driven out of the popu1ation , we need the expected
payoff of any mutant to be 1ess than the expected payoff of a norma1 organism:
(1 ~ E)u(a* ， a*) 十 ε u( 旷 ， b)>(l
ε )u(b， 旷)十 ε u(b， b) for a11 b 并旷
(397.1)
To capture the idea that mutation is extreme1y rare , the notion of evo1utionary
stability requires only that there be some number E such that the inequality ho1ds
whenever ε < E. That is, we make the fo11owing definition.
The action a* is evolutionarily stable if there exists E > 0 such that (397.1)
is satisfied for a11 E with 0 <ε < E.
Intuitive1y, the 1arger is E, the μmore stab1e" is the action a* , because 1arger
mutations are resisted. However, in the current discussion 1 do not attach any
significance to the va1ue of f; for an action to be evo1utionari1y stab1e, 1 require
on1y that there be some va1ue of E such that a11 smaller mutations are resisted.
The condition in this definition of evo1utionary stability is a litt1e awkward to
work with because whenever we app1y it, we need to check whether we can 自nda
suitab1e va1ue of E. It can be reformu1ated to avoid the variab1e f; making it easier
to use.
1 first claim that
if there exists f > 0 such that a* satis且es (397.1) for a11 E with 0 <ε < E,
then (a* , {川 is a Nash equi1ibrium
To reach this conclusion, suppose that (a* , a'可 is not a Nash equilibrium. Then
there exists an action b such that u(b , a*) > u(a* , a*). Hence (397.1) is strictly vio1ated when ε= 0, and thus remains vio1ated for a11 su Hiciently sma11 positive va1
uesof6. (If w <工， and y and z are a町 numbers， then (1 一 ε)ω 十 εy < (1 一 ε )x 十 εz
whenever εis sma11 enough.) Thus there is no va1ue of E > 0 such that 出e inequa1ity ho1ds whenever ε < E. We conclude that 证 the action a* is evo1utionarily stab1e,
then (伊a*非〉
E
1 now claim 由
tha
挝t
>
if (a* , a*) is a strict Nash equilibrium, then there exists f > 0 such that
a* satisfies (397.1) for a11 E with 0 <ε < E.
The argument is that if (a* , u*) is a strict Nash equilibriu叽 then u(b， 矿) <
u(a* , a*) for a11 b, so that the strict inequality in (397.1) is satis自ed for ε 二 O， and
hence a1so for sufficiently sma11 positive va1ues ofε.
What if (a勺1*) is a Nash equi1ibriurn but is not strict? Then some action b 并旷
isabestresponsetoa*: u(b , a*) = u(aγ). Thus (397.1) reduces to the condition
u(a 飞 b) > u(b , b)
We conclude that the action a* is evo1utionari1y stab1e if and on1y if (i) (a* , a*)
is a Nash equ出brium， and (ii) u 忡忡忡 u(b， b) for every b 乒旷 that is a best response to a*. Intuitive1y, for mutant behavior to die out, (i) no mutant can do better
398
Chapter 13. Evolutionary Equilibrium
than an organism us公飞g a* in encountεrs with organisms using a* , and (ii) any mu栅
tant that does as well as an organism using a* în such er飞counters rr飞ustdo 飞电lorse
than an organism using a* in encounters with mutants.
To summarize, the definition of evolutionary stability based on (397.1) 必吨UIV­
alent to the followìng defìnition.
DEFINITION 398.1 (Evolutionarily stable actíon ín strategic game) An actìon a* of
a player in a symmetric two-player strategic game in which th号 payoff of each
player is u(a , a') when the player's acti∞始 a and her opponent' s actìon ís a' is
evolutionarily stable with respect to mutants usi吗 pure strategies îf
• (旷，的 is a Nash equilibrium , and
• u(b , b) < u( 。飞 b) for every action b 乒旷 tt时 is a best response to a 气
Note that if (乱。*) is a strict Nash 叫uilibrium， then the second condition is
automatically satisfied, because no action b 乒 a* is a best response to 旷. Note also
that the ir飞equality in the second condition is strìc t. If it were weak, then situatior.. s
in whìch mutar飞ts neither multiply nor die out, but reproduce at the same rate as
the normal population, would be ìr飞cluded as stable.
13.7 .4 Examples
Both symmetric pure Nash equilibria of the game in Figure 395.1 are strict, so
that both X and Y are evolutionarily stable (confirming our previous analysis).
In the game in Figure 396.1 , (X , X) and 作， Y) are symmetric pure Nash equilib唰
ria also. But iηthis case (X, X) is strict whereas (Y , Y) is no t. Further, because
叫 X， X) > u( γ， X) , the second condition iηthεdefinition of evolutionary stability is not satisfied byγ. Thus in this game onl)' X is evolutionarily stable (again
confirming our previous analysis).
丁he Prisoner's Dilemma (Figure 15.1) has a unique symmetric Nash equilibrium
(FJ) , and this N削 equílíbrium is strict. Thus the action F is the only ev州州仆
arily stable action. The game BoS (Figure 19.1) has no symmettic pure Nash equilibrìum, and hence no evolutionarily stable action. (1 consider mixed strategies in
the next section.)
在飞e following game, which gene主alizes the ideas of th已 game in Exercise 31.2,
严部ents a richer range of possibílities for evolutionarily stable ac柱ons.
EXAMPLE 398.2 (Hawk-Dove) τ\\'0 animals of the samεspecies corr飞严处 for a resource (e.g. fooct or a good nesting site) whose value (in units of" 在tness") is 沙二> O.
(That is , v measures the increase in the expeded number of offspring due to control of the resource.) Each animal can be either aggressíve or passive. If both animals
are aggressive , they fight until one is seriously injured; the winner obtains the resource without sustaining any injury, whereas the loser suffers a loss of c.η沈阳10
animals are equally likely to win, so each one's expected pay。在 15 iu 叶(…c). If
各oth animals are passive , then each obtaÍl飞s the resource with probability with…
t
13.1 Monomorphic pure strategy equilibri极限
399
EVOLUTIONARY GAME THEORY: SOME HISTORY
In hìs book The Descent 吃f Man , Charles Oarwìn gíves a game-theoretic argument
that in sexually reproducing species , the only evolutionarily stable sex ratio is 50:50
(1871 , Vol. t 316). Oarwin's argumεnt is game theo陀tic in appealing to thεfactthat
the number of an animal's descendants depends on the "behavior" of the other
members of the population (the sex ratio of their offsprir丐; see Exercise 416.1).
Coming 50 years before the language and methods of game theory bεgan to develop , however, it is not couched in game-theoretíc terms. In the late 1960s, two
decades after the appearance of von Neumann and Morgenstern's (1944) seminal
book, Hamilton (1967) proposed an 以plicitly game-theoretíc model of the e飞rolu­
tion of the sex ratio that applies to situations more general than that considered by
D盯win.
在ut the key fîgurεin the application of game theory to evolutionary biology is
John Maynard Smíth. Maynard Smìth (1972a) and Maynard Smith and Príce (1 973)
propose thεnotion of an e毛'olutionarily stable strategy, and Maynard Smith's subs吨uent research develops the field in many directions. (Maynard Smith gives signifícant credit to Price: he writes that he would probably not have had the idea of
using game theory had he 1'.ot seen unpublished work by Price. "Unfortunately",
Maynard Smith writes, "Or. Price is better at having ideas than at publishing them"
(1972b, vii).)
Since Maynard Smith and Price's seminal work，台飞rolutionary game theory has
blossomed. 志iological models abound , and the methods of the theory have made
thejr way into economks and other social sciences.
out a figh t. Finally, if one ar飞imal is aggressive whíle the other is passive, then the
aggressor obtains the resource wíthout a figh t. The game is shown in Figure 399.1.
If v > C, then the game has a uniq时 Nash equilibrium (A , A) , which is strict,
so that A is the unique evolutionarily stable action.
If v = C, then also the game has a ur让气ue symmetric r也sh equilibrium (A , A).
But in this case the equilibrium is 只ot stric t: against an opponent that chooses A ,
a player obtains the same payoff whether it chooses A or P. However, the second
v. Thus A is
condition in Oefinition 398.1 Îs satisfied: ~v = u (P , P) <吟4， P)
thεunique evolutionarily stable action in 白is case also曹
In both cases, a population of passive players can be invaded by aggressive
players: an aggressive mutant does be役时也an a passive pl可er when its opA
P
AH卜"叶 ， ~(v … c)
P
O, v
Figure 399.1 The game Hawk-Dove.
Cha抖牵r 13.
400
Evolutionary Equilibrium
ponent is passive, and at least as well as a passive player when its opponent is
aggresslve.
If v < c, then the game has no symmetric Nash 叫出librium in p泣纪 strategies:
ηeither (A A) nor (P , P) is a Nash e气功líbrium. Thus 妇
i 只 t出
hi始sc臼
as优
et白
hega趴附
eh诩as n
ev
飞vol垃
u丘剧
t技ior乳la
盯r川il忖
y stable 武
a ct柱ior口飞. (币le gamε has only ωymmetric Nash 叫uilibrìa î口 this
case.)
(!) EXERCISE 400.1 (Evolutionary stabílity and weak dominatio对 Leta 本 be an evol让
tionarily stable action. Does a* necessarily weakly dominate every other action? Is
it possible th挝 some action weakly dominates a*?
1, EXERCISE 400.2 (Example of evolutionarily stable actions) Pairs of members of a
single population enga萨 in the following game. Each player has three actions ,
corresponding to demands of t 2, or 3 units of payoff. If both players in a pair
make the same demand , each player obtains her demand. 。生herwise 由e player
who demands less obtains the amount demanded by her oppone时， whereas the
playεr who demands more obtains 时， where a is h前 demand and /j îs a r飞umber
i栩栩忖 F
扫in
时
dt舱
检飞冶es蚓
h
e杖tωο
们句
f印炯阳
归川
p
ure 削鸣栩
Yμsymm
州
l
ar口飞d the set of pure 们
e vol挝飞
ut悦iona
盯rí让ly st怡
abl坠
e strategies. What happens if each player
has n actions, corresponding to demands of t 2, ..., n units of payoff (and /j <
l/n)?
To gain an ur飞derstanding of the outcome that evolutionary pressure might induce in games that haveηo evolutionarily stable actíon (e.g. BoS , ar飞dH彼此-Dove
when v < c) , we can take several routes. One is to consider mixed stratεgies
as well as pure strategies; another is to a110w for the possibility of several types
of beha坑。r coexísting ín the popul时ion; a third is to considεr interpretations of
asymmεtric equi1ibria. 1n the next section I discuss the first two approaches; in the
section after that 1 consìder the third approach.
13.2 Mixed strategi公5 and polymorphic equilíbrium
13 .2.1
De药nition
50 far we havεconsidered only sìtuations in whích both "normaγorganìsms and
mutants use pure strategies. If we assume that mixed strategies, as wel1 as pure
strategi时， are passed from parε挝s to theìr offspring and may be thrown up by
mutation, then an argument analogous to the one in the previous section leads to
the conclusion that a口 evolutionarily stable mixed strategy satisfies conditions like
those in Definition 398.1. Precisely, we can dε岳飞eaηevolutionarily stable (mixε句
strategy, known briefly as an
as fo11ows.
产 DEFINITION 400.3 (Evolutionarily stable ntixed strategy) A口 evolutionarily stable
strategy (ESS) in a symmetric two-player strat唔ic game in whích the expected
pay。在 of each player is U(α， lt') when the player's mixed strategy is 泛 and her
opponεηt's mixed strategy isα I is a mixed strategy lt* such that
13.2 Mixed strat告gie量 andpolγmorphic equilibrium
401
• (在气旷) is a Nash equilibrium
• U(卢， ß) < U( o: 气卢) for every mixed strategy ß 乒 IX* that is a best response
to IX 气
Can animals really us台 mixed strategies? John Maynard Smith, the oríginator
of the notion of an ESS, argues as follows.
"If it were selectively advantageous, a randomising device could surely
evolve, either as an entirely neuronal process or by dependence on
functionally irrelevant external stìmulí. Perhaps the one undoubtεd
example of a mixed
is the production of e气ual numbers of X and Y
gametes by the heterogametic sex: if the gonaàs can do it, why not the
braìn?" (1982 , 76).
E飞rìder飞ce that some animals randomìze is presenteà by Brockmann, Grafer飞， and
Dawkins (1979) , who argue that certain wasps pursue mixed strategìes. (For a
dìscussion of Brockmann et a1. 's model , see Section 13 .5.)
13.2.2 Pure strategiε5 and mixed strat.εgies
Definition 400 .3 does no t, of course , preclude the use of pure strategies: every pure
strategy is a special case of a mìxed stratεgy. Th us Definitions 398.1 and 400.3 both
apply to pure strategies. Do thεy yield t1飞e same conclusions?
Suppose that 旷 is an evolutionarily stable action in the sense of Definition
398.1 , and let 扩 be the mixed strategy that assìgns probability 1 to the action a* .
Because a* is evolutionarily stable, (a , a*) is a Nash equilibrium, so (庆气的 is
a mixed strategy Nash equìlìbrium (Proposition 122 .2). Is IX* necessarily an ESS
in the sense of Definition 400.3? 坷。 the second condition in the definition of
an ESS may be violated. That is, a pure strategy may be immune to invasion by
mutants that follow any pure strategy b飞lÌ may not be immune to invasion by r执法
tants that follow some mixed strategy. Stated briefly, though a pure strategy Nash
吨uilìbrium is a mixed strategy Nash equilibrium, an actìon that is evolutionarily stable in the se