# Strategic games (Normal form games) Game theory studies multi-person decision ```Strategic games
(Normal form games)
Lectures in game theory
Fall 2009, Part 1
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
G.B. Asheim, ECON5200-1
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Game theory studies multi-person decision
problems, and analyzes agents that
 are rational (have well-defined preferences)
 reason strategically (take into account their
knowledge and beliefs about what others do)

Classification of games
 non-cooperative vs. cooperative games
 strategic vs. extensive games
 Games with perfect and imperfect information
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Studies the outcome
…of individual … of joint actions
actions when there when there is
is no external
external
enforcement.
enforcement.
(strict sense)
no communication
before the game
non-cooperative
game theory
communication
before the game
(wide sense)
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cooperative
game theory
3
1
Incomplete
information
Imperfect information
Complete information
Perfect information
“Almost
perf.” info.
Static
Dynamic
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
Lecture 1
Strategic
games
Lecture 1
Bayesian
games
Lecture 3
Extensive games
(the general case)
Lecture 2
Extensive games
(multi-stage games)
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Solution concepts
 A systematic description of outcomes
that may emerge in classes of games
 Game theory suggests reasonable solutions for
classes of games and examines their properties

Interpretation of solution concepts
state”) interpretation

The deductive (eductive) interpretation
Bounded rationality
will not be treated in these lectures.
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Rational behavior

Decision-maker i chooses from a set Si of strategies.

where i is a finite set of uncertain states.

The uncertainty may be
Example
 strategic uncertainty relating to the strategies of others.
 exogenous uncertainty relating to the environment.


A state-strategy pair (i, si)  (i  Si)
leads to a consequence c  C.
A consequence function g: i  Si  C assigns a
consequence to each state-strategy pair.
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Rational behavior (cont)


Let the decision-maker be endowed with
 a vNM utility function i : C  .
 a subjective prob. distr. i over i.
Example
A (pure) strategy si is preferred to si if and only if
 i (ωi )i ( g (ωi , si))   i (ωi )i ( g (ωi , si))
ωi  i


ωi  i
Anscombe &amp; Aumann’s decision-theoretic framework. It requires the availability of mixed strategies.
A mixed strategy i is a obj. randomization over Si.

The exp. utility of i:  ( s )
 (ω ) ( g (ω , s )) 

si S i
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i
i
ω 
 i  i
i
i


i
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A strategic game

i
Incomplete
information
Imperfect information
Complete information
Perfect information
Definition 11.1 A strategic game consists of
Lecture 1
Strategic
games
Static
Dynamic
Lecture 2
Extensive games
(multi-stage games)
Lecture 1
Bayesian
games
Lecture 3
Extensive games (the
general case)
 a finite set N (of players)
 for each i  N, a non-empty set Si (of strategies)
 for each i  N, a payoff function ui on S  jNSj:
( si , si )  S , ui ( si , si )  i ( g ( si , si ))




A strategic game is finite if,
for each i  N, Si is finite.

Interpretations:
 the game is played once?
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Consequence
(or outcome)
 simultaneous actions?
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Nash equilibrium for a strategic game

Definition 14.1 A Nash equilibrium of a
strategic game is a strategy profile s  S with the
property that, for each player i  N,
si  S i , u ( si , si )  u ( si , si )
Alternative formulation:

Define a set-valued function Bi (best-response fn):
Bi ( si )  {si  S i | si  S i , u ( si , si )  u ( si , si)}

The strategy profile s  S is a Nash equilibrium if
and only if , for each player i  N, si  Bi ( si )
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Can Nash equilibrium be used as a solution
concept if the game is only played once?
Yes, if each player can predict what each opponent will do.
 For each player, only one strategy survives iterative
elimination of strictly dominated strategies.
 Through communication before the game starts,
the players make a self-enforcing agreement (coordinate on an equilibrium).
 Given a common background, the players are able to
co-ordinate on an equilibrium without communication
before the game starts (Schelling, 1960, focal point).
 A unique Nash equilibrium is not sufficient.
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

N  {1, … , n}
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Example: An auction
Player i’s valuation vi, where v1  …  vn  0.
The players submits bids simultaneously.

The object is given to the players submitting the highest bid (if there are several players with the highest
bid, then the winner is the one with the lowest index.

First price auction: The winner pays his bid.

Second price auction: The winner pays the highest bid
among the non-winners.
Equilibria in a first
 Equilibria in a second
price auction?
price auction?

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Existence of Nash equilibrium

Lemma 20.1 (Kukutani’s fixed point theorem) Let
X be a compact convex subset of n, and let f : X
 X be a set-valued function for which
 for all x  X the set f(x) is non-empty and convex.
 the graph of f is closed
Then there exists x  X such that x  f(x).

Proposition 20.3 A strategic game has
a Nash equilibrium if for all i  N,
Proof of Proposition 20.3
 the strategy set Si is non-empty and compact.
 the payoff function ui is continuous
 the payoff function ui is quasi-concave on Si.
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A Bayesian game

Incomplete
information
Imperfect information
Complete information
Perfect information
Definition 25.1 A Bayesian game consists of
Lecture 1
Strategic
games
Static
Lecture 2
Extensive games
(multi-stage games)
Dynamic
 a finite set N (of players)
Lecture 2
Bayesian
games
Lecture 3
Extensive games (the
general case)
 for each i  N, a set Ai (of actions)
 for each i  N, a set Ti (of types)
 for each i  N, a prob. distr. pi on T  i  N Ti that
satisfies, for all ti  Ti , t   T pi (t i , ti )  0
i
j i
j
 for each i  N, a vNM utility function ui on the
set A  T, where A  jNAj, and where ui(a, t) is
player i’s payoff if (a, t) is realized.
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Nash equilibrium for a Bayesian game
An ex post perspective.

A Nash equilibrium of a Bayesian game is a Nash
equilibrium for the strategic game defined as follows:
 The set of players is the set of all pairs (i, ti)
for i  N and ti  Ti.
 For each player (i, ti), the set of strategies is Ai.
 For each player (i, ti), the payoff function is defined by
u(i ,ti ) (a  ) 

pi (t i , ti )

u (a  , , a(n ,tn ) ), t

 i , ti ) i (1,t1 )
t i  j i T j t   T pi (t 
 i ji j


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Example 27.2
(t1B , t2B ); pi (t1B , t2B )  1 / 4
(t1B , t2S ); pi (t1B , t2S )  1 / 4
B
S
B
S
B
2, 2
0, 0
B
2, 1
0, 0
S
0, 0
1, 1
S
0, 0
1, 2
(t1S , t2B ); pi (t1S , t2B )  1 / 4
(t1S , t2S ); pi (t1S , t2S )  1 / 4
B
S
B
S
B
1, 2
0, 0
B
1, 1
0, 0
S
0, 0
2, 1
S
0, 0
2, 2
Is a(i ,t )  B and a(i ,t )  S for both i a Nash
equilibrium for this Bayesian game?
B
i
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Example: A second price auction

N  {1, … , n}

Ai  

Ti  V

pi (v1 ,  , vn )   nj 1 (v j ) for some probability
distribution  over V.

The payoff u(i ,v ) (a  ) equals the expectation of the
random variable whose value given (v1 , , vn ) is
 vi  max jN \{i} a j (v j ) if player i wins.
 0 otherwise.
 Equilibria?
i
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The mixed extension

Definition 32.1 A mixed
extension of a strategic
game consists of
Incomplete
information
Imperfect information
Complete information
Perfect information
Lecture 1
Strategic
games
Static
Dynamic
Lecture 2
Extensive games
(multi-stage games)
Lecture 1
Bayesian
games
Lecture 3
Extensive games (the
general case)
 a finite set N (of players)
 for each i  N, a set (Si) (of mixed strategies),
where (Si) is the set of prob. distributions on Si.
 for each i  N, a payoff function Ui on jN(Sj)
assigning to each   jN(Sj) the expected value:
U i ( )   sS  jN  j ( s j ) ui ( s )
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Mixed strategy Nash equilibrium



Definition 32.3 A mixed strategy Nash
equilibrium of a strategic game is a Nash
Proof of Proequilibrium of its mixed extension.
position 33.1
Proposition 33.1 Every finite strategy game
Proof of
has a mixed strategy Nash equilibrium.
Lemma 33.2
Lemma 33.2 Let G be a finite strategic game.
Then   iN(Si) is a mixed strategy Nash
equilibrium if and only if, for each player i, every
pure strategy assigned positive probability by  i is
Can be used as primitive
a best response to  i .
definition (cf. Def. 44.1).
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Interpretations of mixed strategies

Mixed strategies as an object of choice

Mixed strategy Nash equilibrium as a

Mixed strategies as pure strategies of
an extended game.

Mixed strategies as pure strategies of
a perturbed game.

Mixed strategies as beliefs.
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Mixed strategy Nash
Mixed strategies as pure
strat. in a perturbed game
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Rationalizability

Definition A set of strategy profiles Z  iN Z i has
the best response property if, for all i  N and si
 Zi, there exists a i   ( j i Z j ) such that si  Bi ( i ).

Definition 55.1 A strategy si  Si is rationalizable
if and only if there exists a set of strategy profiles
with the best resp. prop., Z  iN Z i , such that si  Z i .
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Further results on rationalizability

Lemma Every strategy used with positive
probability by some player in a mixed strategy Nash
equilibrium, is rationalizable.

Proposition Let G be a finite strategic game. Then
there exists, for any i  N, a rationalizable strategy
for i.

Observation Even in a strategic game with a
unique Nash equilibrium, common knowledge of
the players' rationality does not imply that the
players will play this Nash equilibrium.
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Never-best response and strict domination


Definition 59.1 A strategy si  Si of player i in the
finite strategic game G is a never-best response if
there is no i  (jiSj) such that si  Bi(i).
Definition 59.2 A strategy si  Si of player i in the
finite strategic game G is strictly dominated if
there is a mixed strategy i  (Si) such that,
si  S i ,

sS  i (si)ui ( si , si)  ui ( si , si )
i
i
Lemma 60.1 A strategy si  Si of player i in the
finite strategic game G is a never-best response if
and only if it is strictly dominated .
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Iterated elimination of strictly dominated strategies

Definition 60.2 The set of X  S of strategy profiles
in the finite strategic game G survives iterated
elimination of strictly dominated strategies if X
 iNXi and there is a collection of sets (( X tj ) jN )Tt0
that satisfies the following conditions for each j  N.
 X 0j  S j and X Tj  X j
 X tj1  X tj for each t  0, ... , T  1.
 For each t  0, ... , T  1, every strategy of player j
in X tj \ X tj1 is strictly dominated in the game Gt, where,
for each i  N, i’s strategy set is restricted to X it .
 No strategy in X Tj is strictly dominated in the game GT.
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Iterated elimination of strictly dominated strategies (cont.)


Lemma (i) If Z  iNZi has the best response
property, then Z  X. (ii) X  iNXi has the best
response property.
Proposition 61.2 If X  iNXi survives iterated
elimination of strictly dominated strategies in the
finite strategic game G, then, for each j  N, Xj is
the set of player j’s rationalizable strategies.
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Weak domination

R
1, 0
B 1, 0
0, 1
Definition 62.1 A strategy si  Si of player i in the
finite strategic game G is weakly dominated if
there is a mixed strategy i  (Si) such that,
s i  S i ,
s i  S i ,

L
1,
1
T
sS  i (si)ui ( si , si)  ui ( si , si )
sS  i ( si)ui ( s i , si)  ui ( si , si )
i
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Iterated elim. of weakly dom. str. is dubious because
 the result depends on the the order of elimination,
 hard to give the procedure an epistemic foundation.
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