Game Theory
Chapter 10
1
Applications of Game Theory



National Defense – Terrorism and Cold War
Movie Release Dates and Program
Scheduling
Auctions http://en.wikipedia.org/wiki/Spectrum_auction
http://en.wikipedia.org/wiki/United_States_2008_wireless_spectrum_auction





Sports – Cards, Cycling, and race car driving
Politics – positions taken and $$/time spent
on campaigning
Nanny Monitoring
Group of Birds Feeding
Mating Habits
2
Game Theory and Terrorism
Game theory helps insurers to judge the risks of terror
Financial Times Jenny Wiggins September 8, 2004
Shortly after September 11 2001, a small group of companies that
specialise in assessing risk for the insurance industry launched
US terrorism risk models.
These combine technology and data to predict likely terrorist
targets and methods of attack, and possible losses to life and
property.
They are aimed at the insurance and reinsurance industry, which
already uses similar models to assess potential losses from
natural catastrophes such as hurricanes and earthquakes.
"Most major commercial insurers and reinsurers are using
terrorism modelling today," says Robert Hartwig, chief
economist at the Insurance Information Institute.
3
Game Theory and Terrorism (cont.)
Andrew Coburn, director of terrorism research at RMS,
says the company can pinpoint possible targets
because it believes terrorists make rational decisions.
"Their methods and targeting are very systematic," he
says.
RMS uses game theory - analytical tools designed to
observe interactions among people - in its models. It
argues that, as security increases around prime
targets, rational terrorists will seek out softer targets.
Industry participants, however, say the predictive
abilities of the models are limited, given the difficulty of
foreshadowing human behaviour.
The development of the models has attracted the
interest of the US government…
4
Game Theory and Randomization
Random Checks
Newsweek October 22, 2007
Security officials at Los Angeles International Airport
now have a new weapon in their fight against
terrorism: randomness. Anxious to thwart future terror
attacks in the early stages while plotters are casing
the airport, security patrols have begun using a
computer program called ARMOR (Assistant for
Randomized Monitoring of Routes) to make the
placement of security checkpoints completely
unpredictable.
5
Game Theory and Randomization (cont.)
Randomness isn't easy. Even when they want to be
unpredictable, people follow patterns. That's why the
folks at LAX turned to the computer scientists at USC.
The idea began as an academic question in game
theory: how do you find a way for one "agent" (or
robot or company) to react to an adversary who has
perfect information about the agent's decisions? Using
artificial intelligence and game theory, researchers
wrote a set of algorithms to randomize the actions of
the first agent. Academic colleagues couldn't
appreciate how the technology could be useful. "It was
very disappointing," says Milind Tambe, the USC
engineering professor who led the ARMOR team.
6
Applications of Game Theory



National Defense – Terrorism and Cold War
Movie Release Dates and Program
Scheduling
Auctions http://en.wikipedia.org/wiki/Spectrum_auction
http://en.wikipedia.org/wiki/United_States_2008_wireless_spectrum_auction





Sports – Cards, Cycling, and race car driving
Politics – positions taken and $$/time spent
on campaigning
Nanny Monitoring
Group of Birds Feeding
Mating Habits
7
Grey’s Anatomy vs. The Donald
NBC delays 'Apprentice' premiere
By Nellie Andreeva Dec 20, 2007
NBC is taking the premiere of "Celebrity
Apprentice" out of the cross-hairs of the last
original episode of ABC's "Grey's Anatomy"... or
so it seems.
NBC on Wednesday said that it will push the
launch of "Apprentice" from Jan. 3 to Jan. 10,
expanding "Deal or No Deal" to two hours on
Thursday, Jan. 3.
The move follows ABC's midseason schedule
announcement Friday that included the last
original episode of "Grey's" airing Jan. 3,…
8
Grey’s Anatomy vs. The Donald
'Grey' move has NBC red Peacock shifts 'Apprentice' back
By Nellie Andreeva Dec 21, 2007
The Thursday night scheduling tango between NBC and
ABC continued Thursday morning when ABC officially
announced that it will move the last original episode of
"Grey's Anatomy" from Jan. 3 to Jan. 10.
That led to a reversal in NBC's Wednesday decision to
push the premiere of "Celebrity Apprentice" from Jan. 3
toJan. 10 to avoid the first-run "Grey's."
NBC said Thursday afternoon that "Apprentice," hosted
by Donald Trump, will now launch Jan. 3 as originally
planned.
9
Game Theory and Movie Release Dates
The Imperfect Science of Release Dates
New York Times November 9, 2003
On Dec. 25, which this year happens to be a Thursday, five
new movies will be released in theaters -- six, if you count a
new Disney IMAX film called ''Young Black Stallion.'' As with
the Fourth of July and Thanksgiving, there is a special cachet to
opening a film on Christmas Day…. The casual moviegoer
rarely ponders why a particular bubbly romantic comedy, serialkiller thriller, literary costume drama or animated talking-farmanimals movie opens on the day it does. Movies come; movies
go; movies wind up on video. To those responsible for putting
those films on the screen, however, nothing about the timing of
their releases is arbitrary.
10
Game Theory and Movie Release Dates
(cont.)
Last December featured one of the most dramatic games of
chicken in recent memory, when two films starring Leonardo
DiCaprio were both slated to open on Christmas weekend.
Ultimately, Miramax blinked first, moving the release of Martin
Scorsese's ''Gangs of New York'' five days earlier and ceding
the holiday to the other DiCaprio film, DreamWorks' ''Catch Me
if You Can.'' ''We didn't think about moving,'' says Terry Press,
the head of marketing for DreamWorks. ''We had been there
first, and 'Catch Me if You Can' was perfect for that date.'' This
year, DreamWorks chose to schedule a somber psychological
drama, ''House of Sand and Fog,'' for the day after Christmas,
deferring a bit to Miramax. ''I don't want our reviews to run on
the same day as 'Cold Mountain,''' Press says.
Ever wonder why a movie theater shows a preview of an
11
upcoming movie that is to be released in 2 years?
Applications of Game Theory



National Defense – Terrorism and Cold War
Movie Release Dates and Program
Scheduling
Auctions http://en.wikipedia.org/wiki/Spectrum_auction
http://en.wikipedia.org/wiki/United_States_2008_wireless_spectrum_auction





Sports – Cards, Cycling, and race car driving
Politics – positions taken and $$/time spent
on campaigning
Nanny Monitoring
Group of Birds Feeding
Mating Habits
12
FAA Auctions
Blame rests with the FAA
USA TODAY December 18, 2007
The Federal Aviation Administration (FAA) is the gang that
couldn't shoot straight. After years of ignoring airspace that is
too crowded and near-collisions that are too common, the
agency is now plotting a response that would make a bad
problem worse. The problem is of the agency's own making. Air
congestion has increased, but the issue could have been
handled better by federal officials.
Across the country, air traffic control towers are dangerously
understaffed because FAA bean-counters have not prioritized
the hiring of more personnel. As a result, the New York area
airports have 20% fewer controllers on duty than they should.
13
FAA Auctions (cont.)
Blame rests with the FAA
USA TODAY December 18, 2007
Now, the Transportation Department is set to unveil a proposal to
cut flights and sell hourly slots to the highest bidder. But
auctioning flights would raise fares, limit consumer choice and
strike a blow to the economy. It wouldn't shorten the wait at the
gates or increase capacity. It would force airlines to pay a
premium to fly that will surely be passed on to travelers. And it
would reduce options for those flying to small and midsize cities.
Flight rationing, like congestion pricing, is not a viable solution. It
is experimental game theory. America's busiest airports should
not be the guinea pigs for an ideological solution that has never
been tested at any airport, let alone the nation's busiest.
http://www.aviationairportdevelopmentlaw.com/2009/10/articles/faa-1/it-is-official-the-faa-rescinds-slot-auction-rule/
14
Applications of Game Theory



National Defense – Terrorism and Cold War
Movie Release Dates and Program
Scheduling
Auctions http://en.wikipedia.org/wiki/Spectrum_auction
http://en.wikipedia.org/wiki/United_States_2008_wireless_spectrum_auction





Sports – Cards, Cycling, and race car driving
Politics – positions taken and $$/time spent
on campaigning
Nanny Monitoring
Group of Birds Feeding
Mating Habits
15
Game Theory Terminology
Simultaneous Move Game – Game in
which each player makes decisions
without knowledge of the other players’
decisions (ex. Cournot or Bertrand
Oligopoly).
 Sequential Move Game – Game in which
one player makes a move after observing
the other player’s move (ex. Stackelberg
Oligopoly).

16
Game Theory Terminology
Strategy – In game theory, a decision rule
that describes the actions a player will
take at each decision point.
 Normal Form Game – A representation of
a game indicating the players, their
possible strategies, and the payoffs
resulting from alternative strategies.

17
Example 1: Prisoner’s Dilemma
(Normal Form of Simultaneous Move Game)
Martha’s options
Don’t Confess
Peter’s
Don’t Confess
Options
Confess
Confess
M: 2 years
P: 2 years
M: 1 year
P: 10 years
M: 10 years
P: 1 year
M: 6 years
P: 6 years
What is Peter’s best option if Martha doesn’t confess? Confess (1<2)
What is Peter’s best option if Martha confess?
18
Confess (6<10)
Example 1: Prisoner’s Dilemma
Martha’s options
Don’t Confess
Peter’s
Don’t Confess
Options
Confess
Confess
M: 2 years
P: 2 years
M: 1 year
P: 10 years
M: 10 years
P: 1 year
M: 6 years
P: 6 years
What is Martha’s best option if Peter doesn’t confess? Confess (1<2)
19
What is Martha’s best option if Peter Confesses? Confess (6<10)
Example 1: Prisoner’s Dilemma
Martha’s options
First Payoff in each
“Box” is Row Player’s
Payoff .
Don’t Confess
Confess
Peter’s
Don’t Confess 2 years , 2 years 10 years , 1 year
Options
Confess
1 year , 10 years 6 years , 6 years
Dominant Strategy – A strategy that results in the highest payoff to a
player regardless of the opponent’s action.
20
Example 2: Price Setting Game
Firm B’s options
Firm A’s
Options
Low Price
High Price
Low Price
0,0
50 , -10
High Price
-10 , 50
10 , 10
Is there a dominant strategy for Firm B?Low Price
Is there a dominant strategy for Firm A? Low Price
21
Nash Equilibrium

A condition describing a set of strategies
in which no player can improve her payoff
by unilaterally changing her own strategy,
given the other player’s strategy. (Every
player is doing the best they possibly can
given the other player’s strategy.)
22
Example 1: Nash?
Martha’s options
Don’t Confess
Confess
Peter’s
Don’t Confess 2 years , 2 years 10 years , 1 year
Options
Confess
1 year , 10 years 6 years , 6 years
Nash Equilibrium: (Confess, Confess)
23
Example 2: Nash?
Firm B’s options
Firm A’s
Options
Low Price
High Price
Low Price
0,0
50 , -10
High Price
-10 , 50
10 , 10
Nash Equilibrium: (Low Price, Low Price)
24
Chump, Chump, Chump
http://videosift.com/video/Game-Theory-in-BritishGame-Show-is-Tense?loadcomm=1
25
Traffic and Nash Equilibrium
Queuing conundrums; Traffic jams
The Economist, September 13, 2008
Strange as it might seem, closing roads can cut delays
DRIVERS are becoming better informed, thanks to more accurate
and timely advice on traffic conditions. Some services now use
sophisticated computer-modelling which is fed with real-time data
from road sensors, satellite-navigation systems and the analysis of
how quickly anonymous mobile phones pass from one phone mast to
another. Providing motorists with such information is supposed to
help them pick faster routes. But the latest research shows that in
some cases it may slow everybody down.
Hyejin Youn and Hawoong Jeong, of the Korea Advanced Institute of
Science and Technology, and Michael Gastner, of the Santa Fe
Institute, analysed the effects of drivers taking different routes on
journeys in Boston, New York and London. Their study, to be
published in a forthcoming edition of Physical Review Letters, found
that when individual drivers each try to choose the quickest route it
can cause delays for others and even increase hold-ups in the entire
road network.
26
Traffic and Nash Equilibrium (cont.)
The physicists give a simplified example of how this can
happen: trying to reach a destination either by using a short but
narrow bridge or a longer but wide motorway. In their
hypothetical case, the combined travel time of all the drivers is
minimised if half use the bridge and half the motorway. But that
is not what happens. Some drivers will switch to the bridge to
shorten their commute, but as the traffic builds up there the
motorway starts to look like a better bet, so some switch back.
Eventually the traffic flow on the two routes settles into what
game theory calls a Nash equilibrium, named after John Nash,
the mathematician who described it. This is the point where no
individual driver could arrive any faster by switching routes.
27
Traffic and Nash Equilibrium (cont.)
The researchers looked at how this equilibrium could arise if
travelling across Boston from Harvard Square to Boston Common.
They analysed 246 different links in the road network that could be
used for the journey and calculated traffic flows at different volumes
to produce what they call a "price of anarchy" (POA). This is the ratio
of the total cost of the Nash equilibrium to the total cost of an optimal
traffic flow directed by an omniscient traffic controller. In Boston they
found that at high traffic levels drivers face a POA which results in
journey times 30% longer than if motorists were co-ordinated into an
optimal traffic flow. Much the same thing was found in London (a
POA of up to 24% for journeys between Borough and Farringdon
Underground stations) and New York (a POA of up to 28% from
Washington Market Park to Queens Midtown Tunnel).
Modifying the road network could reduce delays. And contrary to
popular belief, a simple way to do that might be to close certain
roads. This is known as Braess’s paradox, after another
mathematician, Dietrich Braess, who found that adding extra capacity
to a network can sometimes reduce its overall efficiency.
28
Game Theory and Politics
Game Theory for Swingers: What states should the candidates visit before
Election Day? Oct. 25, 2004
Some campaign decisions are easy, even near the finish of a
deadlocked race. Bush won't be making campaign stops in Maryland,
and Kerry won't be running ads in Montana. The hot venues are
Florida, Ohio, and Pennsylvania, which have in common rich caches
of electoral votes and a coquettish reluctance to settle on one of their
increasingly fervent suitors. Unsurprisingly, these states have been
the three most frequent stops for both candidates. Conventional
wisdom says Kerry can't win without Pennsylvania, which suggests
he should concentrate all his energy there. But doing that would
leave Florida and Ohio undefended and make it easier for Bush to
win both. Maybe Kerry should foray into Ohio too, which might lead
Bush to try to pick off Pennsylvania, which might divert his
campaign's energy from Florida just enough for Kerry to snatch it
away. ... You see the difficulty: As in any tactical problem, the best
thing for Kerry to do depends on what Bush does, and the best thing
for Bush to do depends on what Kerry does. At times like this, the
division of mathematics that comes to our aid is game theory.
29
Game Theory and Politics (cont.)
To simplify our problem, let's suppose it's the weekend
before Election Day and each candidate can only schedule one
more visit. We'll concede Pennsylvania to Kerry; then for Bush
to win the election, he must win both Florida and Ohio. Let's say
that Bush has a 30 percent chance of winning Ohio and a 70
percent chance at Florida. Furthermore, we'll assume that Bush
can increase his chances by 10 percent in either state by
making a last-minute visit there, and that Kerry can do the
same. If Bush and Kerry both visit the same state, then Bush's
chances remain 30 percent in Ohio and 70 percent in Florida,
and his chance of winning the election is 0.3 x 0.7, or 21
percent. If Bush visits Ohio and Kerry goes to Florida, Bush has
a 40 percent chance in Ohio and a 60 percent chance in
Florida, giving him a 0.4 x 0.6, or 24 percent chance of an
overall win. Finally, if Bush visits Florida and Kerry visits Ohio,
Bush's chances are 20 percent and 80 percent, and his chance
30
of winning drops to 16 percent.
Example 3: Bush and Kerry
Kerry’s options
Bush’s dominant
strategy is to visit Ohio.
Bush’s
Ohio
Options
.3*.7
Florida
.2*.8
Ohio
Florida
21% , 79%
24% , 76%
.4*.6
16% , 84%
21% , 79%
.3*.7
Nash Equilibrium: (Ohio, Ohio)
31
EXAMPLE 4: Entry into a fast food market:
Is there a Nash Equilibrium(ia)?
Yes, there are 2 – (Enter,
Burger King’s options
Don’t Enter) and (Don’t
Enter, Enter). Implies, no
Enter
Don’t Enter
need for a dominant
Skaneateles
Skaneateles
strategy to have NE.
McDonalds’ Enter Skaneateles
Options
Don’t Enter
Skaneateles
PBK = -40
PM = -30
PBK = 0
PM = 50
PBK = 40
PM = 0
PBK = 0
PM = 0
Is there a dominant strategy for BK? NO
Is there a dominant strategy for McD? NO
32
Example 5: Cournot Example from Last
Class
Nash Equilibrium is
90 Q2
Q1=26.67 and Q2=26.6
80
70
r1(Q2)
60
Do Firms have a dominant
Strategy? No, output that maximizes
50
40
profits depends on output
of other firm.
30
r2(Q1)
20
10
26.67
Q1
100
90
80
70
60
50
40
30
20
10
0
0
26.67
33
EXAMPLE 6: Monitoring Workers
Is there a Nash Equilibrium(ia)?
Not a pure strategy Nash Equilibrium–
Worker’s1 options
player chooses to take one action with probability
Randomize the actions yields a Nash = mixed strategy
John Nash proved an equilibrium
alwaysShirk
exists
Work
Manager’s Monitor
Options
Don’t Monitor
W: 1
M: -1
W: -1
M: 1
W: -1
M: 1
W: 1
M: -1
Is there a dominant strategy for the worker? NO
Is there a dominant strategy for the manager? NO
34
Mixed (randomized) Strategy

Definition:
A strategy whereby a player
randomizes over two or more
available actions in order to keep
rivals from being able to predict his
or her actions.
35
Calculating Mixed Strategy
EXAMPLE 6: Monitoring Workers
Manager randomizes (i.e. monitors with
probability PM) in such a way to make the
worker indifferent between working and
shirking.
 Worker randomizes (i.e. works with
probability Pw) in such a way as to make
the manager indifferent between
monitoring and not monitoring.

36
Example 6: Mixed Strategy
Worker’s options
Work
Shirk
PW
Manager’s Monitor
Options
PM
Don’t Monitor
1-PM
1-PW
W: 1
M: -1
W: -1
M: 1
W: -1
M: 1
W: 1
M: -1
37
Manager selects PM to make Worker
indifferent between working and
shirking (i.e., same expected payoff)
Worker’s expected payoff from working
PM*(1)+(1- PM)*(-1) = -1+2*PM
 Worker’s expected payoff from shirking
PM*(-1)+(1- PM)*(1) = 1-2*PM

Worker’s expected payoff the same from working and
shirking if PM=.5. This expected payoff is 0 (-1+2*.5=0
and 1-2*.5=0). Therefore, worker’s best response is to
either work or shirk or randomize between working
38
and shirking.
Worker selects PW to make Manager indifferent
between monitoring and not monitoring.
Manager’s expected payoff from monitoring
PW*(-1)+(1- PW)*(1) = 1-2*PW
 Manager’s expected payoff from not
monitoring
PW*(1)+(1- PW)*(-1) = -1+2*PW

Manager’s expected payoff the same from monitoring
and not monitoring if PW=.5. Therefore, the manager’s
best response is to either monitor or not monitor or
randomize between monitoring or not monitoring .
39
Nash Equilibrium of Example 6
Worker works with probability .5 and
shirks with probability .5 (i.e., PW=.5)
 Manager monitors with probability .5 and
doesn’t monitor with probability .5 (i.e.,
PM=.5)

Neither the Worker nor the Manager can increase
their expected payoff by playing some other
strategy (expected payoff for both is zero). They
are both playing a best response to the other
player’s strategy.
40
Example 6A: What if costs of
Monitoring decreases and Changes
the Payoffs for Manager
Worker’s options
Manager’s Monitor
Options
Don’t Monitor
Work
Shirk
W: 1
M: -1
W: -1
M: 1 1.5
W: -1
M: 1
-.5
W: 1
M: -1
41
Nash Equilibrium of Example 6A
where cost of monitoring decreased
Worker works with probability .625 and
shirks with probability .375 (i.e., PW=.625)
 Same as in Ex. 5, Manager monitors with
probability .5 and doesn’t monitor with
probability .5 (i.e., PM=.5)

The decrease in monitoring costs does not change
the probability that the manager monitors. However, it
increases the probability that the worker works.
42
Example 7

A Beautiful Mind
http://www.youtube.com/watch?v=CemLiSI5ox8
43
Example 7: A Beautiful Mind
Other Student’s Options
Pursue
Blond
John
Nash’s
Pursue
Blond
Options
Pursue
Pursue
Brunnette 1 Brunnette 2
0,0
100 , 50
100 , 50
Pursue
Brunnette 1
50 , 100
0,0
50 , 50
Pursue
Brunnette 2
50 , 100
50 , 50
0,0
Nash Equilibria: (Pursue Blond, Pursue Brunnette 1)
(Pursue Blond, Pursue Brunnette 2)
(Pursue Brunnette 1, Pursue Blond)
(Pursue Brunnette 2, Pursue Blond)
44
Sequential/Multi-Stage Games
Extensive form game: A representation of
a game that summarizes the players, the
information available to them at each
stage, the strategies available to them, the
sequence of moves, and the payoffs
resulting from alternative strategies.
(Often used to depict games with sequential
play.)

45
Example 8
Potential Entrant
Don’t Enter
Enter
Incumbent Firm
Potential Entrant:
0
Incumbent:
+10
Price War
(Hard)
Potential Entrant:
Incumbent:
Share Market
(Soft)
-1
+1
What are the Nash Equilibria?
+5
+5
46
Nash Equilibria
1.
(Potential Entrant Enter,
Incumbent Firm Shares Market)
2.
(Potential Entrant Don’t Enter,
Incumbent Firm Price War)
Is one of the Nash Equilibrium more likely to
occur? Why?
Perhaps (Enter, Share Market)
because it doesn’t rely on a noncredible threat.
47
Subgame Perfect Equilibrium
A condition describing a set of strategies
that constitutes a Nash Equilibrium and
allows no player to improve his own payoff
at any stage of the game by changing
strategies.
(Basically eliminates all Nash Equilibria that
rely on a non-credible threat – like Don’t
Enter, Price War in Prior Game)

48
Example 8
Potential Entrant
Don’t Enter
Enter
Incumbent Firm
Potential Entrant:
0
Incumbent:
+10
Price War
(Hard)
Potential Entrant:
Incumbent:
Share Market
(Soft)
-1
+1
+5
+5
What is the Subgame Perfect Equilibrium?
(Enter, Share Market)
49
Big Ten Burrito
Example 9
Enter
Don’t Enter
Chipotle
Enter
BTB: -25
Chip: -50
Chipotle
Don’t
Enter
Enter
+40
0
0 +70
Don’t
Enter
0
0
50
Big Ten Burrito
Enter
Don’t Enter
Chipotle
Enter
BTB: -25
Chip: -50
Chipotle
Don’t
Enter
Enter
+40
0
0 +70
Don’t
Enter
0
0
Use Backward Induction to Determine
Subgame Perfect Equilibrium.
51
Subgame Perfect Equilibrium
Chipotle should choose Don’t Enter if BTB chooses
Enter and Chipotle should choose Enter if BTB chooses
Don’t Enter.
BTB should choose Enter given Chipotle’s strategy
above.
Subgame Perfect Equilibrium:
(BTB chooses Enter, Chipotle chooses Don’t Enter if
BTB chooses Enter and Enter if BTB chooses Don’t
Enter.)
52
U.S. Postal Service and Anthrax
Is Mail Safer Since Anthrax Attacks?
Questions Remain About Post Office Security 5 Years After 5 Died
HAMILTON, N.J., Sept. 23, 2006 Five years ago next
week, American officials began to suspect that
someone was sending anthrax-tainted letters through
the mail. Five people eventually died and 17 other
became ill as a result. The attacks remain unsolved,
but there have been some security upgrades to the
nation's postal system.
The question remains: are we any safer?
The U.S. Postal Service's Tom Day helped design the
system that now tests for anthrax at all 280 mail
processing centers across the country. He gave CBS
News correspondent Bianca Solarzano a tour of the
John K. Rafferty Hamilton Post Office Building.
53
U.S. Postal Service and Anthrax (cont.)
"This was the first spot where the anthrax was coming out
of the envelopes," Day said, pointing to a mail sorting
machine.
There has been a tunnel-like addition to the machine
where letters collected from mail boxes are checked for
anthrax.
"If anything is escaping from an envelope at this point,
we're collecting it and pulling it out through a system
right here," Day said. "That, then, goes to this box which
is the self contained detection system."
The system's cost: $150 million per year.
So, after all the improvements, is our mail safe?
"I would definitely say the mail in this country is safe,"
Day said. "Can I give a 100 percent guarantee? The
answer is 'no.'"
54
US Postal Service
Example 10
Buy Protector
Don’t Buy Protector
Unstable Person
Unstable Person
Send
Don’t Send
Anthrax
Anth
USPS:
Person:
-600
-10
Subgame Perfect Equilibrium:
Send
Anth
-400 -1000
0
+10
Don’t Send
Anthrax
0
0
(US Postal Service Buys Protector;
Unstable Person Doesn’t Send Anthrax if USPS Buys Protector and Sends Anthrax if USPS
55
Doesn’t Buy Protector)
Example 11
Slide from Oligopoly Lecture
100
Firm 1’s Profits = 60*20-20*20=800
90
80
Firm 2’s Profits = 60*20-20*20=800
70
60
50
D
40
30
MC=AVC=ATC
Q
20
10
0
0
10
20
30
40
50
60
70
80
90
100
If firms collude on Q1=20 and Q2=20
56
Example 11
Slide from Oligopoly Lecture
100
Firm 1’s Profits = 50*30-20*30=900
90
Firm 2’s Profits = 50*20-20*20=600
80
70
60
50
D
40
30
MC=AVC=ATC
Q
20
10
0
0
10
20
30
40
50
60
70
80
90
100
Firms colluding is unlikely if they interact once because firms have
incentive to cheat – in above case Firm 1 increases profits by cheating
57
and producing 30 units.
Slide From Oligopoly Lecture
1.
Repeated Interaction
Suppose Firm 1 thinks Firm 2 won’t deviate from Q2=20 if Firm 1
doesn’t deviate from collusive agreement of Q1=20 and Q2=20. In
addition, Firm 1 thinks Firm 2 will produce at an output of 80 in all
future periods if Firm 1 deviates from collusive agreement of Q1=20
and Q2=20.
Firm 1’s profits from not cheating
Today
In 1 Year
In 2 Years
In 3 Years
In 4 Years
800
800
800
800
800
…
Firm 1’s profits from cheating (by producing Q1=30 Today)
Today
In 1 Year
In 2 Years
In 3 Years
In 4 Years
900
0
0
0
0
…
Does Firm 2’s Strategy Rely on a Non-credible Threat?
Depends on Game –unlikely to be credible even if infinitely
repeated game
58
What if Firms interact for 2 periods as
Cournot Competitors? What is
Subgame Perfect Equilibrium?
Use Backward Induction!!
In the second period, what will happen?
59
Cournot Equilibrium: IN 2ND PERIOD!!!!
Q1=26.67 and Q2=26.67
90
Q2
80
70
r1(Q2)
60
50
40
30
r2(Q1)
20
10
26.67
Q1
100
90
80
70
60
50
40
30
20
10
0
0
26.67
60
Profits from Cournot Equilibrium:
Q1=26.67 and Q2=26.67 so Q=Q1+Q2=53.3
100
Firm 1 Profits=46.66*26.67-20*26.67= 713
90
80
Firm 2 Profits=46.66*26.67-20*26.67= 713
70
60
50
D
46.6640
30
MC =AVC=ATC
Q
20
10
0
0
10
20
30
40
50
60
70
80
90
100
53.33
61
In the 1st period, what will
happen?
If both firms realize that each will produce an
output of 26.67 in the 2nd period (resulting in
profits of $713 for each firm) no matter what
occurs in the 1st period, then the equilibrium
the 1st period should be for both firms to
produce 26.67 and obtain profits of $713 the
1st period.
Using this logic, the Subgame Perfect
Equilibrium is for each firm to produce 26.67
units of output the 1st period and 26.67 units of
62
output the 2nd period.
What if Firms interact for 1000
periods as Cournot Competitors?
What is Subgame Perfect
Equilibrium?
Using similar logic as when the
firms interact 2 periods, the
Subgame Perfect Equilibrium is
for each firm to produce 26.67
units of output each period.
63
Do you really expect this type of
outcome if the firms interact 1000
periods?
Laboratory experiments suggest
that when facing a player a finite
number of times, the players will
“collude” for a number of periods.
Many of these experiments involve a
prisoners dilemma game being
played a finite number of times.
64
In the real world, how do firms (and
individuals) and individuals address
the finite period problem?
Attempt to build in
uncertainty associated
with when the final period
occurs.
Attempt to “change game”.
65
Example 12: The Hold-Up Problem
Dan Conlin
Invest in Firm
Specific Knowledge
Don’t Invest
Dan Conlin
Dan Conlin
and M&M
and M&M
negotiate
negotiate
salary
salary
Dan Conlin:
wI-CI
wDI
Marsh&McClennan:
200-wI
150-wDI
Let wI and wDI denote Dan’s wage if he invests and doesn’t
invest in the firm specific knowledge, respectively. Let the
cost of investing for Dan be CI and let CI=30. Dan Conlin is
worth 200 to M&M if he invests and is worth 150 if he
66
doesn’t.
Example 12: The Hold-Up Problem
Dan Conlin
Invest in Firm
Specific Knowledge
Dan Conlin
and M&M
negotiate
salary
Dan Conlin:
wI-CI
Marsh&McClennan:
200-wI
Don’t Invest
Dan Conlin
and M&M
negotiate
salary
wDI
150-wDI
Assume that Dan’s best “outside option” is a wage of 100
whether or not he invests in the firm specific knowledge and
that the outcome of the negotiations are such that Dan and
67
M&M split the surplus. This means that wI=150 and wDI=125.
Example 12: The Hold-Up Problem
Dan Conlin
Invest in Firm
Specific Knowledge
Don’t Invest
Dan Conlin
Dan Conlin
and M&M
and M&M
negotiate
negotiate
salary
salary
Dan Conlin:
wI-CI=150-30
wDI=125
Marsh&McClennan:
200-wI =200-150
150-wDI=150-125
Subgame Perfect Equilibrium outcome has Dan Conlin not
investing in the firm specific knowledge and receiving a wage
of 125 even though the cost of the knowledge is 30 and it
68
increases his value to the firm by 50.
Example 12: The Hold-Up Problem
Dan Conlin
Invest in Firm
Specific Knowledge
Don’t Invest
Dan Conlin
Dan Conlin
and M&M
and M&M
negotiate
negotiate
salary
salary
Dan Conlin:
wI-CI=150-30
wDI=125
Marsh&McClennan:
200-wI =200-150
150-wDI=150-125
What would you expect to happen in this case?
Dan Conlin and M&M would divide cost of obtaining
the knowledge.
69
Example 13: General Knowledge Investment
Dan Conlin
Invest in
General Knowledge
Don’t Invest
Dan Conlin
Dan Conlin
and M&M
and M&M
negotiate
negotiate
salary
salary
Dan Conlin:
wI-CI=160-30
wDI =125
Marsh&McClennan:
200-wI =200-160
150-wDI=150-125
Assume the game is as in the “hold-up” problem but that
Dan’s best “outside option” is a wage of 120 if he invests in
general knowledge and 100 if he does not. This means that
70
wI=160 and wDI=125 (assuming split surplus when negotiate).
Example 13: General Knowledge Investment
Dan Conlin
Invest in
General Knowledge
Don’t Invest
Dan Conlin
Dan Conlin
and M&M
and M&M
negotiate
negotiate
salary
salary
Dan Conlin:
wI-CI=160-30
wDI =125
Marsh&McClennan:
200-wI =200-160
150-wDI=150-125
Subgame Perfect Equilibrium outcome has Dan Conlin
investing in the general knowledge and receiving a wage of
160.
71
Example 14: Hold-up Problem (same idea
as the Fisher Auto-body / GM situation)
Suppose there are two players: a computer chip maker (MIPS) and a
computer manufacturer (Silicon Graphics). Initially, MIPS decides
whether or not to customize its chip (the quantity of which is
normalized to one) for a specific manufacturing purpose of Silicon
Graphics. The customization costs $75 to MIPS, but adds value of
$100 to the chip only when it is used by Silicon Graphics . The value
of customization is partially lost when the chip is sold to an alternative
buyer, who is willing to pay $60. If MIPS decides not to customize
the chip, it can sell a standardized chip to Silicon Graphics at a price
of zero and Silicon Graphics earns a payoff of zero from using the
chip. If MIPS customizes the chip, the two players enter into a
bargaining game where Silicon Graphics makes a take-it-or-leave-it
price offer to MIPS. In response to this, MIPS can either accept the
offer (in which case the game ends) or reject it (in which case MIPS
approaches an alternative buyer who pays $60).
72
Example 14: Hold-Up Problem
MIPS
Don’t Customize
Customize
Silicone Graphics
0 : MIPS
0 : Silicon Graphics
Offer
Price p
MIPS
Accept
MIPS:
p-75
Silicon Graphics: 100-p
Reject
60-75= -15
0
Subgame Perfect Equilibrium – MIPS accepts price p if p>60.
Silicone Graphics offers a price p=60. MIPS does not
customize. The outcome of this game is that MIPS does not
73
customize even though there is a surplus of $25 to be gained.
Is the Hold-Up Problem Applicable
to other Situations? YES
1.
2.
3.
4.
5.
Upstream Firm Investing in Specific Capital to produce
input for Downstream Firm.
Coal Mines located next to Power Plants.
An academic buying a house before getting tenure or a
big promotion.
Taxing of Oil and Gas Lines by local jurisdictions.
Multinational firms operating in foreign countries
(Foreign Direct Investment)
East Lansing Public Schools allocating a certain
amount of money for capital expenditures and a
certain amount for operating expenditures
74
Using Game Theory to Devise
Strategies in Oligopolies that
Increase Profits
Examples:
1.
Price Matching- advertise a price and promise to
match any lower price offered by a competitor.
100
Bertrand Oligopoly
90
80
In the end, you would
expect both firms to set a
price of $20 (equal to MC)
and have zero profits.
70
60
50
D
40
30
MC
Q
20
10
0
0
10
20
30
40
50
60
70
80
90
100
75
Using Game Theory to Devise
Strategies in Oligopolies that
Increase Profits
Examples:
1. Price Matching- advertise a price and promise
to match an lower price offered by a
competitor. In Bertrand example, perhaps each
firm would set a price of $60 and say will
match.
2. Induce Brand Loyalty – frequent flyer program
3. Randomized pricing – inhibits consumers
learning as to who offers lower price and
reduces ability of competitors to undercut
price.
76