Chapters 29 and 30
Game Theory and Applications
Game Theory
0 Game theory applied to economics by John Von Neuman and Oskar Morgenstern
0 Game theory allows us to analyze different social and economic situations
GAME THEORY
0 Game theory is the study of how people behave in strategic situations.
0 Strategic decisions are those in which each person, in deciding what actions to take, must consider how others might respond to that action.
Games of Strategy Defined
0 Interaction between agents can be represented by a game, when the rewards to each depends on his actions as well as those of the other player
0 A game is comprised of 0 players0 Order to play0 strategies0 Chance 0 Information 0 Payoff
4
0A Nash equilibrium is a situation in which economic actors interacting with one another each choose their best strategy given the strategies that all the others have chosen.
0Each agent is satisfied with (i.e., does not want to change) his strategy (or action) given the strategies of all other agents.
The Nash Equilibrium
John Forbes Nash, Jr. June 13, 1928 --
Simultaneous Games
0 Players choose at the same time and therefore do not know the choices of the other.
0 Also called imperfect information games0 Examples:
0 Rock – paper – scissors0 Cournot competition
Example: Coordination gameAnn’ s Decision
Ballet
Ann gets 8
Jane gets 8
Ann gets 0
Jane gets 0
Ann gets 0
Jane gets 0
Ann gets 10
Jane gets 10
Opera
Jane’sDecision
Ballet Opera
Example 3: The Prisoners’ Dilemma
0 The prisoners’ dilemma provides insight into the difficulty of maintaining cooperation.
0 Often people (firms) fail to cooperate with one another even when cooperation would make them better off.
The Prisoners’ Dilemma
0 The prisoners’ dilemma is a particular “game” between two captured prisoners that illustrates why cooperation is difficult to maintain even when it is mutually beneficial.
The Prisoners’ Dilemma0 Two people committed a crime and are being interrogated
separately.0 They are offered the following:
0 If both confessed, each spends 8 years in jail.0 If both remained silent, each spends 1 year in jail.0 If only one confessed, he will be set free while the other spends
20 years in jail.
Example 1: The Prisoners’ Dilemma Game
Ben’ s Decision
Confess
Confess
Ben gets 8 years
Kyle gets 8 years
Ben gets 20 years
Kyle goes free
Ben goes free
Kyle gets 20 years
Ben gets 1 year
Kyle gets 1 year
Remain Silent
RemainSilent
Kyle’sDecision
Representing Games
0 Games can be represented in 0 Bi matrix/ normal form0 Game tree/ extensive form
Toshiba IBM software game
0 Toshiba and IBM are choosing between two operating systems: UNIX or DOS
0 The two firms move at the same time0 Imperfect information
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Toshiba
DOS UNIX
IBMDOS 600, 200 100, 100
UNIX 100, 100 200, 600
Game of imperfect informationIn normal form
Game of imperfect information
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Toshiba does not know whether IBM moved to the left or to the right, i.e., whether it is located at node 2 or node 3.
1
2 3
IBM
ToshibaToshiba
UNIXDOS
UNIXDOS UNIXDOS
600200
100100
100100
200600
In extensive form
Information set
Toshiba’s strategies:• DOS• UNIX
Sequential Games
0 Players move sequentially 0 A player knows all actions chosen before his move0 Also called perfect information games0 Examples:
0 Chess0 Stackelberg competition
Game of perfect information
17Player 2 (Toshiba) knows whether player 1 (IBM) moved to the left or to the right. Therefore, player 2 knows at which of two nodes it is located
1
2 3
IBM
ToshibaToshiba
UNIXDOS
UNIXDOS UNIXDOS
600200
100100
100100
200600
In extensive form
Strategy
0 A player’s strategy is a plan of action for each of the other player’s possible actions
How to define strategies in sequential games
A strategy is a plan of action for all possible outcomes/ choices made by the previous players0 IBM:
0 Play DOS 0 Play UNIX
0 Toshiba0 Play DOS if he plays DOS and UNIX if he plays UNIX0 Play UNIX if he plays DOS and DOS if he plays UNIX0 Play DOS if he plays DOS and DOS if he plays UNIX0 Play UNIX if he plays DOS and UNIX if he plays UNIX
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Toshiba
(DOS | DOS,DOS | UNIX)
(DOS | DOS,UNIX | UNIX)
(UNIX | DOS,UNIX | UNIX)
(UNIX | DOS,DOS | UNIX)
IBMDOS 600, 200 600, 200 100, 100 100, 100
UNIX 100, 100 200, 600 200, 600 100, 100
Game of perfect informationIn normal form
Note that DOS | UNIX is read as I will play DOS if I observe him play UNIX
Equilibrium for GamesNash Equilibrium
0 Equilibrium 0 state/ outcome0 Set of strategies0 Players – don’t want to change behavior 0 Given - behavior of other players
0 Noncooperative games0 No possibility of communication or binding
commitments
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Nash Equilibria
chosen is *s when i player to payoff
i player of choicestrategy
choicesstrategy ofarray -
i
),...,(
),...,(*
**1
**1
n
*i
n
ss
s
sss
22
ii
nini
n
Ss
ssssss
sss
in all for
If
mequilibriu Nash a is -
ii
ˆ
),...,ˆ,...,(),...,,...,(
),...,(***
1***
1
**1
23
Toshiba
DOS UNIX
IBMDOS 600, 200 100, 100
UNIX 100, 100 200, 600
The strategy pair DOS DOS is a Nash equilibrium as well as UNIX, UNIX
Nash Equilibrium: Toshiba-IBMimperfect Info game
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Toshiba
(DOS | DOS,DOS | UNIX)
(DOS | DOS,UNIX | UNIX)
(UNIX | DOS,UNIX | UNIX)
(UNIX | DOS,DOS | UNIX)
IBMDOS 600, 200 600, 200 100, 100 100, 100
UNIX 100, 100 200, 600 200, 600 100, 100
0 Three Nash equilibria. The following outcomes satisfy the Nash criteria:0 IBM plays DOS, Toshiba plays DOS regardless0 IBM plays DOS, Toshiba matches IBM’s choice0 IBM plays UNIX, Toshiba plays UNIX regardless
Nash Equilibrium: Toshiba-IBMperfect Info game
Nash equilibrium: Coordinating numbers game
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Player 2
1 2 3 4 5 6 7 8 9 10
Player 1
1 1, 1 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0
2 0, 0 2, 2 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0
3 0, 0 0, 0 3, 3 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0
4 0, 0 0, 0 0, 0 4, 4 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0
5 0, 0 0, 0 0, 0 0, 0 5, 5 0, 0 0, 0 0, 0 0, 0 0, 0
6 0, 0 0, 0 0, 0 0, 0 0, 0 6, 6 0, 0 0, 0 0, 0 0, 0
7 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 7, 7 0, 0 0, 0 0, 0
8 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 8, 8 0, 0 0, 0
9 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 9, 9 0, 0
10 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 10, 10
Nash equilibrium: War game
A game with no equilibria in pure strategies
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General 2
Retreat Attack
General 1 Retreat 5, 8 6, 6
Attack 8, 0 2, 3
Nash Equilibrium: The “I Want to Be Like Mike” Game
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Dave
Wear red Wear blue
Michael Wear red (-1, 2) (2, -2)
Wear blue (1, -1) (-2, 1)
A game with no equilibria in pure strategies
Dominant Strategy Equilibria
0 Strategy A dominates strategy B if0 A gives a higher payoff than B 0 No matter what opposing players do
0 Dominant-strategy equilibrium0 All players play their dominant strategies
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Oligopoly Game
29
General Motors
High price Low price
FordHigh price 500, 500 100, 700
Low price 700, 100 300, 300
0 Ford has a dominant strategy to price low 0 If GM prices high, Ford is better of pricing low0 If GM prices low, Ford is better of pricing low
Oligopoly Game
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General Motors
High price Low price
FordHigh price 500, 500 100, 700
Low price 700, 100 300, 300
0 Similarly for GM0 The Nash equilibrium is Price low, Price low
An Equilibrium Refinement
0 Analyzing games in bi-matrix form may result in equilibria that are less satisfactory
0 These equilibria involve a non credible threat0 The Sub Game Perfect Nash Equilibrium is a Nash
equilibrium that involves credible threats only 0 It can be obtained by solving the game in extensive
form using backward induction
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32
Toshiba
(DOS | DOS,DOS | UNIX)
(DOS | DOS,UNIX | UNIX)
(UNIX | DOS,UNIX | UNIX)
(UNIX | DOS,DOS | UNIX)
IBMDOS 600, 200 600, 200 100, 100 100, 100
UNIX 100, 100 200, 600 200, 600 100, 100
Non credible threats: IBM-ToshibaIn normal form
0 Three Nash equilibria0 Some involve non credible threats.0 Example IBM playing UNIX and Toshiba playing UNIX
regardless:0 Toshiba’s threat is non credible
Backward induction
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1
2 3
IBM
ToshibaToshiba
UNIXDOS
UNIXDOS UNIXDOS
600200
100100
100100
200600
Subgame perfect Nash Equilibrium
0 Subgame perfect Nash equilibrium is0 IBM: DOS0 Toshiba: if DOS play DOS and if UNIX play UNIX
0 Toshiba’s threat is credible0 In the interest of Toshiba to execute its threat
Rotten kid game
0 The kid either goes to Aunt Sophie’s house or refuses to go
0 If the kid refuses, the parent has to decide whether to punish him or relent
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Player 2 (a parent)
(punish if the kid refuses)
(relent if the kid refuses)
Player 1(a difficult
child)
Left(go to Aunt Sophie’s House)
1, 1 1, 1
Right(refuse to go to Aunt Sophie’s House)
-1, -1 2, 0
Rotten kid game in extensive form
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• The sub game perfect Nash equilibrium is: Refuse and Relent if refuse• The other Nash equilibrium, Go and Punish if refuse, relies on a non
credible threat by the parent
Kid
Parent
RefuseGo to Aunt Sophie’s House
Relent if refuse
Punish if refuse
-1-1
20
11
1
2
Application 1: Collusive Duopoly
0Example: The European voluntary agreement for washing machines in 1998
0The agreement requires firms to eliminate from the market inefficient models
0Ahmed and Segerson (2011) show that the agreement can raise firm profit, however, it is not stable Firm 2
eliminate Keep
Firm 1eliminate $1,000 $1,000 $200 $1,200
keep $1,200 $200 $500 $500
Application 2: Wal-Mart and CFL bulbs market
0 In 2006 Wal-Mart committed itself to selling 1 million CFL bulbs every year
0 This was part of Wal-Mart’s plan to become more socially responsible
0 Ahmed(2012) shows that this commitment can be an attempt to raise profit.
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1
2 3
Wal-Mart
Small firmSmall firm
Do not commitCommit to output target
Do notCommit Do notCommit
9045
50040
8060
10050
When the target is small
The outcome is similar to a prisoners dilemma
Application 2: Wal-Mart and CFL bulbs market
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1
2 3
Wal-Mart
Small firmSmall firm
Do not commitCommit to output target
Do notCommit Do notCommit
8030
50035
90100
10050
When the target is large
When the target is large enough, we have a game of chicken
Application 2: Wal-Mart and CFL bulbs market