chapters 29 and 30 game theory and applications. game theory 0 game theory applied to economics by...

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

14

Toshiba

DOS UNIX

IBMDOS 600, 200 100, 100

UNIX 100, 100 200, 600

Game of imperfect informationIn normal form

Game of imperfect information

15

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

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

21

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

24

Toshiba





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

25

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

26

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

27

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

28

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

30

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

31

32

Toshiba





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

33

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

35

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

36

• 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.

39

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


40

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


chapters 29 and 30 game theory and applications. game theory 0 game theory applied to economics by...

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