Game Theory

Game Theory

Sherif Khalifa, Ph.D.Department of Economics

California State University, Fullerton

Fall 2007

Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 1 / 22

Introduction

Game theory is a framework to aid in decision making when yourpayo¤ depends on the actions taken by other players.

Strategies are planned decisions of the players.

The payo¤s of the players are the pro�ts or losses that result from thestrategies, and that depend not only on that player�s strategies butalso on the strategies employed by other players.

Simultaneous move game is a game in which each player makesdecisions without knowledge of the other player�s decisions.

Sequential move game is a game in which one player makes a moveafter observing the other player�s move.

One shot game is a game that is played only once.

Repeated game is a game that is played more than once.


Introduction









Introduction









Introduction









Introduction









Introduction









Introduction









Simultaneous Move, One Shot Games

A strategy is a decision rule that describes the actions a player willtake at each decision point.

The normal form representation of a game indicates the players in thegame, the possible strategies of the players, and the payo¤s to theplayers that will result from alternative strategies.

Player BStrategy Left Right

Player A Up 10, 20 15, 8Down �10, 7 10, 10


Simultaneous Move, One Shot Games

A strategy is a dominant strategy if it results in the highest payo¤regardless of the action of the opponent.

A secure strategy is one that guarantees the highest payo¤ given theworst possible scenario.

Nash equilibrium is a condition describing a set of strategies in whichno player can improve her payo¤ by unilaterally changing her ownstrategy, given the other players�strategies.


Applications of One-Shot GamesPricing Decisions

Player BStrategy Low Price High Price

Player A Low Price 0, 0 50,�10High Price �10, 50 10, 10


Applications of One-Shot GamesAdvertising and Quality Decisions

Firm BStrategy Advertise Do not Advertise

Firm A Advertise 4, 4 20, 1Do not Advertise 1, 20 10, 10


Applications of One-Shot GamesCoordination Decisions

Firm BStrategy 120 volt 90 volt

Firm A 120 volt 100, 100 0, 090 volt 0, 0 100, 100


Applications of One-Shot GamesMonitoring Employees Decisions

WorkerStrategy Work Shirk

Manager Monitor �1, 1 1,�1Do not Monitor 1,�1 �1, 1

Mixed strategy is a strategy whereby a player randomizes over two or moreavailable actions in order to keep rivals from being able to predict his orher action.


Applications of One-Shot GamesNash Bargaining

Labor UnionStrategy 0 50 100

Management 0 0, 0 0, 50 0, 10050 50, 0 50, 50 �1,�1100 100, 0 �1,�1 �1,�1


In�nitely Repeated Games

Is a game that is played over and over again forever.

PVFirm = π0 +π11+ i

+π2

(1+ i)2+ .......... =

∞

∑t=0

πt

(1+ i)t

If πt = π, then

PVFirm = π +π

1+ i+

π

(1+ i)2+ .......... =

∞

∑t=0

π

(1+ i)t=

�1+ ii

�π



A trigger strategy is a strategy that is contingent on the past plays ofplayers in a game, and in which some particular past action triggers adi¤erent action by a player.

A player who adopts a trigger strategy continues to choose the sameaction until some other player takes an action that triggers a di¤erentaction by the �rst player.



A trigger strategy is a strategy that is contingent on the past plays ofplayers in a game, and in which some particular past action triggers adi¤erent action by a player.

A player who adopts a trigger strategy continues to choose the sameaction until some other player takes an action that triggers a di¤erentaction by the �rst player.


In�nitely Repeated GamesCheat and Cooperate



PV CheatFirm = 50+ 0+ 0+ .......... = 50

PV CooperateFirm = 10+101+ i

+10

(1+ i)2+ .......... =

10 (1+ i)i

PV CheatFirm = 50 � 10 (1+ i)i

= PV CooperateFirm


In�nitely Repeated GamesCheat and Cooperate

Example

Assume i = 0.4. If the �rm does cheat it earns $40 this period and zeroafterwards. If it does not cheat, it earns $10 every period. Will the �rmcheat or collude?

PV CheatFirm = 40+ 0+ 0+ .......... = 40

PV CooperateFirm = 10+10

1+ 0.4+

10

(1+ 0.4)2+ .......... =

10 (1+ 0.4)0.4

= 35

PV CheatFirm = 40 > 35 = PV CooperateFirm


In�nitely Repeated GamesFactors A¤ecting Collusion

Collusion is easier when there are few �rms rather than many, as thetotal number of monitors needed in the market grow as the number of�rms increases.

If the number of �rms is large, the monitoring costs become so highrelative to collusive pro�ts.

Monitoring costs constitute a greater share of total costs for smallthan large �rms.

Tacit collusion occurs when the �rms do not explicitly conspire tocollude but accomplish collusion indirectly.

In a single price market, the cost of punishing an opponent is higherthan in markets in which di¤erent customers are quoted di¤erentprices.






























Finitely Repeated GamesGames with an Uncertain Final Period



Assume the interest rate=0.

ΠCheatFirmA = 50+ 0+ 0+ .......... = 50

ΠCoopFirmA = 10+ (1�Θ) 10+ (1�Θ)2 10+ .......... =

10Θ

ΠCheatFirmA = 50 �

10Θ= ΠCoop

FirmA


Finitely Repeated GamesGames with an Uncertain Final Period

ExampleAssume Θ = 0.1. If the �rm does cheat it earns $80 this period and zeroafterwards. If it does not cheat, it earns $10 every period. Will the �rmcheat or collude?

ΠCheatFirmA = 80+ 0+ 0+ .......... = 80

ΠCoopFirmA = 10+ (1� 0.1) 10+ (1� 0.1)

2 10+ .......... =100.1

= 100

ΠCheatFirmA = 80 � 100 = ΠCoop

FirmA


Finitely Repeated GamesGames with a Known Final Period




Multistage Games

An extensive form game summarizes the players, the informationavailable to them at each stage, the strategies available to them, thesequence of moves, and the payo¤s resulting from alternativestrategies.

Subgame perfect equilibrium is a condition describing a set ofstrategies that constitutes a Nash equilibrium and allows no player toimprove his own payo¤ at any stage of the game by changingstrategies.


Multistage Games

An extensive form game summarizes the players, the informationavailable to them at each stage, the strategies available to them, thesequence of moves, and the payo¤s resulting from alternativestrategies.

Subgame perfect equilibrium is a condition describing a set ofstrategies that constitutes a Nash equilibrium and allows no player toimprove his own payo¤ at any stage of the game by changingstrategies.


Multistage Games

A

B

B

Up

Down

Up

Down

Up

Down

(10,15)

(5,5)

(0,0)

(6,20)


Multistage Games

A

BIn

Out

Hard

Soft

(-1,1)

(5,5)

(0,10)


Multistage Games

A

BIntroduce

Don’tIntroduce

Clone

Don’tClone

(-5,20)

(100,0)

(1,1)


Multistage Games

M

U

U

$1

$50

Accept

Reject

Accept

Reject

($99,$1)

($0,$0)

($50,$50)

($0,$0)

U($1,$99)

($0,$0)Reject

$99

Accept


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Game Theory