game theory
TRANSCRIPT
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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 2 / 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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 2 / 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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 2 / 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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 2 / 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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 2 / 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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 2 / 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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 2 / 22
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
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 3 / 22
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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 4 / 22
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
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 5 / 22
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
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 6 / 22
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
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 7 / 22
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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 8 / 22
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
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 9 / 22
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
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 10 / 22
In�nitely Repeated Games
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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 11 / 22
In�nitely Repeated Games
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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 11 / 22
In�nitely Repeated GamesCheat and Cooperate
Player BStrategy Low Price High Price
Player A Low Price 0, 0 50,�40High Price �40, 50 10, 10
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
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 12 / 22
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
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 13 / 22
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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 14 / 22
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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 14 / 22
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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 14 / 22
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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 14 / 22
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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 14 / 22
Finitely Repeated GamesGames with an Uncertain Final Period
Player BStrategy Low Price High Price
Player A Low Price 0, 0 50,�40High Price �40, 50 10, 10
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
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 15 / 22
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
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 16 / 22
Finitely Repeated GamesGames with a Known Final Period
Player BStrategy Low Price High Price
Player A Low Price 0, 0 50,�40High Price �40, 50 10, 10
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 17 / 22
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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 18 / 22
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.
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 18 / 22
Multistage Games
A
B
B
Up
Down
Up
Down
Up
Down
(10,15)
(5,5)
(0,0)
(6,20)
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 19 / 22
Multistage Games
A
BIn
Out
Hard
Soft
(-1,1)
(5,5)
(0,10)
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 20 / 22
Multistage Games
A
BIntroduce
Don’tIntroduce
Clone
Don’tClone
(-5,20)
(100,0)
(1,1)
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 21 / 22
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
Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 22 / 22