范微觀經(jīng)濟(jì)學(xué)現(xiàn)代觀點(diǎn)課件ch28-game theory

1、Chapter Twenty-EightGame TheoryGame TheoryGame theory models strategic behavior by agents who understand that their actions affect the actions of other agents. Some Applications of Game TheoryThe study of oligopolies (industries containing only a few firms)The study of cartels; e.g. OPECThe study of

2、 externalities; e.g. using a common resource such as a fishery.The study of military strategies.What is a Game?A game consists ofa set of playersa set of strategies for each playerthe payoffs to each player for every possible list of strategy choices by the players.Two-Player GamesA game with just t

3、wo players is a two-player game.We will study only games in which there are two players, each of whom can choose between only two strategies.An Example of a Two-Player GameThe players are called A and B.Player A has two strategies, called “Up” and “Down”.Player B has two strategies, called “Left” an

4、d “Right”.The table showing the payoffs to both players for each of the four possible strategy combinations is the games payoff matrix.An Example of a Two-Player GameThis is thegamespayoff matrix.Player BPlayer APlayer As payoff is shown first.Player Bs payoff is shown second.LRUD(3,9)(0,0)(1,8)(2,1

5、)An Example of a Two-Player GameE.g. if A plays Up and B plays Right then As payoff is 1 and Bs payoff is 8.This is thegamespayoff matrix.Player BPlayer ALRUD(3,9)(0,0)(1,8)(2,1)An Example of a Two-Player GameAnd if A plays Down and B plays Right then As payoff is 2 and Bs payoff is 1.This is thegam

6、espayoff matrix.Player BPlayer ALRUD(3,9)(0,0)(1,8)(2,1)An Example of a Two-Player GamePlayer BPlayer AA play of the game is a pair such as (U,R)where the 1st element is the strategychosen by Player A and the 2nd is the strategy chosen by Player B.LRUD(3,9)(0,0)(1,8)(2,1)An Example of a Two-Player G

7、ameWhat plays are we likely to see for thisgame?Player BPlayer ALRUD(3,9)(0,0)(1,8)(2,1)An Example of a Two-Player GamePlayer BPlayer AIs (U,R) alikely play?LRUD(3,9)(0,0)(1,8)(2,1)An Example of a Two-Player GamePlayer BPlayer AIf B plays Right then As best reply is Downsince this improves As payoff

8、 from 1 to 2.So (U,R) is not a likely play.Is (U,R) alikely play?LRUD(3,9)(0,0)(1,8)(2,1)An Example of a Two-Player GamePlayer BPlayer AIs (D,R) alikely play?LRUD(3,9)(0,0)(1,8)(2,1)An Example of a Two-Player GamePlayer BPlayer AIs (D,R) alikely play?If B plays Right then As best reply is Down.LRUD(

9、3,9)(0,0)(1,8)(2,1)An Example of a Two-Player GamePlayer BPlayer AIf B plays Right then As best reply is Down.If A plays Down then Bs best reply is Right.So (D,R) is a likely play.Is (D,R) alikely play?LRUD(3,9)(0,0)(1,8)(2,1)An Example of a Two-Player GamePlayer BPlayer AIs (D,L) alikely play?LRUD(

10、3,9)(0,0)(1,8)(2,1)An Example of a Two-Player GamePlayer BPlayer AIf A plays Down then Bs best reply is Right,so (D,L) is not a likely play.Is (D,L) alikely play?LRUD(3,9)(0,0)(1,8)(2,1)An Example of a Two-Player GamePlayer BPlayer AIs (U,L) alikely play?LRUD(3,9)(0,0)(1,8)(2,1)An Example of a Two-P

11、layer GamePlayer BPlayer AIf A plays Up then Bs best reply is Left.Is (U,L) alikely play?LRUD(3,9)(0,0)(1,8)(2,1)An Example of a Two-Player GamePlayer BPlayer AIf A plays Up then Bs best reply is Left.If B plays Left then As best reply is Up.So (U,L) is a likely play.Is (U,L) alikely play?LRUD(3,9)(

12、0,0)(1,8)(2,1)Nash EquilibriumA play of the game where each strategy is a best reply to the other is a Nash equilibrium.Our example has two Nash equilibria; (U,L) and (D,R).An Example of a Two-Player GamePlayer BPlayer A(U,L) and (D,R) are both Nash equilibria forthe game.LRUD(3,9)(0,0)(1,8)(2,1)An

13、Example of a Two-Player GamePlayer BPlayer A(U,L) and (D,R) are both Nash equilibria forthe game. But which will we see? Noticethat (U,L) is preferred to (D,R) by bothplayers. Must we then see (U,L) only?LRUD(3,9)(0,0)(1,8)(2,1)The Prisoners DilemmaTo see if Pareto-preferred es must be what we see i

14、n the play of a game, consider a famous second example of a two-player game called the Prisoners Dilemma.The Prisoners DilemmaWhat plays are we likely to see for thisgame?ClydeBonnie(-5,-5)(-30,-1)(-1,-30)(-10,-10)SCSCThe Prisoners DilemmaIf Bonnie plays Silence then Clydes bestreply is Confess.Clyd

15、eBonnie(-5,-5)(-30,-1)(-1,-30)(-10,-10)SCSCThe Prisoners DilemmaIf Bonnie plays Silence then Clydes bestreply is Confess.If Bonnie plays Confess then Clydesbest reply is Confess.ClydeBonnie(-5,-5)(-30,-1)(-1,-30)(-10,-10)SCSCThe Prisoners DilemmaSo no matter what Bonnie plays, Clydesbest reply is al

16、ways Confess.Confess is a dominant strategy for Clyde.ClydeBonnie(-5,-5)(-30,-1)(-1,-30)(-10,-10)SCSCThe Prisoners DilemmaSimilarly, no matter what Clyde plays,Bonnies best reply is always Confess.Confess is a dominant strategy forBonnie also.ClydeBonnie(-5,-5)(-30,-1)(-1,-30)(-10,-10)SCSCThe Prison

17、ers DilemmaSo the only Nash equilibrium for thisgame is (C,C), even though (S,S) givesboth Bonnie and Clyde better payoffs.The only Nash equilibrium is inefficient.ClydeBonnie(-5,-5)(-30,-1)(-1,-30)(-10,-10)SCSCWho Plays When?In both examples the players chose their strategies simultaneously.Such ga

18、mes are simultaneous play games.Who Plays When?But there are games in which one player plays before another player.Such games are sequential play games.The player who plays first is the leader. The player who plays second is the follower.A Sequential Game ExampleSometimes a game has more than one Na

19、sh equilibrium and it is hard to say which is more likely to occur.When such a game is sequential it is sometimes possible to argue that one of the Nash equilibria is more likely to occur than the other. A Sequential Game ExamplePlayer BPlayer A(U,L) and (D,R) are both Nash equilibriawhen this game

20、is played simultaneouslyand we have no way of deciding whichequilibrium is more likely to occur.LRUD(3,9)(0,0)(1,8)(2,1)A Sequential Game ExamplePlayer BPlayer ASuppose instead that the game is playedsequentially, with A leading and B following.We can rewrite the game in its extensive form.LRUD(3,9)

21、(0,0)(1,8)(2,1)A Sequential Game ExampleUDLLRR(3,9)(1,8)(0,0)(2,1)ABBA plays first.B plays second.A Sequential Game ExampleUDLLRR(3,9)(1,8)(0,0)(2,1)ABBA plays first.B plays second.(U,L) is a Nash equilibrium.A Sequential Game ExampleUDLLRR(3,9)(1,8)(0,0)(2,1)ABBA plays first.B plays second.(U,L) is

22、 a Nash equilibrium.(D,R) is a Nash equilibrium.Which is more likely to occur?A Sequential Game ExampleUDLLRR(3,9)(1,8)(0,0)(2,1)ABBA plays first.B plays second.If A plays U then B plays L; A gets 3.A Sequential Game ExampleUDLLRR(3,9)(1,8)(0,0)(2,1)ABBA plays first.B plays second.If A plays U then

23、B plays L; A gets 3.If A plays D then B plays R; A gets 2.A Sequential Game ExampleUDLLRR(3,9)(1,8)(0,0)(2,1)ABBA plays first.B plays second.If A plays U then B plays L; A gets 3.If A plays D then B plays R; A gets 2.So (U,L) is the likely Nash equilibrium.Pure StrategiesPlayer BPlayer AThis is our

24、original example once more.Suppose again that play is simultaneous.We discovered that the game has two Nashequilibria; (U,L) and (D,R).LRUD(3,9)(0,0)(1,8)(2,1)Pure StrategiesPlayer BPlayer APlayer As has been thought of as choosingto play either U or D, but no combination ofboth; that is, as playing

25、 purely U or D.U and D are Player As pure strategies.LRUD(3,9)(0,0)(1,8)(2,1)Pure StrategiesPlayer BPlayer ASimilarly, L and R are Player Bs purestrategies.LRUD(3,9)(0,0)(1,8)(2,1)Pure StrategiesPlayer BPlayer AConsequently, (U,L) and (D,R) are purestrategy Nash equilibria. Must every gamehave at le

26、ast one pure strategy Nashequilibrium?LRUD(3,9)(0,0)(1,8)(2,1)Pure StrategiesPlayer BPlayer AHere is a new game. Are there any purestrategy Nash equilibria?(1,2)(0,4)(0,5)(3,2)UDLRPure StrategiesPlayer BPlayer AIs (U,L) a Nash equilibrium?(1,2)(0,4)(0,5)(3,2)UDLRPure StrategiesPlayer BPlayer AIs (U,

27、L) a Nash equilibrium? No.Is (U,R) a Nash equilibrium?(1,2)(0,4)(0,5)(3,2)UDLRPure StrategiesPlayer BPlayer AIs (U,L) a Nash equilibrium? No.Is (U,R) a Nash equilibrium? No.Is (D,L) a Nash equilibrium?(1,2)(0,4)(0,5)(3,2)UDLRPure StrategiesPlayer BPlayer AIs (U,L) a Nash equilibrium? No.Is (U,R) a N

28、ash equilibrium? No.Is (D,L) a Nash equilibrium? No.Is (D,R) a Nash equilibrium?(1,2)(0,4)(0,5)(3,2)UDLRPure StrategiesPlayer BPlayer AIs (U,L) a Nash equilibrium? No.Is (U,R) a Nash equilibrium? No.Is (D,L) a Nash equilibrium? No.Is (D,R) a Nash equilibrium? No.(1,2)(0,4)(0,5)(3,2)UDLRPure Strategi

29、esPlayer BPlayer ASo the game has no Nash equilibria in purestrategies. Even so, the game does have aNash equilibrium, but in mixed strategies.(1,2)(0,4)(0,5)(3,2)UDLRMixed StrategiesInstead of playing purely Up or Down, Player A selects a probability distribution (pU,1-pU), meaning that with probab

30、ility pU Player A will play Up and with probability 1-pU will play Down.Player A is mixing over the pure strategies Up and Down.The probability distribution (pU,1-pU) is a mixed strategy for Player A.Mixed StrategiesSimilarly, Player B selects a probability distribution (pL,1-pL), meaning that with

31、probability pL Player B will play Left and with probability 1-pL will play Right.Player B is mixing over the pure strategies Left and Right.The probability distribution (pL,1-pL) is a mixed strategy for Player B.Mixed StrategiesPlayer AThis game has no pure strategy Nash equilibria but it does have

32、a Nash equilibrium in mixed strategies. How is itcomputed?(1,2)(0,4)(0,5)(3,2)UDLRPlayer BMixed StrategiesPlayer A(1,2)(0,4)(0,5)(3,2)U,pUD,1-pUL,pLR,1-pLPlayer BMixed StrategiesPlayer AIf B plays Left her expected payoff is(1,2)(0,4)(0,5)(3,2)U,pUD,1-pUL,pLR,1-pLPlayer BMixed StrategiesPlayer AIf B

33、 plays Left her expected payoff isIf B plays Right her expected payoff is(1,2)(0,4)(0,5)(3,2)U,pUD,1-pUL,pLR,1-pLPlayer BMixed StrategiesPlayer AIfthenB would play only Left. But there are noNash equilibria in which B plays only Left. (1,2)(0,4)(0,5)(3,2)U,pUD,1-pUL,pLR,1-pLPlayer BMixed StrategiesP

34、layer AIfthenB would play only Right. But there are noNash equilibria in which B plays only Right. (1,2)(0,4)(0,5)(3,2)U,pUD,1-pUL,pLR,1-pLPlayer BMixed StrategiesPlayer ASo for there to exist a Nash equilibrium, Bmust be indifferent between playing Left orRight; i.e.(1,2)(0,4)(0,5)(3,2)U,pUD,1-pUL,

35、pLR,1-pLPlayer BMixed StrategiesPlayer ASo for there to exist a Nash equilibrium, Bmust be indifferent between playing Left orRight; i.e.(1,2)(0,4)(0,5)(3,2)U,pUD,1-pUL,pLR,1-pLPlayer BMixed StrategiesPlayer ASo for there to exist a Nash equilibrium, Bmust be indifferent between playing Left orRight

36、; i.e.(1,2)(0,4)(0,5)(3,2)U,D,L,pLR,1-pLPlayer BMixed StrategiesPlayer A(1,2)(0,4)(0,5)(3,2)L,pLR,1-pLU,D,Player BMixed StrategiesPlayer AIf A plays Up his expected payoff is(1,2)(0,4)(0,5)(3,2)L,pLR,1-pLU,D,Player BMixed StrategiesPlayer AIf A plays Up his expected payoff isIf A plays Down his expe

37、cted payoff is(1,2)(0,4)(0,5)(3,2)L,pLR,1-pLU,D,Player BMixed StrategiesPlayer AIfthen A would play only Up.But there are no Nash equilibria in which Aplays only Up. (1,2)(0,4)(0,5)(3,2)L,pLR,1-pLU,D,Player BMixed StrategiesPlayer AIfDown. But there are no Nash equilibria inwhich A plays only Down.

38、then A would play only(1,2)(0,4)(0,5)(3,2)L,pLR,1-pLU,D,Player BMixed StrategiesPlayer ASo for there to exist a Nash equilibrium, Amust be indifferent between playing Up orDown; i.e.(1,2)(0,4)(0,5)(3,2)L,pLR,1-pLU,D,Player BMixed StrategiesPlayer ASo for there to exist a Nash equilibrium, Amust be indifferent between playing Up orDown; i.e.(1,2)(0,4)(0,5)(3,2)L,pLR,1-pLU,D,Player BMixed StrategiesPlayer ASo for there to exist a Nash equilibrium, Amust be indifferent between playing