中級微觀經(jīng)濟學英文版課件：第28章 博弈論

1、Chapter Twenty-EightGame Theory博弈論ContentsDominant strategyNash equilibriumPrisoners dilemma and repeated gamesMultiple equilibria and sequential gamesPure and mixed strategiesGame TheoryGame theory models strategic behavior by agents who understand that their actions affect the actions of other age

2、nts. Some Applications of Game TheoryThe study of oligopolies (industries containing only a few firms)The study of cartels; e.g. OPECThe study of 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 s

3、trategies for each playerthe payoffs to each player for every possible list of strategy choices by the players.Two-Player GamesA game with just two 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

4、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” and “Right”.The table showing the payoffs to both players for each of the four possible strategy combinations is the games payoff matrix (支付矩陣).An Example of

5、 a Two-Player GameA 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.Player BPlayer ALRUD(1,2)(2,1)(0,1)(1,0)An Example of a Two-Player GameWhat plays are we likely to see for thisgame?Player BPlayer ALRUD(

6、1,2)(2,1)(0,1)(1,0)An Example of a Two-Player GameIf B plays Left then As best reply is Down.Player BPlayer ALRUD(1,2)(2,1)(0,1)(1,0)An Example of a Two-Player GameIf B plays Right then As best reply is Down.Player BPlayer ALRUD(1,2)(2,1)(0,1)(1,0)An Example of a Two-Player GameSo no matter what B p

7、lays, Asbest reply is always Down. Down is As dominant strategy （超優(yōu)策略）. Player BPlayer ALRUD(1,2)(2,1)(0,1)(1,0)An Example of a Two-Player GameSimilarly, Left is Bs dominant strategy. Player BPlayer ALRUD(1,2)(2,1)(0,1)(1,0)An Example of a Two-Player GameTherefore, (Down, Left) is dominant strategyf

8、or both players. It is the only equilibrium. Player BPlayer ALRUD(1,2)(2,1)(0,1)(1,0)No Dominant Strategy for One PlayerWhen Strength Is Weakness大豬小豬PWPW(-1,10)(6,4)(-1,10)(0,0)No Dominant Strategy for BothThe Battle of Sexes小紅小東CSCS(2,1)(0,0)(0,0)(1,2)Nash EquilibriumA play of the game where each s

9、trategy is a best reply to the other is a Nash equilibrium.A dominant strategy equilibrium is a Nash equilibrium; In the “strength is weakness” example, (W,P) is a Nash equilibrium.In the “battle of sexes” example, there are two Nash equilibria.The Prisoners DilemmaA Nash equilibrium may not be Pare

10、to optimal/efficient.Consider a famous second example of a two-player game called the Prisoners Dilemma （囚徒困境）.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 Confes

11、s.ClydeBonnie(-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

12、 is always 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

13、Prisoners 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)SCSCOther Examples of Prisoners DilemmaCheating in a Cartel.Price competition.Military

14、competition in the cold war.How to Avoid Prisoners DilemmaRepeated gamesFinite number of periodsInfinite number of periods or uncertain about the (finite number of periods)Tit-for-tatThe demand factor (water meters and airlines)Binding contractMultiple EquilibriaChicken gameYouth 2Youth 1SwerveStrai

15、ghtSwerve(0,0)(1,-1)(-1,1)(-2,-2)StraightMultiple EquilibriaSometimes a game has more than one Nash equilibrium and it is hard to say which is more likely to occur.Solutions:CoordinationStrategic behavior; establish reputationSequential movesA Sequential Game ExampleWhen such a game is sequential it

16、 is sometimes possible to argue that one of the Nash equilibria is more likely to occur than the other. A Sequential Game ExampleIncumbentEntrant(Enter,dont fight) and (stay out, fight) are both Nash equilibria when this game is played simultaneouslyand we have no way of deciding whichequilibrium is

17、 more likely to occur.FightDont fightEnterStay out(1,9)(0,0)(1,8)(2,1)Suppose instead that the game is playedsequentially, with incumbent leading and entrant following.We can rewrite the game in its extensive form.A Sequential Game ExampleIncumbentEntrantFightDont fightEnterStay out(1,9)(0,0)(1,8)(2

18、,1)A Sequential Game ExampleFightDont fightEnterStay out(1,9)(1,8)(0,0)(2,1)EntrantIncumbentIncumbentDont fightFight(Stay out, Fight) is a Nash equilibrium.A Sequential Game ExampleFightDont fightEnterStay out(1,9)(1,8)(0,0)(2,1)EntrantIncumbentIncumbentDont fightFight(Stay out, Fight) is a Nash equ

19、ilibrium.(Enter, Dont Fight) is a Nash equilibrium.Which is more likely to occur?The entrant prefers (Enter, Dont Fight), but the incumbent may threat to fight.Is the threat credible?Can make it credible.A Sequential Game ExampleA Sequential Game ExampleFightDont fightEnterStay out(1,9)(1,8)(0,2)(2,

20、1)EntrantIncumbentIncumbentDont fightFightBy building up excess capacity, the threat becomes credible. The potential entrant stays out. Pure Strategies In all previous examples, players are thought of as choosing to play either one or the other, but no combination ofboth; that is, as playing purely

21、one or the other. The strategies presented so far are players pure strategies （純粹策略）.Consequently, equilibria are pure strategy Nash equilibria. Must every game have at least one pure strategy Nash equilibrium?Pure StrategiesPlayer BPlayer AHere is a new game. Are there any purestrategy Nash equilib

22、ria?(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,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

23、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 Nash 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 e

24、quilibrium? 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)UDLRMore ExamplesMatching PenniesPlayer BPlayer A(1,-1)(-1, 1)(-1,1)(1, -1)HTHTMore Examples點球進攻球員守門員(1,0)(0, 1)(0,1)(1, 0)左右左右Pure StrategiesPlayer BPlayer ASo the game

25、 has no Nash equilibria in purestrategies. Even so, the game does have aNash equilibrium, but in mixed strategies(混合策略).(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 probability pU Player A will

26、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 probability pL Player B

27、 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 a Nash equilibrium in m

28、ixed 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 plays Left her expecte

29、d 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 StrategiesPlayer AIfthenB would pl

30、ay 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,pLR,1-pLPlayer BMixed S

31、trategiesPlayer 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; i.e.(1,2)(0,4)(0,5)(3

32、,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 expected payoff is(1,2)(0,4

33、)(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. then A would play only(

34、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 play

35、ing 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,R,U,D,Player BMixed StrategiesPlayer BPlayer ASo the games only Nash equilibrium has Aplaying the mixed strategy (3/5, 2/5) and hasB playing the mixed strategy (3/4, 1/4).(1,2)(0,4)(0,5)(3,2)U,D,L,R,Mixed StrategiesPlayer BPlayer AThe p