专业课资料 练习题 真题 范里安ppt带中文注释 第28章

1、Chapter Twenty-Eight,Game Theory 博弈论,Contents,Dominant strategy Nash equilibrium Prisoners dilemma and repeated games Multiple equilibria and sequential games Pure and mixed strategies,Game Theory,Game theory models strategic behavior by agents who understand that their actions affect the actions of

2、 other agents.,Some Applications of Game Theory,The study of oligopolies (industries containing only a few firms) The study of cartels; e.g. OPEC The 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 of a set of

3、players a set of strategies for each player the payoffs to each player for every possible list of strategy choices by the players.,Two-Player Games,A 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 str

4、ategies.,An Example of a Two-Player Game,The 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

5、matrix (支付矩阵).,An Example of a Two-Player Game,A play of the game is a pair such as (U,R) where the 1st element is the strategy chosen by Player A and the 2nd is the strategy chosen by Player B.,An Example of a Two-Player Game,What plays are we likely to see for this game?,An Example of a Two-Player

6、 Game,If B plays Left then As best reply is Down.,An Example of a Two-Player Game,If B plays Right then As best reply is Down.,An Example of a Two-Player Game,So no matter what B plays, Asbest reply is always Down. Down is As dominant strategy （超优 策略）.,An Example of a Two-Player Game,Similarly, Left

7、 is Bs dominant strategy.,An Example of a Two-Player Game,Therefore, (Down, Left) is dominant strategy for both players. It is the only equilibrium.,No Dominant Strategy for One Player,When Strength Is Weakness,No Dominant Strategy for Both,The Battle of Sexes,Nash Equilibrium,A play of the game whe

8、re each strategy 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 Dilemma,A Nash equilibrium ma

9、y not be Pareto optimal/efficient. Consider a famous second example of a two-player game called the Prisoners Dilemma （囚徒困境）.,The Prisoners Dilemma,What plays are we likely to see for this game?,Clyde,Bonnie,(-5,-5),(-30,-1),(-1,-30),(-10,-10),S,C,S,C,The Prisoners Dilemma,If Bonnie plays Silence th

10、en Clydes bestreply is Confess.,Clyde,Bonnie,(-5,-5),(-30,-1),(-1,-30),(-10,-10),S,C,S,C,The Prisoners Dilemma,If Bonnie plays Silence then Clydes bestreply is Confess. If Bonnie plays Confess then Clydesbest reply is Confess.,Clyde,Bonnie,(-5,-5),(-30,-1),(-1,-30),(-10,-10),S,C,S,C,The Prisoners Di

11、lemma,So no matter what Bonnie plays, Clydesbest reply is always Confess. Confess is a dominant strategy for Clyde.,Clyde,Bonnie,(-5,-5),(-30,-1),(-1,-30),(-10,-10),S,C,S,C,The Prisoners Dilemma,Similarly, no matter what Clyde plays,Bonnies best reply is always Confess. Confess is a dominant strateg

12、y forBonnie also.,Clyde,Bonnie,(-5,-5),(-30,-1),(-1,-30),(-10,-10),S,C,S,C,The Prisoners Dilemma,So 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.,Clyde,Bonnie,(-5,-5),(-30,-1),(-1,-30),(-10,-10)

13、,S,C,S,C,Other Examples of Prisoners Dilemma,Cheating in a Cartel. Price competition. Military competition in the cold war.,How to Avoid Prisoners Dilemma,Repeated games Finite number of periods Infinite number of periods or uncertain about the (finite number of periods) Tit-for-tat The demand facto

14、r (water meters and airlines) Binding contract,Multiple Equilibria,Chicken game,Youth 2,Youth 1,Swerve,Straight,Swerve,(0,0),(1,-1),(-1,1),(-2,-2),Straight,Multiple Equilibria,Sometimes a game has more than one Nash equilibrium and it is hard to say which is more likely to occur. Solutions: Coordina

15、tion Strategic behavior; establish reputation Sequential moves,Sequential Game,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.,Who Plays When?,When players chose their strategies simultaneously, a game is a simu

16、ltaneous play game. 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 Example,Player B,Player A,(U,L) and (D,R) are both Nash equilibriawh

17、en this game is played simultaneouslyand we have no way of deciding whichequilibrium is more likely to occur.,L,R,U,D,(3,9),(0,0),(1,8),(2,1),A Sequential Game Example,Player B,Player A,Suppose instead that the game is playedsequentially, with A leading and B following. We can rewrite the game in it

18、s extensive form.,L,R,U,D,(3,9),(0,0),(1,8),(2,1),A Sequential Game Example,U,D,L,L,R,R,(3,9),(1,8),(0,0),(2,1),A,B,B,A plays first.B plays second.,A Sequential Game Example,U,D,L,L,R,R,(3,9),(1,8),(0,0),(2,1),A,B,B,A plays first.B plays second.,(U,L) is a Nash equilibrium.,A Sequential Game Example

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

20、Sequential Game Example,U,D,L,L,R,R,(3,9),(1,8),(0,0),(2,1),A,B,B,A 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.,A Sequential Game Example,U,D,L,L,R,R,(3,9),(1,8),(0,0),(2,1),A,B,B,A plays first.B plays second.,If A plays U then B plays L;

21、 A gets 3. If A plays D then B plays R; A gets 2.So (U,L) is the likely Nash equilibrium.,Fight,Dont fight,Enter,Stay out,(1,9),(1,9),(0,0),(2,1),Entrant,Incumbent,Incumbent,Dont fight,Fight,Entrant plays first.Incumbent plays second.,A Game of Entry Deterrence,Fight,Dont fight,Enter,Stay out,(1,9),

22、(1,9),(0,0),(2,1),Entrant,Incumbent,Incumbent,Dont fight,Fight,IF the entrant stays out, then payoff is (1,9).,A Game of Entry Deterrence,Fight,Dont fight,Enter,Stay out,(1,9),(1,9),(0,0),(2,1),Entrant,Incumbent,Incumbent,Dont fight,Fight,(Enter, Dont Fight) is a Nash equilibrium.,A Game of Entry De

23、terrence,The entrant prefers (Enter, Dont Fight), but the incumbent may threat to fight. Is the threat credible? Can make it credible.,A Game of Entry Deterrence,Fight,Dont fight,Enter,Stay out,(1,9),(1,9),(0,2),(2,1),Entrant,Incumbent,Incumbent,Dont fight,Fight,By building up excess capacity, the t

24、hreat becomes credible. The potential entrant stays out.,A Game of Entry Deterrence,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 one or the other. The strategies presented so far ar

25、e players pure strategies （纯粹策略）. Consequently, equilibria are pure strategy Nash equilibria. Must every game have at least one pure strategy Nash equilibrium?,Pure Strategies,Player B,Player A,Here is a new game. Are there any purestrategy Nash equilibria?,(1,2),(0,4),(0,5),(3,2),U,D,L,R,Pure Strat

26、egies,Player B,Player A,Is (U,L) a Nash equilibrium?,(1,2),(0,4),(0,5),(3,2),U,D,L,R,Pure Strategies,Player B,Player A,Is (U,L) a Nash equilibrium? No. Is (U,R) a Nash equilibrium?,(1,2),(0,4),(0,5),(3,2),U,D,L,R,Pure Strategies,Player B,Player A,Is (U,L) a Nash equilibrium? No. Is (U,R) a Nash equi

27、librium? No.Is (D,L) a Nash equilibrium?,(1,2),(0,4),(0,5),(3,2),U,D,L,R,Pure Strategies,Player B,Player A,Is (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),U,D,L,R,Pure Strategies,Player B,Player A,I

28、s (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),U,D,L,R,More Examples,Matching Pennies,Player B,Player A,(1,-1),(-1, 1),(-1,1),(1, -1),H,T,H,T,More Examples,点球,进攻球员,守门员,(1,0),(0, 1),(0,1),(1, 0),

29、左,右,左,右,Pure Strategies,Player B,Player A,So the game 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),U,D,L,R,Mixed Strategies,Instead of playing purely Up or Down, Player A selects a probability distrib

30、ution (pU,1-pU), meaning that with probability 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 Strategies,Similarly, Player B selects a probabili

31、ty distribution (pL,1-pL), meaning that with 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 Strategies,Player A,This game has

32、no pure strategy Nash equilibria but it does have a Nash equilibrium in mixed strategies. How is itcomputed?,(1,2),(0,4),(0,5),(3,2),U,D,L,R,Player B,Mixed Strategies,Player A,(1,2),(0,4),(0,5),(3,2),U,pU,D,1-pU,L,pL,R,1-pL,Player B,Mixed Strategies,Player A,If B plays Left her expected payoff is,(1

33、,2),(0,4),(0,5),(3,2),U,pU,D,1-pU,L,pL,R,1-pL,Player B,Mixed Strategies,Player A,If B plays Left her expected payoff isIf B plays Right her expected payoff is,(1,2),(0,4),(0,5),(3,2),U,pU,D,1-pU,L,pL,R,1-pL,Player B,Mixed Strategies,Player A,If,then,B would play only Left. But there are no Nash equi

34、libria in which B plays only Left.,(1,2),(0,4),(0,5),(3,2),U,pU,D,1-pU,L,pL,R,1-pL,Player B,Mixed Strategies,Player A,If,then,B would play only Right. But there are no Nash equilibria in which B plays only Right.,(1,2),(0,4),(0,5),(3,2),U,pU,D,1-pU,L,pL,R,1-pL,Player B,Mixed Strategies,Player A,So f

35、or there to exist a Nash equilibrium, Bmust be indifferent between playing Left orRight; i.e.,(1,2),(0,4),(0,5),(3,2),U,pU,D,1-pU,L,pL,R,1-pL,Player B,Mixed Strategies,Player A,So 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

36、,pU,D,1-pU,L,pL,R,1-pL,Player B,Mixed Strategies,Player A,So 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,D,L,pL,R,1-pL,Player B,Mixed Strategies,Player A,(1,2),(0,4),(0,5),(3,2),L,pL,R,1-pL,U,D,Player B,Mixed Strategies,Pla

37、yer A,If A plays Up his expected payoff is,(1,2),(0,4),(0,5),(3,2),L,pL,R,1-pL,U,D,Player B,Mixed Strategies,Player A,If A plays Up his expected payoff isIf A plays Down his expected payoff is,(1,2),(0,4),(0,5),(3,2),L,pL,R,1-pL,U,D,Player B,Mixed Strategies,Player A,If,then A would play only Up.,Bu

38、t there are no Nash equilibria in which Aplays only Up.,(1,2),(0,4),(0,5),(3,2),L,pL,R,1-pL,U,D,Player B,Mixed Strategies,Player A,If,Down. But there are no Nash equilibria in which A plays only Down.,then A would play only,(1,2),(0,4),(0,5),(3,2),L,pL,R,1-pL,U,D,Player B,Mixed Strategies,Player A,S

39、o 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,pL,R,1-pL,U,D,Player B,Mixed Strategies,Player A,So 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,pL,R,1-pL,

40、U,D,Player B,Mixed Strategies,Player A,So 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 B,Mixed Strategies,Player B,Player A,So the games only Nash equilibrium has Aplaying the mixed strategy (3/5, 2/5) and hasB playing the mixed strategy (3/4, 1/4