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

2、y of externalities; e.g. using a common resource such as a fishery.uThe study of military strategies.What is a Game?uA 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 GamesuA game with

3、 just two players is a two-player game.uWe 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 GameuThe players are called A and B.uPlayer A has two strategies, called “Up” and “Down”.uPlayer B has two strategies, calle

4、d “Left” and “Right”.uThe 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

5、,0)(1,8)(2,1)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 GamePlayer BPlayer AA play of the game is a pair such as (U,R)where the 1st element

6、 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 GameWhat 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)

7、(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 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

8、)(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(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

9、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(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

10、(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-Player 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 p

11、lays 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)(0,0)(1,8)(2,1)Nash EquilibriumuA play of the game where each strategy is a best reply to the other is a Nash equilibrium.uOur example has two Nash equilibria; (U,L) a

12、nd (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 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 b

13、othplayers. Must we then see (U,L) only?LRUD(3,9)(0,0)(1,8)(2,1)The Prisoners DilemmauTo see if Pareto-preferred outcomes must be what we see in 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

14、 for thisgame?ClydeBonnie(-5,-5)(-30,-1)(-1,-30)(-10,-10)SCSCThe Prisoners DilemmaIf Bonnie plays Silence then Clydes bestreply is Confess.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 Clydesb

15、est reply is Confess.ClydeBonnie(-5,-5)(-30,-1)(-1,-30)(-10,-10)SCSCThe Prisoners DilemmaSo no matter what Bonnie plays, Clydesbest reply 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 pla

16、ys,Bonnies best reply is always Confess.Confess is a dominant strategy forBonnie also.ClydeBonnie(-5,-5)(-30,-1)(-1,-30)(-10,-10)SCSCThe 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 ineff

17、icient.ClydeBonnie(-5,-5)(-30,-1)(-1,-30)(-10,-10)SCSCWho Plays When?uIn both examples the players chose their strategies simultaneously.uSuch games are simultaneous play games.Who Plays When?uBut there are games in which one player plays before another player.uSuch games are sequential play games.u

18、The player who plays first is the leader. The player who plays second is the follower.A Sequential Game ExampleuSometimes a game has more than one Nash equilibrium and it is hard to say which is more likely to occur.uWhen such a game is sequential it is sometimes possible to argue that one of the Na

19、sh 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 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 ExamplePlaye

20、r 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)(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

21、)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 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.

22、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 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

23、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 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)

24、(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 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 pur

25、estrategies.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 least 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

26、 equilibria?(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 StrategiesPlayer BPlayer ASo the game has no Nash equilibria in purestrategi

27、es. Even so, the game does have aNash equilibrium, but in mixed strategies.(1,2)(0,4)(0,5)(3,2)UDLRMixed StrategiesuInstead of playing purely Up or Down, Player A selects a probability distribution (p pU,1-p pU), meaning that with probability p pU Player A will play Up and with probability 1-p pU wi

28、ll play Down.uPlayer A is mixing over the pure strategies Up and Down.uThe probability distribution (p pU,1-p pU) is a mixed strategy for Player A.Mixed StrategiesuSimilarly, Player B selects a probability distribution (p pL,1-p pL), meaning that with probability p pL Player B will play Left and wit

29、h probability 1-p pL will play Right.uPlayer B is mixing over the pure strategies Left and Right.uThe probability distribution (p pL,1-p 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 mixed strategies

30、. How is itcomputed?(1,2)(0,4)(0,5)(3,2)UDLRPlayer BMixed StrategiesPlayer A(1,2)(0,4)(0,5)(3,2)U,p pUD,1-p pUL,p pLR,1-p pLPlayer BMixed StrategiesPlayer AIf B plays Left her expected payoff is25 1p pp pUU ()(1,2)(0,4)(0,5)(3,2)U,p pUD,1-p pUL,p pLR,1-p pLPlayer BMixed StrategiesPlayer AIf B plays

31、Left her expected payoff isIf B plays Right her expected payoff is25 1p pp pUU ().42 1p pp pUU ().(1,2)(0,4)(0,5)(3,2)U,p pUD,1-p pUL,p pLR,1-p pLPlayer BMixed StrategiesPlayer A25 142 1p pp pp pp pUUUU ()()IfthenB would play only Left. But there are noNash equilibria in which B plays only Left. (1,

32、2)(0,4)(0,5)(3,2)U,p pUD,1-p pUL,p pLR,1-p pLPlayer BMixed StrategiesPlayer A25 142 1p pp pp pp pUUUU ()()IfthenB would play only Right. But there are noNash equilibria in which B plays only Right. (1,2)(0,4)(0,5)(3,2)U,p pUD,1-p pUL,p pLR,1-p pLPlayer BMixed StrategiesPlayer ASo for there to exist

33、a Nash equilibrium, Bmust be indifferent between playing Left orRight; i.e.25 142 13 5p pp pp pp pp pUUUUU ()()/ .(1,2)(0,4)(0,5)(3,2)U,p pUD,1-p pUL,p pLR,1-p pLPlayer BMixed StrategiesPlayer ASo for there to exist a Nash equilibrium, Bmust be indifferent between playing Left orRight; i.e.25 142 13

34、 5p pp pp pp pp pUUUUU ()()/ .(1,2)(0,4)(0,5)(3,2)U,D,L,p pLR,1-p pL5352Player BMixed StrategiesPlayer A(1,2)(0,4)(0,5)(3,2)L,p pLR,1-p pLU,D,5352Player BMixed StrategiesPlayer AIf A plays Up his expected payoff is.)1(01LLLp p p p p p (1,2)(0,4)(0,5)(3,2)L,p pLR,1-p pLU,D,5352Player BMixed Strategie

35、sPlayer AIf A plays Up his expected payoff isIf A plays Down his expected payoff is).1(3)1(30LLLp p p p p p (1,2)(0,4)(0,5)(3,2)L,p pLR,1-p pLU,D,5352.)1(01LLLp p p p p p Player BMixed StrategiesPlayer Ap pp pLL 3 1()Ifthen A would play only Up.But there are no Nash equilibria in which Aplays only U

36、p. (1,2)(0,4)(0,5)(3,2)L,p pLR,1-p pLU,D,5352Player BMixed StrategiesPlayer AIfDown. But there are no Nash equilibria inwhich A plays only Down. p pp pLL 3 1()then A would play only(1,2)(0,4)(0,5)(3,2)L,p pLR,1-p pLU,D,5352Player BMixed StrategiesPlayer ASo for there to exist a Nash equilibrium, Amu

37、st be indifferent between playing Up orDown; i.e.p pp pLL 3 1()(1,2)(0,4)(0,5)(3,2)L,p pLR,1-p pLU,D,5352Player BMixed StrategiesPlayer ASo for there to exist a Nash equilibrium, Amust be indifferent between playing Up orDown; i.e.p pp pp pLLL 3 13 4()/ .(1,2)(0,4)(0,5)(3,2)L,R,U,D,53524341Player BMixed StrategiesPlayer BPlayer ASo the games only Nash equilibrium has Aplaying the mixed strategy (3/5, 2/5) and hasB playing the mixed strategy