上海财经大学中级微观经济学课程课件(叶正茂)第五章

第五章博弈论GameTheory

§5.1博弈论简介

Gametheorymodelsstrategicbehaviorbyagentswhounderstandthattheiractionsaffecttheactionsofotheragents.博弈论是关于包含相互依存情况中理性人理性行为的研究。例:齐王与田忌赛马的博弈齐王田忌

A1

好B1

好

A2

中B2

中

A3

下B3

下

A1

好

B3

下

A2

中

B1

好

A3

下

B2

中SomeApplicationsofGameTheoryThestudyofoligopolies(industriescontainingonlyafewfirms)Thestudyofcartels;e.g.OPECThestudyofexternalities;e.g.usingacommonresourcesuchasafishery.Thestudyofmilitarystrategies.WhatisaGame?Agameconsistsofasetofplayersasetofstrategiesforeachplayerthepayoffstoeachplayerforeverypossiblelistofstrategychoicesbytheplayers.怎样描述一个博弈？博弈三大要素参与者或局中人参与者的策略〔空间〕报酬或收益或支付函数：作为博弈的结局，每个参与者都得到各自的报酬或收益可以用支付矩阵〔报酬矩阵〕描述一个博弈乙合作不合作合作10，106，12甲不合作12，68，8第一个数字

代表甲的报酬；第二个数字

代表乙的报酬；Two-PlayerGamesAgamewithjusttwoplayersisatwo-playergame.Wewillstudyonlygamesinwhichtherearetwoplayers,eachofwhomcanchoosebetweenonlytwostrategies.TheplayersarecalledAandB.PlayerAhastwostrategies,called“Up〞and“Down〞.PlayerBhastwostrategies,called“Left〞and“Right〞.Thetableshowingthepayoffstobothplayersforeachofthefourpossiblestrategycombinationsisthegame’spayoffmatrix.AnExampleofaTwo-PlayerGameThisisthe

game’spayoffmatrix.PlayerBPlayerAPlayerA’spayoffisshownfirst.

PlayerB’spayoffisshownsecond.LRUD(3,9)(0,0)(1,8)(2,1)AnExampleofaTwo-PlayerGameE.g.ifAplaysUpandBplaysRightthenA’spayoffis1andB’spayoffis8.Thisisthe

game’spayoffmatrix.PlayerBPlayerALRUD(3,9)(0,0)(1,8)(2,1)AnExampleofaTwo-PlayerGameAndifAplaysDownandBplaysRightthenA’spayoffis2andB’spayoffis1.Thisisthe

game’spayoffmatrix.PlayerBPlayerALRUD(3,9)(0,0)(1,8)(2,1)PlayerBPlayerAAplayofthegameisapairsuchas(U,R)wherethe1stelementisthestrategychosenbyPlayerAandthe2ndisthestrategychosenbyPlayerB.LRUD(3,9)(0,0)(1,8)(2,1)Whatplaysarewelikelytoseeforthisgame?PlayerBPlayerALRUD(3,9)(0,0)(1,8)(2,1)AnExampleofaTwo-PlayerGamePlayerBPlayerAIs(U,R)a

likelyplay?LRUD(3,9)(0,0)(1,8)(2,1)AnExampleofaTwo-PlayerGamePlayerBPlayerAIfBplaysRightthenA’sbestreplyisDown

sincethisimprovesA’spayofffrom1to2.

So(U,R)isnotalikelyplay.Is(U,R)a

likelyplay?LRUD(3,9)(0,0)(1,8)(2,1)AnExampleofaTwo-PlayerGamePlayerBPlayerAIs(D,R)a

likelyplay?LRUD(3,9)(0,0)(1,8)(2,1)AnExampleofaTwo-PlayerGamePlayerBPlayerAIs(D,R)a

likelyplay?IfBplaysRightthenA’sbestreplyisDown.LRUD(3,9)(0,0)(1,8)(2,1)AnExampleofaTwo-PlayerGamePlayerBPlayerAIfBplaysRightthenA’sbestreplyisDown.IfAplaysDownthenB’sbestreplyisRight.So(D,R)isalikelyplay.Is(D,R)a

likelyplay?LRUD(3,9)(0,0)(1,8)(2,1)AnExampleofaTwo-PlayerGamePlayerBPlayerAIs(D,L)a

likelyplay?LRUD(3,9)(0,0)(1,8)(2,1)AnExampleofaTwo-PlayerGamePlayerBPlayerAIfAplaysDownthenB’sbestreplyisRight,

so(D,L)isnotalikelyplay.Is(D,L)a

likelyplay?LRUD(3,9)(0,0)(1,8)(2,1)AnExampleofaTwo-PlayerGamePlayerBPlayerAIs(U,L)a

likelyplay?LRUD(3,9)(0,0)(1,8)(2,1)AnExampleofaTwo-PlayerGamePlayerBPlayerAIfAplaysUpthenB’sbestreplyisLeft.Is(U,L)a

likelyplay?LRUD(3,9)(0,0)(1,8)(2,1)AnExampleofaTwo-PlayerGamePlayerBPlayerAIfAplaysUpthenB’sbestreplyisLeft.IfBplaysLeftthenA’sbestreplyisUp.So(U,L)isalikelyplay.Is(U,L)a

likelyplay?LRUD(3,9)(0,0)(1,8)(2,1)§5.2NashEquilibrium

-参与人同时采取行动AplayofthegamewhereeachstrategyisabestreplytotheotherisaNashequilibrium.OuraboveexamplehastwoNashequilibria;(U,L)and(D,R).纳什均衡是指假设其他参与者不改变策略，任何一个参加者都不会改变自己的策略的均衡状态。AnExampleofaTwo-PlayerGamePlayerBPlayerA(U,L)and(D,R)arebothNashequilibriafor

thegame.LRUD(3,9)(0,0)(1,8)(2,1)AnExampleofaTwo-PlayerGamePlayerBPlayerA(U,L)and(D,R)arebothNashequilibriafor

thegame.Butwhichwillwesee?Notice

that(U,L)ispreferredto(D,R)byboth

players.Mustwethensee(U,L)only?后面的序贯博弈可以说明这个问题。LRUD(3,9)(0,0)(1,8)(2,1)§5.3占优策略均衡

-参与人同时采取行动

乙合作不合作合作10，106，12甲不合作12，68，8

无论对方采取什么策略叫占优策略，某参与者的唯一最优的策略

博弈均衡

指博弈中的所有参与者都不想改变自己的策略的一种状态。

由博弈中的所有参与者的占优策略组合所构成的均衡就是占优策略均衡。只要每一参与者都具有占优策略的话，那么，该博弈一定存在占优策略均衡。占优策略均衡与纳什均衡的比较：占优策略均衡一定是纳什均衡；但纳什均衡不一定是占优策略均衡。占优策略均衡是比纳什均衡更强的一个博弈均衡概念。ThePrisoner’sDilemma

-占优策略均衡的一个例子ToseeifPareto-preferredoutcomesmustbewhatweseeintheplayofagame,considerafamoussecondexampleofatwo-playergamecalledthePrisoner’sDilemma.ThePrisoner’sDilemmaWhatplaysarewelikelytoseeforthisgame?ClydeBonnie(-5,-5)(-15,-1)(-1,-15)(-10,-10)SCSCThePrisoner’sDilemmaIfBonnieplaysSilencethenClyde’sbest

replyisConfess.ClydeBonnie(-5,-5)(-15,-1)(-1,-15)(-10,-10)SCSCThePrisoner’sDilemmaIfBonnieplaysSilencethenClyde’sbest

replyisConfess.IfBonnieplaysConfessthenClyde’s

bestreplyisConfess.ClydeBonnie(-5,-5)(-15,-1)(-1,-15)(-10,-10)SCSCThePrisoner’sDilemmaSonomatterwhatBonnieplays,Clyde’s

bestreplyisalwaysConfess.ConfessisadominantstrategyforClyde.ClydeBonnie(-5,-5)(-15,-1)(-1,-15)(-10,-10)SCSCThePrisoner’sDilemmaSimilarly,nomatterwhatClydeplays,

Bonnie’sbestreplyisalwaysConfess.Confessisadominantstrategyfor

Bonniealso.ClydeBonnie(-5,-5)(-15,-1)(-1,-15)(-10,-10)SCSCThePrisoner’sDilemmaSotheonlyNashequilibriumforthis

gameis(C,C),eventhough(S,S)gives

bothBonnieandClydebetterpayoffs.TheonlyNashequilibriumisinefficient.ClydeBonnie(-5,-5)(-15,-1)(-1,-15)(-10,-10)SCSCThePrisoner’sDilemma囚徒困境反映了一个问题，从个人角度出发选择的占优策略，从整体来看，却是最差的结局。即个人理性与团体理性的冲突。ClydeBonnie(-5,-5)(-15,-1)(-1,-15)(-10,-10)SCSC是否存在合作解？一次博弈肯定不存在。重复博弈无限期重复博弈中存在囚徒困境中的合作均衡解。有限期重复博弈中就不存在囚徒困境中的合作均衡解。〔见书上410页的说明〕但如果是不能确定终止期的有限期重复博弈模型中，纳什均衡的合作解是存在的。§5.4序贯博弈

-参与人的行动有先后Inbothexamplestheplayerschosetheirstrategiessimultaneously.Suchgamesaresimultaneousplaygames.Buttherearegamesinwhichoneplayerplaysbeforeanotherplayer.Suchgamesaresequentialplaygames.Theplayerwhoplaysfirstistheleader.Theplayerwhoplayssecondisthefollower.ASequentialGameExampleSometimesagamehasmorethanoneNashequilibriumanditishardtosaywhichismorelikelytooccur.WhensuchagameissequentialitissometimespossibletoarguethatoneoftheNashequilibriaismorelikelytooccurthantheother.ASequentialGameExamplePlayerBPlayerA(U,L)and(D,R)arebothNashequilibria

whenthisgameisplayedsimultaneously

andwehavenowayofdecidingwhich

equilibriumismorelikelytooccur.LRUD(1,9)(0,0)(1,9)(2,1)ASequentialGameExamplePlayerBPlayerASupposeinsteadthatthegameisplayed

sequentially,withAleadingandBfollowing.Wecanrewritethegameinitsextensiveform.LRUD(1,9)(0,0)(1,9)(2,1)ASequentialGameExampleUDLLRR(1,9)(1,9)(0,0)(2,1)ABBAplaysfirst.

Bplayssecond.ASequentialGameExampleUDLLRR(1,9)(1,9)(0,0)(2,1)ABBAplaysfirst.

Bplayssecond.(U,L)isaNashequilibrium.ASequentialGameExampleUDLLRR(1,9)(1,9)(0,0)(2,1)ABBAplaysfirst.

Bplayssecond.(U,L)isaNashequilibrium.(D,R)isaNashequilibrium.

Whichismorelikelytooccur?ASequentialGameExampleUDLLRR(1,9)(1,9)(0,0)(2,1)ABBAplaysfirst.

Bplayssecond.IfAplaysUthenBplaysL;Agets1.ASequentialGameExampleUDLLRR(1,9)(1,9)(0,0)(2,1)ABBAplaysfirst.

Bplayssecond.IfAplaysUthenBplaysL;Agets1.IfAplaysDthenBplaysR;Agets2.ASequentialGameExampleUDLLRR(1,9)(1,9)(0,0)(2,1)ABBAplaysfirst.

Bplayssecond.IfAplaysUthenBplaysL;Agets1.IfAplaysDthenBplaysR;Agets2.

So(D,R)isthelikelyNashequilibrium.(D,R)isthelikelyNashequilibrium因此，如果让A先行动，(D,R)isthelikelyNashequilibrium，而(上,左)就不是一个合理的均衡，否那么A就是不理性的。进一步分析：从参与人B的角度来看，他只能得到收益1而不是9.他可以威胁A说，如果A采取“下〞，他就采取“左〞，A只能得到0.此时A选择“上〞是合理的。(上,左)就是博弈的均衡。结论是：如果B的威胁是可置信的，(上,左)就是博弈的均衡;如果B的威胁是不可置信的，那么在A先行动的情况下(下,右)就是博弈的均衡。一个具体的例子-遏制进入的博弈假设一个垄断厂商正面临着另一家厂商的进入威胁。进入者先决定是否进入市场，然后，在位者决定是否降价作为斗争遏制其进入。ASequentialGameExample不进入进入斗争斗争不斗争不斗争(1,9)(1,9)(0,0)(2,1)进入者在位者选择在位者选择由于进入者先行动，如果它选择进入，获得的收益为2。〔进入，不斗争〕是一个均衡结果。但在位者可以威胁，如果你进入，我就选择“斗争〞，大家的收益都为0.如果斗争的威胁是可置信的话，例如在位者拥有多余的生产能力，而且此时如果进入者进入，它选择“斗争〞策略，它可以获得的收益是2而不是0〔因为它做好了准备〕见书上415页图28.2ASequentialGameExample不进入进入斗争斗争不斗争不斗争(1,9)(1,9)(0,2)(2,1)进入者在位者选择在位者选择既然进入者知道，在位者“斗争〞是可置信的，那么进入者理性选择是“不进入〞，因为“进入〞收益为0，“不进入〞收益为1.而此时在位者获得的收益为9.这就意味着，在位者将维持现有的垄断地位，并永远不会利用额外的生产能力！尽管如此，在位者投资额外的生产能力仍是一件值得做的事情，因为它使得要进入的进入者觉得它的“斗争〞策略可置信。§5.5纯策略与混合策略PlayerBPlayerAThisisouroriginalexampleoncemore.

Supposeagainthatplayissimultaneous.

WediscoveredthatthegamehastwoNash

equilibria;(U,L)and(D,R).LRUD(3,9)(0,0)(1,8)(2,1)PureStrategiesPlayerBPlayerAPlayerA’shasbeenthoughtofaschoosing

toplayeitherUorD,butnocombinationof

both;thatis,asplayingpurelyUorD.

UandDarePlayerA’spurestrategies.LRUD(3,9)(0,0)(1,8)(2,1)PureStrategiesPlayerBPlayerASimilarly,LandRarePlayerB’spure

strategies.每个参与人只选择一种策略并始终坚持这个选择，这种策略称为纯策略。LRUD(3,9)(0,0)(1,8)(2,1)PureStrategiesPlayerBPlayerAConsequently,(U,L)and(D,R)arepure

strategyNashequilibria.Musteverygame

haveatleastonepurestrategyNash

equilibrium?LRUD(3,9)(0,0)(1,8)(2,1)PureStrategiesPlayerBPlayerAHereisanewgame.Arethereanypure

strategyNashequilibria?(1,2)(0,4)(0,5)(3,2)UDLRPureStrategiesPlayerBPlayerAIs(U,L)aNashequilibrium?(1,2)(0,4)(0,5)(3,2)UDLRPureStrategiesPlayerBPlayerAIs(U,L)aNashequilibrium?No.Is(U,R)aNashequilibrium?(1,2)(0,4)(0,5)(3,2)UDLRPureStrategiesPlayerBPlayerAIs(U,L)aNashequilibrium?No.Is(U,R)aNashequilibrium?No.

Is(D,L)aNashequilibrium?(1,2)(0,4)(0,5)(3,2)UDLRPureStrategiesPlayerBPlayerAIs(U,L)aNashequilibrium?No.Is(U,R)aNashequilibrium?No.

Is(D,L)aNashequilibrium?No.

Is(D,R)aNashequilibrium?(1,2)(0,4)(0,5)(3,2)UDLRPureStrategiesPlayerBPlayerAIs(U,L)aNashequilibrium?No.Is(U,R)aNashequilibrium?No.

Is(D,L)aNashequilibrium?No.

Is(D,R)aNashequilibrium?No.(1,2)(0,4)(0,5)(3,2)UDLRPureStrategiesPlayerBPlayerASothegamehasnoNashequilibriainpure

strategies.Evenso,thegamedoeshavea

Nashequilibrium,butinmixedstrategies.(1,2)(0,4)(0,5)(3,2)UDLRMixedStrategiesInsteadofplayingpurelyUporDown,PlayerAselectsaprobabilitydistribution(pU,1-pU),meaningthatwithprobabilitypUPlayerAwillplayUpandwithprobability1-pUwillplayDown.PlayerAismixingoverthepurestrategiesUpandDown.Theprobabilitydistribution(pU,1-pU),isamixedstrategyforPlayerA.MixedStrategiesSimilarly,PlayerBselectsaprobabilitydistribution(pL,1-pL),meaningthatwithprobabilitypLPlayerBwillplayLeftandwithprobability1-pLwillplayRight.PlayerBismixingoverthepurestrategiesLeftandRight.Theprobabilitydistribution(pL,1-pL)isamixedstrategyforPlayerB.PlayerAThisgamehasnopurestrategyNashequilibriabutitdoeshaveaNashequilibriuminmixedstrategies.Howisit

computed?(1,2)(0,4)(0,5)(3,2)UDLRPlayerBMixedStrategiesPlayerA(1,2)(0,4)(0,5)(3,2)U,pUD,1-pUL,pLR,1-pLPlayerBMixedStrategiesPlayerAIfBplaysLefther〔指Ｂ〕expectedpayoffis：(1,2)(0,4)(0,5)(3,2)U,pUD,1-pUL,pLR,1-pLPlayerBMixedStrategiesPlayerAIfBplaysLeftherexpectedpayoffis

IfBplaysRightherexpectedpayoffis(1,2)(0,4)(0,5)(3,2)U,pUD,1-pUL,pLR,1-pLPlayerBMixedStrategiesPlayerAIfthenBwouldplayonlyLeft.ButtherearenoNashequilibriainwhichBplaysonlyLeft.(1,2)(0,4)(0,5)(3,2)U,pUD,1-pUL,pLR,1-pLPlayerBMixedStrategiesPlayerAIfthenBwouldplayonlyRight.ButtherearenoNashequilibriainwhichBplaysonlyRight.(1,2)(0,4)(0,5)(3,2)U,pUD,1-pUL,pLR,1-pLPlayerBMixedStrategiesPlayerASofortheretoexistaNashequilibrium,B

mustbeindifferentbetweenplayingLeftor

Right;i.e.(1,2)(0,4)(0,5)(3,2)U,pUD,1-pUL,pLR,1-pLPlayerBMixedStrategiesPlayerASofortheretoexistaNashequilibrium,B

mustbeindifferentbetweenplayingLeftor

Right;i.e.(1,2)(0,4)(0,5)(3,2)U,pUD,1-pUL,pLR,1-pLPlayerBMixedStrategiesPlayerASofortheretoexistaNashequilibrium,B

mustbeindifferentbetweenplayingLeftor

Right;i.e.(1,2)(0,4)(0,5)(3,2)U,D,L,pLR,1-pLPlayerBMixedStrategiesPlayerA(1,2)(0,4)(0,5)(3,2)L,pLR,1-pLU,D,PlayerBMixedStrategiesPlayerAIfAplaysUphisexpectedpayoffis(1,2)(0,4)(0,5)(3,2)L,pLR,1-pLU,D,PlayerBMixedStrategiesPlayerAIfAplaysUphisexpectedpayoffis

IfAplaysDownhisexpectedpayoffis(1,2)(0,4)(0,5)(3,2)L,pLR,1-pLU,D,PlayerBMixedStrategiesPlayerAIfthenAwouldplayonlyUp.ButtherearenoNashequilibriainwhichA

playsonlyUp.(1,2)(0,4)(0,5)(3,2)L,pLR,1-pLU,D,PlayerBMixedStrategiesPlayerAIfDown.ButtherearenoNashequilibriainwhichAplaysonlyDown.thenAwouldplayonly(1,2)(0,4)(0,5)(3,2)L,pLR,1-pLU,D,PlayerBMixedStrategiesPlayerASofortheretoexistaNashequilibrium,A

mustbeindifferentbetweenplayingUpor

Down;i.e.(1,2)(0,4)(0,5)(3,2)L,pLR,1-pLU,D,PlayerBMixedStrategiesPlayerASofortheretoexistaNashequilibrium,A

mustbeindifferentbetweenplayingUpor

Down;i.e.(1,2)(0,4)(0,5)(3,2)L,pLR,1-pLU,D,PlayerBMixedStrategiesPlayerASofortheretoexistaNashequilibrium,A

mustbeindifferentbetweenplayingUpor

Down;i.e.(1,2)(0,4)(0,5)(3,2)L,R,U,D,PlayerBMixedStrategiesPlayerBPlayerASothegame’sonlyNashequilibriumhasA

playingthemixedstrategy(3/5,2/5)andhas

Bplayingthemixedstrategy(3/4,1/4).(1,2)(0,4)(0,5)(3,2)U,D,L,R,MixedStrategiesPlayerBPlayerAThepayoffswillbe(1,2)withprobability(1,2)(0,4)(0,5)(3,2)U,D,L,R,9/20MixedStrategiesPlayerBPlayerAThepayoffswillbe(0,4)withprobability(0,4)(0,5)(3,2)U,D,L,R,(1,2)9/203/20MixedStrategiesPlayerBPlayerAThepayoffswillbe(0,5)withprobability(0,4)(0,5)U,D,L,R,(1,2)9/203/206/20(3,2)MixedStrategiesPlayerBPlayerAThepayoffswillbe(3,2)withprobability(0,4)U,D,L,R,(1,2)9/203/20(0,5)(3,2)6/202/20MixedStrategiesPlayerBPlayerA(0,4)U,D,L,R,(1,2)9/203/20(0,5)(3,2)6/202/20Mix