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1、VonNeumann’sCardModel:BluffingAveryclassicexample11.TheSetupoftheGameTocheckissimplytobetzerochips,I;Toraiseistobetmorethanthelastplayer,I;Tocallistobetthesameasthelastplayer,II;Aplayerwhodoesnotcallmustfold,II2VonNeumann’scardmodelonlycontainingK,Q,Jofr
2、edhearts,i.e.,onlythreecards.TheGamerulesare:KbeatsQ,QbeatsJ.Beforethedeal,eachplayermustputananteof$1intothepot.Afterthedeal,thereisaroundofbetting,duringwhichplayerIImayfold.IfIIfolds,thenplayerIwinsthepot,nomatterwhohasthebetterhand.IfplayerIIdoesn’tf
3、old,thereisashowdown,afterwhichtheplayerwiththehighercardwinsthepot.TheshowdownoccurswhenplayerIIcallsPlayerI’slastbetbymakinghistotalcontributiontothepotequaltoplayerI’s.3InVonNeumann’smodel,playerImovesfirst.PlayerIcaneithercheckorraise(byadding$1tothe
4、pot).SoplayerIchecks,player2mustcall,andthenthereisalsoashowdown.IfplayerIraises,playerIIhasachoice:foldorcall.Tocheckissimplytobetzerochips;Toraiseistobetmorethanthelastplayer;Tocallistobetthesameasthelastplayer;Aplayerwhodoesnotcallmustfold.4Thechancem
5、ovethatbeginsthedeckintooneofsixequallylikelyorders.ThefirstcardisdealttoplayerIandthensecondcardtoplayerII.Pleasefindbehaviouralstrategiesforeachplayerofequilibrium.(Youshouldgivefullprocessofsolvingthisgamebythemethodsofbehaviouralstrategy,backwardindu
6、ction).5DealKQJKJQQKJQJKJKQJQK(1/6)(1/6)(1/6)(1/6)(1/6)(1/6)III<½>K<½><½>Q<½><½>J<½>(ε)(δ)checkcheck(ε)checkcheckcheckcheck(δ)(+1,-1)(+1,-1)(-1,+1)(+1,-1)(-1,+1)(-1,+1)raiseraiseraiseraiseraise(1-δ)raiseIIK(1-ε)(1-ε)call(1-δ)foldfoldcall(+1,-1)(-2,+2)(+1
7、,-1)(-2,+2)IIJcallcallfoldfold(+1,-1)(+2,-2)(+1,-1)(+2,-2)II<α>Q<1-α>foldcallfold(q)call(q)(1-q)(1-q)6(+1,-1)(+2,-2)(+1,-1)(-2,+2)2.ThePreliminaryAnalysisoftheGameWeshallstartfromtheinformationsets.WhenIholdingtheKing,thingsarecleartoI.WhenIIholdingtheKi
8、ngortheJack,thingsarealsocleartoII.7WhenPlayerIholdingtheQueen,therecanbetwocircumstances.(1).IfplayerIIholdingtheKing,hewilldefinitelycall.Underthiscircumstances,playerImustcheck.(2).IfplayerIIholdingtheJack,hewilldefinit
