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1、SolutionConcept:SequentialEquilibriaofExtensive-FormGames(II)12.SequentialRationalityatInformationStateswithPositiveProbabilities2.1AFewKeyDefinitions2SequentiallyRationalIfplayeriwouldactuallywanttodoThestrategyforwhathisstrategyspecifiesforhimatsplayeriisinSwheninformationstatesactually⇒sequent
2、iallyioccurs.rational.3ChanceProbabilityForanyextensive-formgameGe,thegivenstructureofthegamespecifiesachanceprobabilityforeverybranchimmediatelyfollowingachancenode.tee2,-21.a2.0MRaisedFoP1,-1reldas1,-1s.5000.5teble-2,2aMck1.braise2.0foldPa1,-1-1,1ss4MoveProbabilitytee2,-21.a2.0MAbehavioural-str
3、ategyRaiseprofileσspecifiesdFoP1,-1reldas1,-1s0.50amoveprobability0.5teble-2,2aMforeverybranchck1.braise2.0immediatelyfollowingfoldPa1,-1adecisionnode.-1,1ssσ(m)isthemoveprobabilityofabranchwithmovelabelmi.simmediatelyfollowinganodecontrolledbyplayeriwithinformationstates.5⎯P(y
4、σ,x)Foranybehaviou
5、ralstrategyprofileσ,andanytwonodesxandy,ifyfollowsx,let⎯P(y
6、σ,x)denotethemultiplicativeproductofallthechanceprobabilitiesandmoveprobabilitiesspecifiedbyσforthebranchesonthepathfromxtoy.tee2,-21.a2.0MRaisedFoP1,-1reldas1,-1s.5000.5teble-2,2aMck1.braise2.0foldPa1,-1-1,1ss6Ifydoesnotfollowsx,then⎯P(
7、y
8、σ,x)=0.So,ifallplayerschosetheirmovesaccordingtoσandifalltheplayofthegamestartedatx,⎯P(y
9、σ,x)istheconditionalprobabilitythatthepathofplaywouldgothroughyafterx.tee2,-2M1.aRaise2.0dFoP1,-1reldas1,-1s.5000.5teble-2,2aMck1.braise2.0foldPa1,-1-1,1ss7UtilitiesdenotethesetofallterminalnodesintheΩegame
10、G.denotethepayofftoplayeriatterminalwy()inodey.Uxii()σσ=∑y∈ΩP(,)()yxwy.theexpectedpayofftoplayeriiftheplayofthegamebeganinnodexUx()σratherthantheroot,iallplayersthereafterchosetheirmovesasspecifiedbyσ.82.2What’stheMeaningofSequentialRationality?9Supposethatplayerihasaninformationstatesthatoccursa
11、tonlyonenodex,so,Y={x}.sThenthebehavioural-⇔σσ∈argmaxUx(,ρ).strategyprofileσisis..i−isssequentiallyrationalρss∈Δ()Dtee2,-2M1.aRaise2.0dFoP1,-1reldas1,-1s.5000.5teble-2,2aMck1.braise2.0foldPa1,-1-1,1ss10σwouldmaximizepl
