game theory4 博弈论 英文

,IntroductiontoGameTheoryLecture4,ReviewMixedstrategyNashequilibriumreviewexamplebestresponsefunctionsgraphsGameEliminationofstrategiesthatarestrictlydominatedbymixedstrategiesillustrationexample,2,Preview,/32,ReviewMSNEEliminationByMixingSummary,Strictvs.weakdominanceEliminationofweaklydominatedstrategiesleadsto:strictNashequilibriabutcaneliminatenonstrictNashequilibriaThatiswhyweonlyeliminatestrictlydominatedstrategies,3,Review,ReviewMSNEEliminationByMixingSummary,/32,MixedstrategyNEneedformakingoneselfunpredictableleadstomixingstrategiesMixedstrategy:playerchoosesaprobabilitydistribution(p1,p2,.,pN)overhersetofactionsratherthanasingleactionIfthereisnoNEwithoutmixing,wewillfindatleastoneMSNE(Nash-proof)IfNEwithoutmixingexists,wemayfindadditionalMSNE,4,Review,ReviewMSNEEliminationByMixingSummary,/32,ReviewMixedstrategyNashequilibriumreviewexamplebestresponsefunctionsgraphsGameEliminationofstrategiesthatarestrictlydominatedbymixedstrategiesillustrationexample,5,Preview,ReviewMSNEEliminationByMixingSummary,/32,twoNE:(B,B)and(S,S)anyMSNE?P1mustbeindifferentbetweenBandS(otherwisenotmixing,playingpurestrategy):q*2+(1-q)*0=q*0+(1-q)*1=q=1/3BSP2mustbeindifferentbetweenBandS:p*1+(1-p)*0=p*0+(1-p)*2=p=2/3,6,MixedStrategyNE,ReviewMSNEEliminationByMixingSummary,/32,Ifq1/3:BisbetterthanSIfq=1/3:BisasgoodasS0ifq1/30ifp2/3(2/3,1/3);(1/3,2/3),7,MixedStrategyNE,ReviewMSNEEliminationByMixingSummary,/32,player1(2)choosesBwithprobabilityp(q)andSwithprobability1-p(1-q),8,MixedStrategyNE,ReviewMSNEEliminationByMixingSummary,/32,twoNE:(T,R)and(B,L)anyMSNE?P1mustbeindifferentbetweenTandB(otherwisenotmixing,playingpurestrategy):q*0+(1-q)*0=q*2+(1-q)*0=q=0TBP2mustbeindifferentbetweenLandR:p*1+(1-p)*2=p*2+(1-p)*1=p=1/2,9,MixedStrategyNE,ReviewMSNEEliminationByMixingSummary,/32,Ifq0:BisbetterthanTIfq=0:BisasgoodasS0ifq0B1(q)=p:0p1ifq=01ifp1/2p1/2,10,MixedStrategyNE,ReviewMSNEEliminationByMixingSummary,/32,player1choosesTwithprobabilitypandBwithprobability1-pplayer2choosesLwithprobabilityqandRwithprobability1-q,11,MixedStrategyNE,ReviewMSNEEliminationByMixingSummary,/32,notjustmathematicalexerciseexamples:matchingpenniesrockpaperscissorspenaltykicksbaseballpitchestennisservicetravelagenciespricingpoliciesmakingyourselfunpredictable,12,MixedStrategyNE,ReviewMSNEEliminationByMixingSummary,/32,Holmesvs.MoriartyHolmes(agenius)getsonthetrainLondon-Canterbury-DovertogettoDoverMoriarty(equallysmartguy)rentsaspecialandfollowsHolmesHolmespreferstogetoffondifferentstationMoriartyprefersthesamestation,13,MixedStrategyNE,ReviewMSNEEliminationByMixingSummary,/32,Holmesvs.MoriartyHolmes:MoriartyknowsthatIwanttogotoD,soIdbettergetoffinCHolmes:MoriartyisalmostassmartasIamheknowsthisandgoestoC,soIdbettergotoDHolmes:ButMoriartyknowsthatIknow,14,MixedStrategyNE,ReviewMSNEEliminationByMixingSummary,/32,sowhatevermyreasoningis,Moriartywillfigureitoutandgetme,15,MixedStrategyNE,ReviewMSNEEliminationByMixingSummary,/32,SolutiontoHolmesdilemma:IfHolmeshimselfdoesnotknowwhichactionhewillchoose,MoriartycannottakeadvantageofknowingHolmessaction=Ignoranceisabliss,16,MixedStrategyNE,ReviewMSNEEliminationByMixingSummary,/32,nopurestrategyNEplayershavetomix:forexample:(,),(,)couldthiswork?stillnoNE,weneeddifferentprobabilitiesformixing,17,MixedStrategyNE,ReviewMSNEEliminationByMixingSummary,/32,howabout:(1/3,2/3),(3/4,1/4)couldthiswork?Yes,thisleadstooneMixedstrategyNE,18,MixedStrategyNE,ReviewMSNEEliminationByMixingSummary,/32,HolmesmustbeindifferentbetweenDandC(otherwisenotmixing,playingpurestrategy):q*0+(1-q)*8=q*4+(1-q)*(-4)=q=3/4DCMoriartymustbeindifferentbetweenDandC:p*8+(1-p)*(-4)=p*(-4)+(1-p)*2=p=1/3,19,MixedStrategyNE,ReviewMSNEEliminationByMixingSummary,/32,1ifq3/40ifp1/3MSNE:(1/3,2/3);(3/4,1/4),20,MixedStrategyNE,ReviewMSNEEliminationByMixingSummary,/32,ReviewMixedstrategyNashequilibriumreviewexamplebestresponsefunctionsgraphsGameEliminationofstrategiesthatarestrictlydominatedbymixedstrategiesillustrationexample,21,Preview,ReviewMSNEEliminationByMixingSummary,/32,nopurestrategyisdominatedbyanotherpurestrategyhowever,D+EstrictlydominatesF(3.5,5.5,2.5)(3,4,2),22,EliminationbyMixedStrategies,ReviewMSNEEliminationByMixingSummary,/32,Example:onlystrategythatisneverbestresponsetoopponentsactionsisTthereexistspand(1-p)suchthat:pM+(1-p)BT,23,EliminationbyMixedStrategies,ReviewMSNEEliminationByMixingSummary,/32,pM+(1-p)BTp*1+(1-p)*43=p4=p1/5p*3+(1-p)*21=alwaystruewecanchooseforexamplep=1/4,24,EliminationbyMixedStrategies,ReviewMSNEEliminationByMixingSummary,/32,pM+(1-p)BT1/4M+3/4B=(13/4,17/4,9/4)(3,4,1)=T,25,EliminationbyMixedStrategies,ReviewMSNEEliminationByMixingSummary,/32,onlystrategythatisneverbestresponsetoopponentsactionsisRthereexistspand(1-p)suchthat:pL+(1-p)CR,26,EliminationbyMixedStrategies,ReviewMSNEEliminationByMixingSummary,/32,pL+(1-p)CRp*7+(1-p)*25=p3/5p*4+(1-p)*54=p(5,4)=R,27,EliminationbyMixedStrategies,ReviewMSNEEliminationByMixingSummary,/32,afteriterativeeliminationofdominatedstrategiesweget:nofurthereliminationispossiblebecauseeveryactionisbestresponsetosomeofopponentsactions,28,EliminationbyMixedStrategies,ReviewMSNEEliminationByMixingSummary,/32,MixedstrategiesNashequilibriummakingyouractionsunpredictableduopoly,sportIterativeeliminationofstrictlydominatedstrategiesstrategycanbedominatedbypurestrategystrategycanbedominatedbymixedstrategyHomeworkdeadlinenextWednesday,29,Summary,ReviewMSNEEliminationByMixingSummary,/32,GameParticipation:youcangetupto1pointphonenumbers+IDswillbestrictlyprotectedparticipationisanonymous,nobodywillknowwhoplayedwhatinformationaboutpayoffwillbeinISIS,30,Game,ReviewMSNEEliminationByMixingSummary,/32,2playerssplit1point(tobeaddedtothefinalexam