ConstraintsinRepeatedGames-UniversityofPennsylvania在重复博弈的约束-宾夕法尼亚大学

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1、ConstraintsinRepeatedGamesRationalLearningLeadstoNashEquilibrium…sowhatisrationallearning?Kalai&Lehrer,1993Rationallearningis…BayesianUpdatingfrequentistvs.BayesianstatisticsWhatisRationalLearning?frequentistvs.BayesianstatisticsFrequentistApproachAssumeacoin10timesanditcomesupheads8timesAfrequ

2、entistapproachwouldconcludethatthecoincomesupheads80%ofthetimeUsingtherelativefrequencyasaprobabilityestimate,wecancalculatethemaximumlikelihoodestimate(MLE)FrequentistMLEnotalwaysaccurateinallcontextsFormthemodelassertingP(head)=m,andsanobservedsequence,theMLEis:argmaxmP(s

3、m)BayesianApproach

4、Allowsustoincorporatepriorbeliefse.g.,thatourcoinisfair(whynot?)Wecanmeasuredegreesofbelief,whichcanbeupdatedinthefaceofevidenceusingBayes’theoremP(m

5、s)=(P(s

6、m)*P(m))/P(s)WealreadyhaveP(s

7、m),wecanquantifyP(m)andignorethenormalizationfactorP(s)ArgmaxmP(m

8、s)=.75forP(m)=6m(1-m)UnderWhatCond

9、itions?InfinitelyrepeatedgamesubjectivebeliefsaboutothersarecompatiblewithtruestrategiesPlayersknowtheirownpayoffmatricesPlayerschoosestrategiestomaximizetheirexpectedutilityPerfectlymonitoredDiscountedpayoffs…musteventuallyplayaccordingtoaNashequilibriumoftherepeatedgameWhatIsn’tNeededassumpti

10、onsabouttherationalityofotherplayersknowledgeofthepayoffmatricesofotherplayersDefinitionsAgameisperfectlymonitoredifallplayershaveaccesstothecompletehistoryofthegameuptothepointwheretheyarecurrentlyat.discountingintroducesafactorthatfuturepayoffsaremultipliedby:ui(f)=(1-i)∑t=0∞Ef(xit+1)itnote

11、therelationtogeometricseries…continuedbeliefsarecompatiblewithtruestrategiesifthedistributionoverinfiniteplaypathsinducedbythebeliefisabsolutelycontinuouswithrespecttothatofthetruestrategiesAmeasurefisabsolutelycontinuouswithrespecttog(denotedf<<g)ifeveryeventhavingapositivemeasureaccording

12、tofalsohasapositivemeasureaccordingtog.MoreDefinitionsLet>0andletand’betwoprobabilitymeasuresdefinedonthesamespace.is-closeto’ifthereisameasurablesetQsatisfying:(Q)and’(Q)aregreaterthan1-foreverymeasurablesetAQ(1-)’(