11/20/20081The Prisoner’s DilemmaIt’s so famous, the payoffs have names!T>R>P>SClyde BarrowCooperateDefectT – temptationR – rewardP – punishmentS – suckerBonnie ParkerCooperateR , RS , TDefectT , SP, PFinitely Repeated Prisoner’s DilemmaClyde BarrowCooperateDefectBonnie ParkerCooperate3 , 31 , 4Defect4 , 12, 2“Solving” the Prisoner’s Dilemma means finding ways of achieving cooperationTit-for-Tat invented by Anatol RapoportBegin by playing Cooperate (Nice strategy)Each round, do what opponent did last roundFinitely Repeated Prisoner’s DilemmaTit-for-Tat performed well in Axelrod’s tournamentTit-for-Tat vs. Tit-for-Tat is not an SPE in a finitely repeated PD, no matter how many times it is repeated.Tit-for-Tat vs. Tit-for-Tat is not a Nash Equilibrium in the finitely repeated PDAxelrod’s results are not robustWhat if we repeated the PD infinitely many times?Repeated Play of Prisoner’s DilemmaHow do we represent preferences over outcomes of repetitions of a game? The Present Value of $x n years from now:PV = $x/(1+r)nPV $x/(1 r)We assume that a player “discounts” future payoffs the same way we discount future moneyLet denote the discount factor for future payoffs. Here, is like 1/(1+r) We value a string of payoffs 2, 2, 3, 2 as10<<δ232232δδδ+++δRepeated Play of Prisoner’s DilemmaDiscounted sums:10<<δnnSδδδ++++="211211++++++=nnSδδδ"Solving for yieldsExercise: What is nnnSS⋅+=+=+δδ11nSδδ−−=+111nnSnnTδδδ+++="2Repeated Play of Prisoner’s DilemmaAnswer: , so that1)(1−==−nnnSSTδδδδ−−=+11nnT
has intentionally blurred sections.
Sign up to view the full version.