Strategy21Handout

# Strategy21Handout - The Prisoner's Dilemma It's so famous...

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11/20/2008 1 The Prisoner’s Dilemma It’s so famous, the payoffs have names! T>R>P>S Clyde Barrow Cooperate Defect T – temptation R – reward P – punishment S – sucker Bonnie Parker Cooperate R , R S , T Defect T , S P, P Finitely Repeated Prisoner’s Dilemma Clyde Barrow Cooperate Defect Bonnie Parker Cooperate 3 , 3 1 , 4 Defect 4 , 1 2, 2 “Solving” the Prisoner’s Dilemma means finding ways of achieving cooperation Tit-for-Tat invented by Anatol Rapoport Begin by playing Cooperate (Nice strategy) Each round, do what opponent did last round Finitely Repeated Prisoner’s Dilemma Tit-for-Tat performed well in Axelrod’s tournament Tit-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 PD Axelrod’s results are not robust What if we repeated the PD infinitely many times? Repeated Play of Prisoner’s Dilemma How do we represent preferences over outcomes of repetitions of a game? The Present Value of \$x n years from now: PV = \$ x/(1+r) n PV \$ x/(1 r) We assume that a player “discounts” future payoffs the same way we discount future money Let denote the discount factor for future payoffs. Here, is like 1/(1+r) We value a string of payoffs 2, 2, 3, 2 as 1 0 < < δ 2 3 2 2 3 2 δ δ δ + + + δ Repeated Play of Prisoner’s Dilemma Discounted sums: 1 0 < < δ n n S δ δ δ + + + + = " 2 1 1 2 1 1 + + + + + + = n n S δ δ δ " Solving for yields Exercise: What is n n n S S + = + = + δ δ 1 1 n S δ δ = + 1 1 1 n n S n n T δ δ δ + + + = " 2 Repeated Play of Prisoner’s Dilemma Answer: , so that 1 ) ( 1 = = n n n S S T δ δ δ δ = + 1 1 n n T

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