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Unformatted text preview: 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 Cooperate Defect 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 ¡ TitforTat invented by Anatol Rapoport ¡ Begin by playing Cooperate (Nice strategy) ¡ Each round, do what opponent did last round Finitely Repeated Prisoner’s Dilemma ¡ TitforTat performed well in Axelrod’s tournament ¡ TitforTat vs. TitforTat is not an SPE in a finitely repeated PD, no matter how many times it is repeated. ¡ TitforTat vs. TitforTat 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 < < δ 2 3 2 2 3 2 δ δ δ + + + δ Repeated Play of Prisoner’s Dilemma ¡ Discounted sums: 1 < < 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...
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This note was uploaded on 04/14/2010 for the course ECON 398 taught by Professor Emre during the Spring '07 term at University of Michigan.
 Spring '07
 Emre

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