Question

3. (Level A) Suppose the following Prisoner's Dilemma is repeated infinitely 112 C D C 2, 2 0, 3 D 3, 0|1, 1 Let uļ be the payoff to player i in period t. Player i (i = 1, 2) maximizes her. average discounted sum of payoffs, given by ( where δ is the common discount factor of both players Suppose the players try to sustain (C, C) in each period by the Grim Trigger strategy. That is, each player plays the following strategy: Play C in the first period . In any other period, play C if both players have played C in all previous periods; and play D otherwise (i.e., if any one player has ever played D before) Find the minimal discount factor δ for which the adoption of the above grim trigger strategy by both players constitutes a subgame perfect equilibrium of the infinitely repeated game.

Answer 1

No of stage Game: < D, D > Now using Grim Trigger strategy, to sustain cooperation rightarrow If both cooperate, then Present discounted value of Payoffs V^C = 2 + 2 delta + 2 delta^2 + 2 delat^3 + = 2/1 - delta, Now If any one defects, then PDV of deviation payoffs V^D = 3 + 1 delta + 1 delta^2 + 1 delat^3 + = 3 + delta/1 - delta Now cooperate if, V^C greaterthanorequalto V^D, 2/1 - delta greaterthanorequalto 3 + delta/1 - delta = > 2 greaterthanorequalto 3 - 3 delta + delta so 2 delta greaterthanorequalto 1 = > delta greaterthanorequalto 1/2 thus Min delta = 1/2.