3 A Repeated Prisoner\'s Dilemma Game Suppose that two players play an infinite-

Question

3 A Repeated Prisoner's Dilemma Game Suppose that two players play an infinite-horizon Prisoner's Dilemma game, with discount factor ? > 0 The payoffs during each stage of the game are given below Player 2 Player 11,1 DI 4,--4 0.0 If player 1 receives the payoff pk n stage k then Player l's total payoff in the repeated game is Similarly, if Player 2 receives the payoff qk in stage k then player 2's total payoff is 1 +62+8s+. Find the conditions on the discount factor ? such that cooperation can be supported in the infinitely- repeated game (you can use the grim-trigger strategy profile discussed in Watson Chapter 22)

Explanation / Answer

We consider the strategy with punishment factor i.e. If player i doesn't cooperate in stage , then he is punished by other player at stage t+1

Strategy with cooperation: (C,C,C,....), payoff = 1 + delta+ delta^2 +..... = S1

Stategy with deviation: let the players deviate to D at some time t (from 0 to infinity):

(C,C,....,D,D,D,....) = 1+ delta + delta^2 +....+ 4*delta^t + 0*delta^{t+1} + 0*delta^{t+2} + .... = S2

To achieve cooperation => S1 >= S2

Now, till time t-1 both the strategies are same, we start from time 't'.

delta^t + delta^{t+1} + ...... >= 4*delta^t + 0*delta^{t+1} + ....

1 + delta + delta^2 + ..... >= 4 + 0*delta+ 0*delta^2 + .....

1/(1-delta) >= 4      ====>    1-delta <= 1/4. ====> delta >= 3/4

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