Consider the following Prisoner’s Dilemma game with utilities (UJohn Wayne, UMontgomery Clift). Table 1: The Prisoner’s Dilemma Game.
Montgomery Clift
cooperate defect
John Wayne cooperate 9,9 2,12
defect 12,2 8,8
(a) Find the Nash equilibrium if the game is played once.
(b) Suppose Mr. Walter Brennan offers to (credibly) enforce cooperation between John Wayne and Montgomery Clift. How costly can Mr. Brennan’s services be before it makes sense for Mr. Wayne and Clift to hire Mr. Brennan’s services? If the cost is exactly that much, how much will each of Mr. Wayne and Clift have to pay Mr. Brennan?
(c) Suppose this game is played an infinite amount of times, and that a dollar earned one period in the future is worth β, where 0 < β < 1, so that the present value of playing forever a strategy that yields payoff p in every period is p/(1 −β). Suppose also that if a player chooses cooperate in every previous round, the other player chooses to cooperate in the next round, but if a player chooses defect, the other player chooses defect in all future periods. For what values of β would you expect cooperation to emerge voluntarily between Mr. Wayne and Mr. Clift?
(d) Suppose the game were played twice, rather than an infinite number of times. What would you expect the equilibrium to be in period 2? How does what happens in period 2 affect the ability of a punishment strategy like in part (c) to work in enforcing cooperation in period 1. Does the value of β matter in this case? Explain.