Grim Trigger in the repeated Prisoner's Dilemma

1. Grim 'D'igger in the Repeated Prisoner's Dilemma. In one instance of the prisoner's dilemma, each player in order to confer a benet b > c onto the other player this prisoner's dilemma. are therefore: choosw whether to pay some cost c > 0 . The payos from a single iteration of The repeated prisoner's dilemma is built out of several stages, each of which is a copy of the above game. At the end of each stage, the two players repeat the prisoner's dilemma again in the repeated prisoner's dilemma. is a rule with probability 6, where 0 S 6 S 1. A streteg which determines whether a player will cooperate or defect in each given stage. This rule 's auctions in previous rounds. may depend on which round it is, and on either player following rule: cooperate if both For example, the grim. trigger strategy is described by the of this problem is to show that players have never defected, and defect. otherwise. The goal the strategy pair in which both players play grim trigger is a Nash equilibrium if 6 > f. (8) Suppose that player 1 and player 2 are both following the grim trigger strategy. What actions will be played in each stage of the repeated game? What are the payoffs to players 1 and 2 in each stage? (b) Using your result from part la, write down the expected payo' to player entire repeated prisoner's dilemma in terms of c, b, and 6. 1 from the 1 Hint: Remember that, if |6| ;. So far we have focused on the Grim Trigger because it is a relatively simple strategy to understand, but not necessarily because we think it is used in practice. Importantly, many of the insights we have learned from studying the Grim Trigger generalize to any Nash equilibrium. It Faller (f) Show that in any Nash equilibrium in which both players play C in each period, player 2 must cooperate less in the future if player 1 were to deviate and play D at any period instead of C. Interpret this result in terms of 'reciprocity