Cooperation, Secret Handshakes, and Imitation in the Prisoners' Dilemma

In the prisoners’ dilemma game, the only evolutionarily stable strategy is defection, even though mutual cooperation yields a higher payoff. Building on a paper by Robson (1990), we introduce mutants who have the ability to send a (costly) signal, i.e., the “secret handshake,” before each round of the game and to condition their actions on whether or not they observe the same signal from their opponent. A population playing the strategy “always defect” is vulnerable to secret handshake mutants who cooperate when they meet other secret handshakers and defect against other opponents. However, these secret handshakers are in turn vulnerable to a second round of mutants who imitate the secret handshake and then defect against all opponents. But now a new group of secret handshakers with a different secret handshake can arise. Thus, play can cycle between cooperation and defection. We study the dynamics of that cycling. We show that in the limit, as the probability of mutation goes to zero, cooperation occurs on average half the time. Using simulations to study the implications of our model when the mutation probability is larger than zero, we find that it is possible for cooperation to be sustained for long periods. In general, cooperation is favored when mutual cooperation has a large payoff advantage over mutual defection, and when the payoff advantage of unilateral defection is small. Surprisingly, however, there are cases where an increased payoff to unilateral defection actually raises the level of cooperation. Journal of Economic Literature Classification Numbers: C70, C72, D60. ∗ We thank Eddie Dekel for suggesting this line of research and for helpful conversations. We also thank Jeff Ely, Juuso Valimaki, Asher Wolinsky, and John Woodruff for their suggestions and comments. For computer assistance, we thank Mehmet Sayal and especially Cem Evrendilek. All remaining errors and imperfections are ours. ** Department of Economics, Northwestern University, 2003 Sheridan Road, Evanston, IL 60208; [email protected], [email protected].