Cooperation in the Repeated Prisoners' Dilemma when the Number of Stages Is Not Commonly Known

Abraham Neyman

It has often been observed that cooperative behavior emerges in actual play of the repeated prisoners' dilemma. This observation seems to be in conflict with the fact that, in any finite repetition of the prisoners' dilemma, all Nash equilibria (and even all correlated equilibria) lead to the non-cooperative outcome in each stage. In this paper we show that a very small departure from the common knowledge assumption on the number, T, of repetitions already enables cooperation. More generally, with such a departure, any feasible individually-rational outcome of any one-shot game can be approximated by a Nash equilibrium of a finitely-repeated version of that game. The sense in which the departure from common knowledge is "small" is as follows: (i) With probability one, the players know T with precision +- 1. (ii) With probability 1 - %, the players know T precisely; moreover, this knowledge is mutual to degree %T. (iii) the deviation of T from its expectation is extremely small.

January, 1995
Published in: 
(revised in DP #162)