Cooperation in Repeated Games When the Number of Stages is not Commonly Known - Neyman, Abraham (Econometrica, 1999)
An exponentially small departure from the common knowledge assumption on the number T of repetitions of the prisoners' dilemma already enables cooperation. More generally, with such a departure, any feasible individually rational outcome of any one-shot game can be approximated by an equilibrium of a finitely repeated version of that game. The departure from common knowledge is small in the following sense:(1) the players know T with precision +/-K; (2) with probability 1 - epsilon, the players know T precisely; moreover, this knowledge is mutual of order epsilon T; and (3) the deviation of T from its finite expectation is exponentially small.