Cooperation in repeated games when the number of stages is not commonly known

It is shown that 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 a subgame perfect 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 ±K. (ii) With probability 1 - ε, the players know T precisely; moreover, this knowledge is mutual of order εT. (iii) The deviation of T from its finite expectation is exponentially small.