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 alread y 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 follo ws: (i) With probability one, the players know T with precision +/-K. (ii) With probability 1- epsilon, the players know T precisely; moreover, this k nowledge is mutual of order epsilon T. (iii) The deviation of T from its fi nite expectation is exponentially small.