Rationality and Bounded Information in Repeated Games

Actions in a repeated or stochastic game can in principle depend on all previous outcomes. Given this vast policy space, human players may often be forced to use heuristics that base actions on incomplete information, such as the outcomes of only the most recent trials. Here it is proven that such bounded rationality is often fully rational, in that the optimal policy with respect to some limited information about the game's history will coincide with the universally optimal policy, provided that the information used by all players satisfies two widely inclusive axioms. It is then shown how this result allows explicit calculation of subgame-perfect equilibria (SPEs) for any stochastic game. The technique is applied to the iterated Prisoner's Dilemma for the case of 1-back memory, and two sets of SPEs are derived. These solutions exhibit varying degrees of (individually rational) cooperation, which can be seen via the present framework to be a direct result of repeated interaction.