Reputation with long run players and imperfect observation

Previous work shows that reputation results may fail in repeated games between two long-run players with equal discount factors. We restrict attention to an infinitely repeated game where two players with equal discount factors play a simultaneous move stage game where actions of player 2 are imperfectly observed. The set of commitment types for player 1 is taken as any (countable) set of finite automata. In this context, for a restricted class of stage games, we provide a one sided reputation result. If player 1 is a particular commitment type with positive probability and player 2's actions are imperfectly observed, then player 1 receives his highest pay-off, compatible with individual rationality, in any Bayes-Nash equilibria, as agents become patient.