Maintaining a permanent reputation with replacements

We study the impact of unobservable replacements on the sustainability of reputation effects in frequently repeated games played by a long run player facing a sequence of short run players. At the beginning of every period the long-run player is replaced with a new long run player with probability. The new long run player is either a commitment type who plays the same strategy in every period when he is in the game, or a normal type. The long run player's choice of stage game strategy is imperfectly observed by the short run players. We show that the long run player's payoff, in any Nash equilibrium, is bounded below by what he could get by committing to his most favorite commitment type strategy after every history of the game, even as his rate of impatience vanishes at the same rate as his replacement probability. Hence arbitrarily infrequent replacements are sufficient to prevent reputations and their effects from disappearing when the stage game is played frequently enough.