Reputation Effects under Short Memories

I analyze a novel reputation game between a patient seller and a sequence of myopic consumers, in which the consumers have limited memories and do not know the exact sequence of the seller's actions. I focus on the case where each consumer only observes the number of times that the seller took each of his actions in the last K periods and players have monotone-supermodular payoffs, namely, the patient player's payoff is strictly increasing in his opponent's action and is strictly decreasing in his own action, and both players' stage-game payoff functions have strictly increasing differences. With positive probability, the seller is a commitment type that exerts high effort in every period.