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. When payoffs are monotone-supermodular, I show that the patient seller can approximately secure his commitment payoff in all equilibria as long as K is at least one. I also show that the consumers can approximately attain their first-best welfare in all equilibria if and only if their memory length K is lower than some cutoff. Although a longer memory enables more consumers to punish the seller once the seller shirks, it weakens their incentives to punish the seller once they observe him shirking