Building Reputations via Summary Statistics

A patient seller interacts with a sequence of myopic consumers. Each consumer decides whether to trust the seller after she observes the number of times that the seller took each of his actions in the last K periods, but not the order with which these actions were taken. I assume that the seller's effort and consumers' trust are strategic complements, and that with positive probability, the seller is a commitment type who exerts the highest effort in every period. I show that the seller sustains his reputation for exerting the highest effort in all equilibria if and only if K is below some cutoff. Although a larger K allows more consumers to observe the seller's opportunistic behavior, it weakens consumers' incentives to punish the seller after they observe opportunistic behaviors. This effect undermines the seller's reputational incentives and lowers consumers' welfare. I also show that coarsening the summary statistics observed by the consumers may encourage the seller to sustain his reputation and may improve consumers' welfare.