A patient seller decides whether to build a reputation for exerting high effort in front of a sequence of 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 show that (i) the seller’s payoff from building a reputation is at least his commitment payoff for all K and in all equilibria, and (ii) the seller sustains his reputation for exert high 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 their incentives to punish the seller after they observe opportunistic behavior. This effect undermines the seller’s reputational incentives and lowers consumers’ welfare. I also provide conditions under which coarsening the summary statistics observed by the consumers encourages the seller to sustain his reputation and improves consumers’ welfare.
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