Marginal Reputation

We study reputation formation where a long-run player repeatedly observes private signals and takes actions. Short-run players observe the long-run player's past actions but not her past signals. The long-run player can thus develop a reputation for playing a distribution over actions, but not necessarily for playing a particular mapping from signals to actions. Nonetheless, we show that the long-run player can secure her Stackelberg payoff if distinct commitment types are statistically distinguishable and the Stackelberg strategy is confound-defeating. This property holds if and only if the Stackelberg strategy is the unique solution to an optimal transport problem. If the long-run player's payoff is supermodular in one-dimensional signals and actions, she secures the Stackelberg payoff if and only if the Stackelberg strategy is monotone. Applications include deterrence, delegation, signaling, and persuasion. Our results extend to the case where distinct commitment types may be indistinguishable but the Stackelberg type is salient under the prior.