Information design and reputations in the frequent-interaction limit

Consumer welfare may be higher when platforms sometimes conceal product quality information, because this can incentivize firms to invest more in quality. I study an infinite horizon model in which a long-run firm with a persistent type interacts with a sequence of short-run consumers. The focus is on the case where there are frequent consumer purchases. When a designer commits in advance to an information policy to maximize average consumer welfare, the approximately optimal policy and the resulting welfare depend on whether the policy may condition on the full history or only the publicly observable history. An approximately optimal full history policy approximates the upper bound welfare that is consistent with firm participation. It may do so by privately keeping track of a firm rating, threatening a hypothetical punishment (revealing firm type) off the path of play, while never giving consumers information on the path of play. However, learning must occur on the path of play in an approximately optimal public history policy. A public history policy can only achieve the upper bound welfare conditional on beliefs being above some cutoff. The approximately optimal public history policy features a learning cutoff: The firm must pass an initial test, driving beliefs up to the cutoff, which is essential for rewarding the firm.