Dynamic Persuasion: Managing Reputation under Censored Feedback

Abstract

We study a long-run sender who repeatedly communicates about a state to a sequence of myopic receivers and whose credibility is a public score updated from past interactions. Receivers face a lemon loss and an opportunity loss, and their behavior depends on these only through a single index, the tolerance. Because the score records only detected misrepresentations, it rests above the sender's true honesty rate; we characterize this credibility premium in closed form. In a fluid limit in which the score adjusts slowly, no policy earns more than operating at a single credibility level, and the sender's gain from managing credibility equals the value of the premium: against receivers who correct for imperfect detection, that gain vanishes. At a fixed adjustment speed, every policy is bounded by the fluid value plus a term linear in the speed. We then study three policy classes, linear, (s,S), and threshold, obtain the optimal parameters of the linear and threshold classes in closed form, show how receiver losses shape them, and show that, under an approximation, the optimal (s,S) band collapses to a threshold. Simulations of the dynamics agree with the closed forms and indicate where the single-level conclusion stops: bands pay when switching modes is costly, and cycles pay when demand is seasonal.