Persuasion and Optimal Stopping

We develop a duality-based first-order approach to dynamic persuasion in optimal stopping problems with general action-, state-, and time-dependent preferences. A direct-communication reduction recasts the design problem as a semi-static program over joint distributions of stopping beliefs and times; strong duality and a near-necessary first-order condition then reduce it to a one-dimensional differential equation in a multiplier that prices the agent's continuation incentive, characterizing the optimum as a \emph{concavification} of a multiplier-augmented payoff. We demonstrate the method in three applications: dynamic binary persuasion, where optimal policies combine \emph{suspense generation} with \emph{action-targeting}; a structural result by which the principal's time-risk preferences alone determine whether suspense is optimal; and dynamic linear persuasion, where the optimum is \emph{dynamic tail-censorship}.