Persuasion and Matching: Optimal Productive Transport

We consider general Bayesian persuasion problems where the receiver’s utility is single-peaked in a one-dimensional action. We show that a signal that pools at most two states in each realization is always optimal and that such “pairwise” signals are the only solutions under a nonsingularity condition on utilities. Our core results provide conditions under which the induced receiver action is single-dipped or single-peaked on each set of nested signal realizations. We also provide conditions for the optimality of either full disclosure or negative assortative disclosure, where all signal realizations are nested. Methodologically, our results rely on novel duality and complementary slackness theorems. Our analysis extends to a general problem of assigning one-dimensional inputs to productive units, which we call “optimal productive transport.” This problem covers additional applications including matching with peer effects (assigning workers to firms, students to schools, or residents to neighborhoods), robust option pricing (assigning future asset prices to price distributions), and partisan gerrymandering (assigning voters to districts).