Optimality of Double Intervals in Persuasion: A Convex Programming Framework

We study information design problems, where the payoff of the information designer is an increasing step function of the posterior mean of the state her signals induce. Settings where the designer employs public signaling mechanisms to persuade multiple receivers (who have (i) finitely many actions that exhibit strategic complementarities, and (ii) best responses that are increasing in the state), and where her payoff is increasing in the receivers’ actions, are special cases. We provide a novel and tractable convex optimization framework for obtaining an optimal signaling mechanism. This mechanism is deterministic and admits a double-interval structure: it pools some “good” and “bad” states, and with each signal realization it associates at most two subintervals of the set of states.Our framework is broadly applicable, and we illustrate this through two very different applications. In the first application, we focus on receivers in a network who take binary actions (that can represent, e.g., adoption of a product of unknown quality), and characterize the optimal mechanism in terms of the network’s cores. Interestingly, as a consequence of the double-interval structure, the set of states for which a receiver takes action 1 is in general nonconvex. We show that in large random networks the designer can obtain asymptotically optimal mechanisms even in the absence of detailed network information, by using only the degree distribution. In our second application, we focus on persuading a single receiver whose type is private. Using a slight modification of our framework, we establish that the optimal mechanism is based on a laminar interval structure – a generalization of the double-interval structure – where the number of subintervals associated with each signal realization increases with the number of types.