Reduced Form Information Design: Persuading a Privately Informed Receiver

We study information design problems where the designer’s payoff is a step function of the posterior mean of the state induced by her signals. Settings where the designer’s payoff depends on the receiver’s actions, the receiver’s payoff is affine in the state, and the receiver’s actions belong to a finite set are special cases. To maximize her payoff, the designer needs to induce certain atomic distributions over posterior means. We show that the relevant set of posterior means can be characterized in terms of a collection of convex constraints. Leveraging this characterization, we provide a reduced form approach to information design. In this approach, the designer first solves a convex optimization problem, where she optimizes over the aforementioned set and then she constructs an information structure that is consistent with the optimal solution. The approach is versatile and tractable. We apply it to characterize the optimal information structures when the receiver is privately informed and establish the optimality of information structures based on a laminar partition of the set of states.