The persuasion duality

We present a unified duality approach to Bayesian persuasion. The optimal dual variable, interpreted as a price function on the state space, is shown to be a supergradient of the concave closure of the objective function at the prior belief. Strong duality holds when the objective function is Lipschitz continuous. When the objective depends on the posterior belief through a set of moments, the price function induces prices for posterior moments that solve the corresponding dual problem. Thus, our general approach unifies known results for one-dimensional moment persuasion, while yielding new results for the multidimensional case. In particular, we provide a condition for the optimality of convex-partitional signals, derive structural properties of solutions, and characterize the optimal persuasion scheme when the state is two-dimensional and the objective is quadratic.