The Persuasion Duality

TL;DR

This paper introduces a duality framework for Bayesian persuasion, unifying existing results and providing new insights into optimal signaling schemes, especially in multi-dimensional settings with moment-based objectives.

Contribution

It develops a unified duality approach for Bayesian persuasion, extending known results to multi-dimensional cases and characterizing optimal signals and schemes.

Findings

Supergradient interpretation of the optimal dual variable.

Strong duality holds under Lipschitz continuity.

Characterization of optimal persuasion schemes in multi-dimensional settings.

Abstract

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 multi-dimensional case. In particular, we provide a necessary and sufficient condition for the optimality of convex-partitional signals, derive structural properties of solutions, and characterize the optimal persuasion scheme in the case when the state is two-dimensional and the objective is quadratic.