Optimal Transport of Information

TL;DR

This paper introduces a novel optimal transport framework for Bayesian persuasion with continuous states and actions, providing new theoretical insights and explicit solutions for complex multidimensional problems.

Contribution

It characterizes optimal information design as a partition with finite signals and links the problem to Monge-Kantorovich transport, deriving new optimality conditions and PDEs.

Findings

Optimal design is a partition with finite signals.

Characterization of solutions via Monge-Kantorovich transport.

Explicit solutions for multidimensional persuasion problems.

Abstract

We study the general problem of Bayesian persuasion (optimal information design) with continuous actions and continuous state space in arbitrary dimensions. First, we show that with a finite signal space, the optimal information design is always given by a partition. Second, we take the limit of an infinite signal space and characterize the solution in terms of a Monge-Kantorovich optimal transport problem with an endogenous information transport cost. We use our novel approach to: 1. Derive necessary and sufficient conditions for optimality based on Bregman divergences for non-convex functions. 2. Compute exact bounds for the Hausdorff dimension of the support of an optimal policy. 3. Derive a non-linear, second-order partial differential equation whose solutions correspond to regular optimal policies. We illustrate the power of our approach by providing explicit solutions to several non-linear, multidimensional Bayesian persuasion problems.