Optimal Transport of Information
72 Pages Posted: 23 Feb 2021 Last revised: 8 Mar 2021
Date Written: February 22, 2021
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.
Keywords: Bayesian Persuasion, Information Design, Signalling, Optimal Transport
JEL Classification: D82, D83, E52, E58, E61
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