Optimal Persuasion via Bi-Pooling

Abstract

The canonical Bayesian persuasion setting studies a model where an informed agent, the Sender, can partially share his information with an uninformed agent, the Receiver. The Receiver's utility is a function of the state of nature and the Receiver's action while the Sender's is only a function of the Receiver's action. The classical results characterize the Sender's optimal information disclosure policy whenever the state space is finite. In this paper we study the same setting where the state space is an interval on the real line. We introduce the class of {\it bi-pooling policies} and the induced distribution over posteriors which we refer to as \emph{bi-pooling distributions}. We show that this class of distributions characterizes the set of optimal distributions in the aforementioned setting. Every persuasion problem admits an optimal bi-pooling distribution as a solution. Conversely, for every bi-pooling distribution there exists a persuasion problem in which the given distribution is the \emph{unique} optimal one. We leverage this result to study the structure of the price function (see \cite{dworczak2019simple}) in this setting and to identify optimal information disclosure policies.