Optimal persuasion via bi-pooling

Mean-preserving contractions are critical for studying Bayesian models of information design. We introduce the class of bi-pooling policies, and the class of bi-pooling distributions as their induced distributions over posteriors. We show that every extreme point in the set of all mean-preserving contractions of any given prior over an interval takes the form of a bi-pooling distribution. By implication, every Bayesian persuasion problem with an interval state space admits an optimal bi-pooling distribution as a solution, and conversely, for every bi-pooling distribution, there is a Bayesian persuasion problem for which that distribution is the unique solution.