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.