The Bounds of Mediated Communication

We study the bounds of mediated communication in sender-receiver games in which the sender's payoff is state-independent. We show that the feasible distributions over the receiver's beliefs under mediation are those that induce zero correlation, but not necessarily independence, between the sender's payoff and the receiver's belief. Mediation attains the upper bound on the sender's value, i.e., the Bayesian persuasion value, if and only if this value is attainable under unmediated communication, i.e., cheap talk. The lower bound is given by the cheap talk payoff. We provide a geometric characterization of when mediation strictly improves on this using the quasiconcave and quasiconvex envelopes of the sender's value function. In canonical environments, mediation is strictly valuable when the sender has countervailing incentives in the space of the receiver's belief. We apply our results to asymmetric-information settings such as bilateral trade and lobbying and explicitly construct mediation policies that increase the surplus of the informed and uninformed parties with respect to unmediated communication.