Mediated Cheap Talk Design (with proofs)

We study an information design problem with two informed senders and a receiver in which, in contrast to traditional Bayesian persuasion settings, senders do not have commitment power. In our setting, a trusted mediator/platform gathers data from the senders and recommends the receiver which action to play. We characterize the set of implementable action distributions that can be obtained in equilibrium, and provide an $O(n\log n)$ algorithm (where $n$ is the number of states) that computes the optimal equilibrium for the senders. Additionally, we show that the optimal equilibrium for the receiver can be obtained by a simple revelation mechanism.