Existence of Equilibrium in Bayesian Persuasion Models

This note provides sufficient conditions for the existence of equilibrium in a class of extended Bayesian persuasion models. We consider a framework in which a single sender discloses public information to n receivers. After observing the sender’s signal, the receivers update their beliefs to an interim belief and decide whether to gather additional (private) costly information. We show that when no information is costless, and the cost function is consistent with strict Blackwell informativeness, continuous in the interim belief and continuous and quasi-convex in the choice of the receiver, an equilibrium always exists. We show that focus on sender’s preferred equilibrium (SPE) alone does not guarantee existence, and we provide an example with a posterior separable cost function in which an optimal sender’s strategy does not exist despite the focus on SPE