Generic Finiteness of Equilibrium Outcome Distributions in Sender-Received Cheap-Talk Games

This paper establishes the generic finiteness of equilibrium outcome distributions for Sender-Receiver cheap-talk games. An equilibrium in a Sender-Receiver cheap-talk game is said to be in reduced form if every message is used by at least one type and no two messages provoke the same response. It is shown that, for a generic set of utilities on outcomes, a Sender-Receiver cheap-talk game has a finite number of reduced form equilibria. A corollary is that, for generic utilities, the set of probability distributions over outcomes generated by equilibria is finite. Because of the identification of terminal nodes for utility purposes, Sard's theorem is not applicable in the way it was used in Kreps and Wilson (1982), and a structurally different proof strategy is developed. Some additional characterization of the equilibria are obtained in the process of the proof.