On the cardinality of the message space in sender–receiver games

Abstract

We study sender–receiver games in which a privately informed sender sends a message to N receivers, who then take an action. The sender’s type space T has finite cardinality (i.e., |T|<∞). We show that every equilibrium payoff vector (resp. every Pareto efficient equilibrium payoff vector) is achieved by an equilibrium in which the sender sends at most |T|+N (resp. |T|+N−1) messages with positive probability. We also show that such bounds do not exist when two privately informed senders simultaneously send a message to a receiver.