Not so cheap talk: Costly and discrete communication

We model an interaction between an informed sender and an uninformed receiver. Like the classic cheap talk setup, the informed player sends a message to an uninformed receiver who is to take an action which affects the payoffs of both players. However, unlike the classic cheap talk setup, the sender can communicate only through the use of discrete messages. In particular, the sender has a finite set of message elements with which to compose messages. The sender incurs a communication cost which is increasing in the number of elements contained in the message. We characterize the resulting equilibria with a permissive out-of-equilibrium restriction. We introduce a stronger out-of-equilibrium requirement and show that if the sender and receiver have aligned preferences regarding the action of the receiver then only the most informative equilibrium exists. When the preferences between players are not aligned, we show that our stronger condition does not guarantee uniqueness and we provide an example where an increase in communication costs can improve communication. As we show in an example, this improvement can occur to such an extent that an equilibrium can outperform the Goltsman et. al. (2009) upper bound for payoffs in mediated communication.