The art of brevity

We analyze a class of sender-receiver games with quadratic payoffs, which includes the communication games in Alonso, Dessein and Matouschek (2008) and Rantakari (2008) as special cases, for which the receiver's maximum expected payoff when players have access to arbitrary, mediated communication protocols is attained in one-round of face-to-face, unmediated cheap talk. This result is based on the existence for these games of a communication equilibrium with an infinite number of partitions of the state space. We provide explicit expressions for the maximum expected payoff of the receiver, and illustrate its use by deriving new comparative statics of the quality of optimal communication. For instance, a shift in the underlying uncertainty that reduces expected conflict can worsen the quality of communication.