Language Games

A language game is a communication game in which players are uncertain about each other’s ability to produce and interpret messages. We consider two types of language games, public-announcement games and pre-play communication games. A public-announcement game is a finite static complete-information game, the base game, preceded by the observation of a public signal, interpreted as an announcement. In a pre-play communication game the base game is preceded by one round of simultaneous public pre-play communication. We show that for any base game all rational correlated equilibrium (CE) distributions of that game can be realized as Bayesian Nash equilibrium distributions of a language game with that base game. If we define a language equilibrium (LE) of a game as a Bayesian Nash equilibrium of a language game with that base game, our result can be succinctly stated as “The sets of rational CE and LE distributions coincide.” Thus, uncertainty about the meaning of messages can fully substitute for correlation. ∗I am grateful for comments received from audiences at the Workshop on Knightian Uncertainty and Robustness in Games at the Center for Mathematical Economics (IMW, Universtät Bielefeld) and the Chinese University of Hong Kong as well as for helpful suggestions by Kim-Sau Chung, Ehud Lehrer, Wooyoung Lim, Shih En Lu, Navin Kartik, Gregory Pavlov, Joel Sobel and Péter Vida. . . . I can direct a man who has learned only German, only by using the German language. Describe the aroma of coffee. — Why can’t it be done? Do we lack the words? Ludwig Wittgenstein, Philosophical Investigations