**6. Conclusions**

Coordination games of various forms, from actual rendezvous games to super-modular games and complementarity games, have received increasing attention in the game theory literature. Most equilibrium selection in such games, however, has been relatively informal, appealing to such concepts as focal points, initial conditions, or competition (essentially an evolutionary argument). Cheap talk, meaning costless and nonbinding preplay communication, has presented an intuitively pleasing method for formally attacking the equilibrium selection problem. The model of *conversations* presented here attempts to provide one possible resolution to this question of equilibrium selection, as well as to the even older question of justifying the Nash equilibrium concept.

The model assumes that players meet for the first time and communicate in order to allay their uncertainty about the future actions of their opponents. Since they have no knowledge of the cheap talk strategies used by the other players, we do not look for an actual equilibrium of the extended game. Instead, we look for all outcomes that could reasonably occur as the result of rational communication on the part of the players. Messages are defined to be credible in the context of a particular conversation. If at the end of a conversation, a player has put forward a consistent and credible appearance, this is assumed to in fact be the other players' belief about his or her future actions. From this base, it is proven that meaningful communication (that is, in which there is convergence) must end up at a Nash equilibrium. This is a partial justification for the Nash concept. It is then proven that optimal

<sup>18</sup> See, for example, the survey paper by Bergin and MacLeod (1993) [26].

communication, meaning that all players make strategic and rational announcements, leads to the deselection of inefficient equilibria.

A strength of the paper is that it gives a decisive answer to these two issues within the context of a single model. It also applies to games with more than two players or that do not necessarily exhibit common interests. There are, however, several qualifications to the model. First, the results do not prove that convergence must take place, only that if it does, then it takes a certain form. Secondly, since by no means all applications allow the possibility for preplay communication, this cannot be a general justification for the Nash concept. Indeed, this also potentially predicts a distinction between environments where one would expect Nash equilibrium to obtain versus others where one would not necessarily expect it. Finally, the model does put restrictions on the belief formation process, in that it requires some very small amount of faith to be put in credible announcements, at least over the long run. Note that this is not a departure from full rationality; traditional models have simply left this process unmodeled. There are also a number of possible relevant extensions of this model, notably to correlated equilibrium and to introducing a stochastic element in the conversation.

Calvin Coolidge once wisely said, "It is better to remain silent and be thought a fool than to speak and prove it." However, that applies only to fools: the moral of this paper is, "It is worse to remain silent and only be supposed rational than to speak and confirm it."

**Funding:** This research received no external funding.

**Acknowledgments:** I would like to thank Max Amarante, In-Koo Cho, Lynn Conell-Price, Peter Diamond, Glenn Ellison, Ehud Kalai, Elizabeth Murry, Tom Palfrey, Lones Smith, Radhanjali Shukla, and the seminar participants at Caltech, MIT, Penn State, Rochester, and the 17th Arne Ryde Symposium for valuable comments. All remaining shortcomings are mine only.

**Conflicts of Interest:** The author reports no conflict of interest.
