**2. Previous Literature**

The concept of cheap talk was introduced into the economics literature by Crawford and Sobel (1982) and Farrell (1987) [2,3]. Since then, a sizable literature has developed related to this topic, with such examples as Farrell and Gibbons (1989), Forges (1990), Farrell (1993), Aumann and Hart (1993), Blume and Sobel (1995), and a survey in Farrell and Rabin (1996) [4–9]. The paper that perhaps is closest to the present one is Rabin (1994) [10]. It models a finite instead of an infinite opportunity for communication, but also seeks a notion of optimality rather than equilibrium in the analysis of the extended game. The specific form of cheap talk assumed by Rabin is different from the one presented below, in particular with respect to the element of choice between strategies against which to credibly best respond. The results can be framed in terms of the two central questions posed here, but are generally less conclusive in either. Both papers adhere to the full rationality paradigm of classical game theory and previous work on cheap talk, as opposed to, say, the evolution literature.

There are a number of papers that study a more limited class of games. For instance, Matsui (1991) [11] applied cheap talk to common interest games, and in this context, his notion of *cyclic stability* yields efficiency. Canning (1997) [12] studied signaling games of common interest, although the messages do not necessarily constitute cheap talk per se. He found that off-path beliefs are vital to the question of whether or not efficiency is eventually realized; randomly drawn off-path beliefs encourage experimentation and lead to efficiency. Finally, Sandroni (2000) [13] studied two person repeated coordination games without cheap talk. He introduced the concept of *blurry beliefs*, which is a less restrictive (that is, more fully rational) belief dynamic than those used in evolutionary game theory, although it is stronger than anything used here. Sandroni showed that if the belief classes of the players satisfy *reciprocity*, then cooperation will be achieved. Overall, the current paper pins down the link between communication and (efficient) equilibrium outcomes more concretely than the previous literature. In particular, it explores a specific empirically-consistent model of belief formation and shows a two way equivalence between that process and the optimality of the resulting behaviors.

A fairly large class of papers has studied repeated games and the emergence of Nash equilibria without introducing cheap talk, including Crawford and Haller (1990), Young (1993), and Kalai and Lehrer (1993) [14–16]. Finally, there have been some experimental studies of communication and equilibrium selection in various coordination games; see, for example,Cooper et al (1992), Brandts and Cooper 125 (2007), and Cachon and Camerer (1996) [17–19]. The results can be summarized (and oversimplified) as finding that two way pregame communication greatly increases the chances of observing efficient equilibrium outcomes. Pertinently, this holds even if the efficient equilibrium is not risk-dominant, in contradistinction to some previous results. Meanwhile, some experimental studies found that preplay communication can actually induce fewer choices consistent with Nash equilibria, e.g., Boulu-Reshef et al (2020) [20] in the context of public goods games. This could either be due to the limited opportunity for communication and/or the possibility of social preferences (which would change the set of NE).
