**Theorem 1.**

*1. NE*(*G*) ⊆ *AccE*(*G*)

*2. AccE*(*G*) ⊆ *εNE*(*G*)

**Proof.** (1) Let *σ* ∈ *NE*(*G*), and consider prior forecasts *π* very close to *σ*. By the definition of a Nash equilibrium, any *si* ∈ *supp*(*σi*) is in *<sup>ε</sup>BRi*(*π*−*i*). Now, let the players announce actions in the support

<sup>14</sup> Sequential announcements lead to a forced asymmetry regarding who speaks when. The effects of this generalized first-mover advantage are irrelevant for the present discussion.

<sup>15</sup> Unless player *i* has only one possible credible announcement, as discussed in Section 3.

of *σ* in such a way as to match as nearly as possible the actual distribution prescribed by *σ*. Initially, all these actions will be credible as stated. Of course, the forecasts will change over time, but since the updating process is slow and the cheap talk announcements are matching the given distribution, the forecasts will always stay near *σ*. Hence, the actions in the support of the announcements will remain credible forever. In this manner, lim *<sup>t</sup>*→∞*p<sup>t</sup> <sup>i</sup>* exists ∀*i*, and moreover, lim *<sup>t</sup>*→∞*p<sup>t</sup> <sup>i</sup>* = *σi*. Thus, *σ* is indeed an acceptable equilibrium.

(2) If *σ* ∈ *AccE*(*G*) and so is the limit of a convergent conversation, it must be that all *si* ∈ supp(*σi*) are credibly announced infinitely often during the preplay cheap talk stage.16 Since in the limit, both the forecasts and the appearances are arbitrarily near *σ*, each such *si* must be in *εBRi*(*σ*−1), and therefore, *<sup>σ</sup><sup>i</sup>* ∈ *<sup>ε</sup>BRi*(*σ*−*i*)∀*i*.

Among other things, this result justifies the possibility that after a convergent conversation, players both rationally and self-consistently hold the beliefs that are given by the model. Theorem 1 in some sense clarifies the relationship between cheap talk (as has been modeled here) and Nash equilibrium. If the communication is meaningful, that is if the cheap talk has a limit, then it must lead to a Nash outcome. Of course, there is no guarantee that the conversation will converge, and it is quite possible that it will not.17 Furthermore, no Nash equilibrium, even if inefficient, can yet be ruled out. Something stronger than an acceptable equilibrium is required.

We next turn to defining the appropriate efficiency concept in this setting.

**Definition 2.** *Call σ* ∈ *NE*(*G*) *directly attainable from σ* ∈ *NE*(*G*) *by the coalition S if σ<sup>s</sup> is a Nash equilibrium in the induced game fixing all players outside of S to play as in σ , and if also* ∀*i* ∈/ *S, we have ui*(*σi*, *σS*, *σ* <sup>−</sup>*i*,*S*) <sup>&</sup>gt; *ui*(*σ <sup>i</sup>* , *σS*, *σ* <sup>−</sup>*i*,*S*)*.*

This is a strenuous definition: the first condition asks that the members of *S* be able to "jump" to *σ* from *σ* , and the second condition requires that once they have done so, they can force the rest of the players to follow them.

**Definition 3.** *Call σ* ∈ *NE*(*G*) *attainable from σ* ∈ *NE*(*G*) *by the coalition S if there is a chain of equilibria, each directly attainable by S, leading from σ to σ; if also,* ∀*i* ∈ *S ui*(*σ*) > *ui*(*σ* )*; and if finally, there is no similar such chain (for any coalition) leading away from σ.*

These are once again fairly strict requirements. The second one states that all members of *S* must strictly prefer the new equilibrium, and the third states that the new equilibrium itself is immune to these sorts of deviations.

**Definition 4.** *A Nash equilibrium of G is stably efficient if nothing is attainable from it; the set of these equilibria is denoted StE f f*(*G*)*.*

By considering the grand coalition of all players, it is clear that an equilibrium exhibiting stable efficiency will tend to be efficient. In games with distinct payoffs, no singleton coalitions can ever attain alternate equilibria (this follows from the first condition of the first definition), and hence, in two-person games, stable efficiency is generically equivalent to efficiency. It is clear that stably efficient equilibria always exist (since whatever is attained must itself be stably efficient). In most games, efficiency and stable efficiency will coincide, but when they do not, it is important that we use the latter concept. Stable efficiency is related to the coalition-proof concept introduced by [21], but is more

<sup>16</sup> In particular, since the conversation converges, there must be some round after which nobody ever cleans their slate and starts over.

<sup>17</sup> Consider, as one example, fictitious play in the rock-paper-scissors game.

farsighted in that it looks at the full implications of a coalitional deviation; it turns out that neither definition is a refinement of the other.

Recall that a cheap talk strategy is **optimal** if it is not weakly dominated.

**Definition 5.** *An agreeable equilibrium (of <sup>G</sup>) is a profile <sup>σ</sup>* <sup>∈</sup> *<sup>n</sup>* × *i*=1 Δ(*Si*) *such that σ* = *b for some belief vector b resulting from a convergent optimal conversation starting at some prior forecasts π; the set of agreeable equilibria is denoted AgrE*(*G*)*.*
