**Theorem 2.**

*1. StE f f*(*G*) ⊆ *AgrE*(*G*)

*2. AgrE*(*G*) ⊆ *εStEf f*(*G*)

**Proof.** (1) Consider *σ* ∈ *StEf f*(*G*), and let the prior forecasts *π* be very close to *σ*. Since the forecasts favor *σ* so heavily, the only way that another equilibrium can ever be reached during the conversation is if it is directly attainable or the result of a chain of directly attainable equilibria. Thus, all of the players know that these are the only feasible outcomes, and in fact (see the strategies below), they can be reached in a conversation. However, since *σ* is stably efficient, it is not possible for any player (as a member of any coalition) to be sure that by deviating from one of these alternates, a superior payoff can be achieved. It must be the case that either not all members of the coalition will profit by the switch (in which case, those who do not profit will not participate in the deviation) or if they do, that then, there is another coalition that can profitably and successfully deviate away from this new point. Of course, it is possible that one's payoff will be increased by attempting to switch equilibria, but there will always be circumstances in which it is not profitable. Thus, there is no strategy that weakly dominates the strategy "emulate *σ*", which is always available due to the prior forecasts. This implies that one optimal strategy for all players is to follow *σ*, and the result of this will be that the conversation converges with *σ*. There may be other optimal strategies, and there may be other possible results to the conversation; however, this is sufficient to show that *σ* ∈ *AgrE*(*G*), as desired.

(2) Suppose that a conversation is converging toward an equilibrium *σ* that is not stably efficient (even up to *ε*-indifference). If there is just one coalition that can attain a superior equilibrium for itself, it can pursue the following strategy: (a) Erase its current appearance and start over, and then, (b) announce the actions that lead to the first equilibrium along the chain. If all members of the coalition have done likewise, then they will be able to credibly repeat those announcements in the next round, since these are mutual best responses given the forecasts near *σ* for the other players. If the other members have not done this, each individual can start over again and try once more. If eventually they coordinate, then they can continue to make these announcements indefinitely. At some point, the forecasts and appearances will then be very close to this new equilibrium, and the only credible choice for the other players will be to switch to it as well (this follows from the definition of directly attainable). They can continue in this fashion until the final equilibrium in the chain, where the process will conclude (by the argument in Part (1) above).

Of course, this attempted deviation will not always work, but it is safe in that either it works (that is, all members of the coalition coordinate) and a higher payoff is realized or it does not and the conversation stays at *σ* instead. Therefore, the deviation strategy weakly dominates the "emulate *σ* and stay where you are" strategy. Since this is true for all members of the coalition, optimality implies that all of them will attempt to force the switch to the preferred attainable equilibrium, and with probability one they will eventually coordinate (since they always have the opportunity to start over). Therefore, *σ* was not in fact an agreeable equilibrium.

Similarly, if there were several coalitions that could attain superior equilibria, each member of each coalition can start over at each round and attempt to coordinate with his or her coalition. Any player who is a member of several coalitions or who has a choice between attainable equilibria can randomize between these possibilities. If the player puts almost all weight on his or her individually preferred outcome among all these choices and spreads *σ*(*ε*) weight across the others, then this will be *ε*-optimal, but will at the same time guarantee that with probability one, coordination takes place at some point. This weakly dominates "emulate *σ*" because either the conversation converges to *σ* anyway (though this never actually happens with optimality), or another coalition coordinates (which could not be helped), or one of the attempted coalitions coordinates first (which increases payoffs). Therefore, once again, no optimal conversation will remain at *σ*, and thus, it could not have been agreeable.

The intuition behind Part (1) is particularly simple in two player games. In this case, given a strong prior forecast, either player can insist on the original equilibrium *σ* for longer than the other player can credibly hold out against it (by the definition of Nash). Therefore, both players must optimally be able to get at least their payoff from *σ*. However, since *σ* is efficient, this means that both players get exactly this payoff under any optimal strategies, and thus, staying at *σ* itself is as good as anything else. The examples in the next section serve to illustrate the mechanisms behind both the definitions and the proof of the theorem. It should be pointed out that in most specific cases, very little of the somewhat complex machinery developed above is necessary or applicable; the process is often hopefully quite natural and intuitive.
