**5. Examples**

The most obvious example of an equilibrium selection problem is posed by the following coordination game:

Of the three Nash equilibria in the game, only one is efficient. There is also an inefficient equilibrium, and this type of coordination problem comes up often in many contexts—including viral pandemics (see, e.g., Jnawali et al. 2017 [22]). Although in scenarios without communication, it is possible for (B,B) to occur, Theorem 2 implies that the efficient equilibrium (*A*, *A*) is the only possible outcome after rational nonbinding communication takes place among the players, no matter the prior forecasts. This is easy to see if either of the forecasts puts significant weight on *A*. In that case, the other player can credibly repeatedly announce *A* as the best response and, in this manner, eventually force the only credible announcement by either player to be *A*. Since this yields the highest possible payoff, it is optimal, and the conversation will converge to *A*.

If instead the prior forecasts are both heavily skewed toward *B*, then each player can reason as follows: "If I announce *B*, we will be stuck there forever, and I will get a payoff of one. If I announce *A*, there is some chance that my opponent will announce *B*, in which case, we will get stuck, and I will receive one. However, there is also some chance that my opponent will announce *A*. If we both continue to do this, these will remain credible announcements (since they each best respond to the other's appearance), and we will converge to the efficient equilibrium, delivering me a payoff of two instead of zero or one. I can always go back to announcing *B* and force that equilibrium (or start over altogether), so there is no risk of ending up at the really inefficient mixed equilibrium. Since there are no instantaneous payoffs lost from miscoordination along the way, the only possible optimal strategy is for me to announce *A*."

Both players are rational, so they will in fact both announce *A* at all rounds of the cheap talk communication, and the conversation will end up converging to the efficient equilibrium. Given that the forecasts were heavily skewed toward *B*, it may be a long time before the two players have truly convinced each other of their intention to play *A*, but they have all the time in the world and every reason to make use of it. If we looked instead at the pure coordination game in which (*A*, *A*) also yields payoffs of one to each player, the analysis is slightly changed. If the prior forecasts lean toward either of the symmetric and efficient pure equilibria, the conversation will converge in that direction. However, if the priors miscoordinate just right (for example, they are completely uniform

for both players), it will be necessary for both players to randomize their initial announcement. If they coordinate at that point, successful convergence follows. If not, they simply clean their slate, start over, and try again. At some point, they must (that is, with probability one) both choose the same action (this is why it is necessary to randomize rather than to try to coordinate in some deterministic pattern), and then, they are done.

A less clear-cut example with a unique efficient equilibrium is found in the following version of the "stag-hunt" game:


Here, the unique efficient equilibrium involves choosing a risk-dominated action, perhaps making it more difficult to reach. Allowing communication, however, will afford the players an opportunity to convince each other that it is safe to play action *S*. [23] has argued to the contrary that cheap talk may not help in this game. His reasoning is that since each player would prefer the other to take action *S*, they should each attempt to convince the other player to choose it. The way to do this is by claiming that you yourself are also going to pick *S*. Therefore, hearing the other player announce *S* should be discounted as purely manipulative and ignored.

It seems that Aumann's argument is not self-evident, at least when there is an unlimited chance to communicate. Rational players know that they will eventually agree on a Nash equilibrium; there is zero probability of suckering the other player or miscoordinating. At this point, it comes down to a choice among equilibria. Knowing this perfectly in advance, if a player announces *S*, it must be because he or she is hoping to eventually end up at the efficient equilibrium, that is to end up playing *S*. It is, after all, the best response at that point. In any case, the data clearly support the idea that allowing preplay messages increases the probability of observing the efficient, but risk-dominated equilibrium; see Charness (2000) and Miller and Moser (2004) [24,25].

We turn our attention next to the Battle of the Sexes, which is not at all a game with common interests:


In this case, it is not immediately obvious that even with communication, efficiency can necessarily be achieved. If the prior forecasts favor either one of the pure equilibria, then the player who prefers that equilibrium will be able to credibly "insist" on it, and it will be the ultimate limit of the conversation. If the forecasts are balanced, however, neither player can be assured of getting his/her preferred outcome. Insisting on it whenever possible may lead the conversation to converge toward the inefficient mixed equilibrium, which is worse for both players. Therefore, this strategy is not optimal. If instead, the players "yield" to the other player with some extremely small probability at each round, this will always be achieved within *ε* of any other strategy, and since it always leads to one of the efficient equilibria, it weakly dominates the strategy by a player that forever insists on getting his or her way. Thus, under this scenario, the players are behaving optimally and can achieve efficiency with certainty.

As a final example, we turn to games with three players in order to explain some of the added complexity that arises. First, consider the following game in which the matrix player's payoffs are listed last:


This game has two pure Nash equilibria, namely (*U*, *L*, *A*) and (*D*, *R*, *B*), only the first of which is efficient. The second equilibrium is directly attainable from the first through a coalition of the row and column players, but it is not fully attainable because they enjoy a lower payoff in this equilibrium. Thus, the first equilibrium is stably efficient (and hence, the second, dominated one cannot be) and will be the result of rational communication. Nevertheless, since the row and column payoffs would be higher at the intermediate point along the chain fixing the matrix player at *A*, the original efficient equilibrium is not coalition-proof. Now, modify the payoffs slightly:


Only the equilibrium payoffs have been changed, but the analysis has been affected greatly. Both pure equilibria are now efficient, but for exactly the reasons outlined above, only the second one, (*D*, *R*, *B*), is stably efficient and can be the result of cheap talk. On the other hand, the original equilibrium is now coalition-proof, showing the discrepancy between the two concepts.

One of the (unavoidable) limitations of this model is that it can say nothing about zero-sum games, except that communication can only converge to a Nash equilibrium. Other games in which all equilibria are efficient, and so for which Theorem 2 is vacuous, are games with a unique Nash equilibrium. These include matching pennies, rock-paper-scissors (where many of the convergence problems of fictitious play show up), and the game-theoretic standby of the prisoner's dilemma. Of course, we cannot expect that simple communication would lead to cooperation, a strictly dominated strategy. We have assumed throughout that there is only a single (though unlimited) chance for the players to talk for playing a game. If **G** is a repeated game and the players have a full conversation between each stage, then optimal speech should lead to efficient outcomes all along the extensive form game tree, both on and off the equilibrium path. This gives rise to the difficult problem of finding renegotiation-proof equilibria18.
