**3. Motivation**

This section provides some intuition and justification for the structure of the model that follows; the impatient reader can skip to the next section. The model assumes that there is an action game to be played, about which the players are assumed to have full information (in order to abstract away from any signaling incentives during the conversation). Each player begins with a common forecast about what actions he or she will take in the upcoming game. These expectations can be interpreted as vague initial ideas about how the game might be played, arising perhaps from societal conventions or from

focal points (hence the assumption that the forecasts are common and known). They are not beliefs in the formal sense, although they will be updated throughout the conversation.4 Since a priori, nothing can be absolutely ruled out by any of the players, the prior forecasts are totally mixed.<sup>5</sup> Needless to say, the forecasts are not in any way binding: players ignore what they themselves are "expected" to do, although they can take into account the influence this expectation has on their opponents.6 The key distinction between forecasts and standard beliefs is that the forecasts are about the general environment (how might this game typically be played by others?), whereas beliefs are about the actual decisions by the specific players interacting in a given concrete situation. Thus, among other implications, it makes sense to reason about players trying to influence the beliefs that their opponents have about them, whereas they cannot influence the more broadly prevalent forecasts. Of course, then, we need to model from whence the forecasts come (social norms, news media, evolutionary psychology, etc), but that is outside the scope of the present paper.

During the conversation stage, before playing the action game, players send public messages to each other. Since we are attempting to understand what such preplay communication can achieve, we assume that there is an unlimited (but countable) opportunity to send these messages. For simplicity and without loss of generality, the messages are taken simply to be announcements of a player's own expected actions in the game. One could assume instead that players announce mixtures of their possible actions, but this is an unnecessary complication. Essentially, given infinite riskless communication, this slight limitation on the flexibility of messages imposes no loss in the long run. Implicitly, we are assuming that players can understand one another and that they take messages at face value (not in a strategic sense, but in a linguistic sense). If the message "action L" is sent, everyone understands that to mean "action L" and not "action R". Thus, there is a *natural language* for speech; the players share enough common history or cultural affiliation that they are able to talk and understand one another in a previously unencountered situation.

Of course, not all announcements should be considered seriously. We need to define a notion of *credibility* or believability. The first requirement is that a player's announced action should be *self-committing*, in the sense that if it were believed and best responded to, the original announcer would still be willing to carry through with it (within the confines of the action game). This requirement is equivalent, then, to being in the support of some Nash equilibrium of the action game. At the beginning of the preplay conversation, any self-committing action is credible, so players have a chance to guide the discussion. In general, there will be some tradeoff between allowing the players leeway to influence the conversation at the beginning, but requiring them at some point to pay attention to what the others are saying and to reflect that updated information in their own announcements. Unlike in the deterministic best response dynamics of evolutionary models, it is important in this model that players have a choice over what to say; this is the hallmark of a conversation. It is this choice, along with the lack of payoffs until the action game is at the very end, that differentiates this paper's model from an evolutionary learning model.

The common forecast is very slowly updated by each credible announcement. We can think of the prior forecast as the result of a long, but finite fictitious history of credible announcements, with each new stated action adding to the average.<sup>7</sup> As beliefs get updated, the initial forecast can be ignored and only the actual credible announcements counted toward an average forecast: this forecast constitutes a

<sup>4</sup> The players do not have beliefs about the full strategies of their opponents, only ideas about what might actually occur in the game. Thus, the preplay forecasts are distributions over actions, not distributions over mixed strategies (which themselves are distributions over actions). This is not crucial to the conclusions reached.

<sup>5</sup> It is not strictly necessary for the results that the priors be totally mixed.

<sup>6</sup> The author performed the analysis under the seemingly weaker assumption that all that is known about the prior forecasts is that they place a certain minimum weight on each action, but the results carry over. Since this assumption adds complexity, but is no sounder in justification (Why can the entire distributions not be known if the minimum weights are?), it has been left out.

<sup>7</sup> Recall that the average of multiple sets of actions is equivalent to a mixed strategy.

player's *appearance*. In general, we recursively define an announced action to be credible if it is the best response (within *ε*) to either the current forecast of an opponent's behavior or to an opponent's appearance.<sup>8</sup> If there are more than two players, either the common updated forecast or a player's appearance may be substituted for each. The intuition here is that a player can either say something like, "This is what I think you are going to do, and if so then I would plan to do such-and-such," or something along the lines of, "Okay—for the moment I'll take you at your face value, and in that case I'll want to do so-and-so." Of course, he or she only needs to consider credible announcements in making these plans.

At any time during the preplay conversation, a player can make any announcement desired, but only those statements that are credible will have an impact on the conversation. Since all players know the prior forecasts and all previous announcements, they can calculate which of these announcements were actually credible and hence also which of their own announcements will be perceived as credible by others. If at any point, there is but a single action that is credible for a particular player, it must be that this player can only seriously be considering that action (at that point in time). Therefore, in effect, it does not matter whether or not he or she actually announces that action; everyone knows that it is being considered, and hence, it should count toward the forecast and appearance of that player, regardless of what may or may not be announced. This argument implies that without loss of generality, we may assume that all players make credible announcements during each round of the conversation.<sup>9</sup> Finally, we assume that at each point in time, any player can start over; that is, declare a clean slate and remake their appearance anew. This is the equivalent of declaring that the conversation has broken down from his or her perspective and, among other implications, allows the players to attempt to coordinate. Although it may seem like an overly strong possibility, in fact, a player's appearance is a powerful commitment device, and so, giving up on it involves a significant loss.10 In any case, of course, the clean slate option is available equally to all of the players. This completes the description of the cheap talk conversation.

One last remaining question about the credibility concept concerns the infinite durability of credible announcements. That is, a credible announcement always "counts" even if it is no longer credible. The reason for this is that any credible announcement indicates evidence of a desire for that action if possible, and there is no reason to think that the desire will change or that the desired action may not once again become plausible. In effect, each announcement has a small impact that builds toward the whole impression, rather than the fads of currently credible actions. In fact, if only those actions that are credible at the moment are averaged into the player's appearance, at each communication stage, one can observe swings back and forth of what is and is not believable. Furthermore, in this updated setting, eventually, only one pure strategy will be credible, and so, it is essentially impossible for players to converge to a mixed strategy.

Once the preplay conversation is complete, we have a countably infinite sequence of announcements for each player, with an associated sequence of appearances (the average credible announcement to date). This latter sequence may or may not have a limit.11 Because of the infinite horizon and the nature of the updating process, if the limit does exist for a given player, then the forecasts made by the other players about this player will also converge, and to the same point. In this case, we specify that entering the concrete action game, the beliefs held by the other players about this player are also this same point in the strategy space. In this way, the conversation is a model of belief formation. If the appearance does not converge, then the appropriate forecast will not converge either,

<sup>8</sup> We assume that players only care about payoffs up to some arbitrarily small constant *ε*, either because they cannot perceive finer differences or because they are indifferent over this range.

<sup>9</sup> We make the standard assumptions on the action game so that a best response always exists.

<sup>10</sup> In particular, continually starting over inhibits convergence, in which case, the player has no influence on the ultimate course of the discussion. This is never optimal, as shown below.

<sup>11</sup> If no credible announcements were made after some finite stage, this is taken to mean that the limit does not exist. However, as above, we may assume that this does not occur.

and beliefs remain open for the time being. Of course, it may be true in general that appearances have a limit only for some (possibly empty) subset of the group of players.

If the appearances of all players converge, then we say that the conversation itself converges. However, in this case, every player continues to make credible announcements, and hence, at the limit, these announcements must be near the best responses to the actions stated by the other players, and hence to the limits of the other players. Since by definition, the latter are the beliefs held by the given player upon entering the action game, his or her limit must be an action that is (near) the best response to his or her beliefs and is therefore one optimal strategy to pursue in the action game. Therefore, we may assume that this limit action is indeed chosen, validating the beliefs of the other players. Of course, since this is true for all players, the limits must be mutual best responses, and thus, the play arrives at a Nash equilibrium. This is Theorem 1 below.

We next turn our attention to the question of optimality in the cheap talk stage of the overall game. Stepping back for a moment, we consider the question of whether or not to participate in the conversation at all, given the opportunity to do so. Since there is a natural language with which to communicate, any player can initiate a conversation. Whether or not they choose to participate, other players will hear and be influenced by the announcements of this player. Therefore, if they do not also make announcements, this player (or players) will have free reign to drive beliefs toward the equilibrium of their choosing (by announcing it ad infinitum). Since this outcome is at least weakly bad for other players, it cannot hurt them to also join in the conversation and attempt to guide the discussion in a direction favorable for them. For instance, in the Battle of the Sexes game, played between one man and one woman, Player 1 conversing with himself will continuously announce the equilibrium that he prefers. Entering the action game, the other player believes these announcements and best responds to them, so that the play will in fact be at that equilibrium. In this case, it would have been a good idea for Player 2 to at least try to promote her favored outcome, that is to participate in the conversation. Thus, we may assume, without any loss of generality, that all players converse.

Players do not know the cheap talk strategies employed by their opponents (if they did, we should instead be modeling what occurred before this conversation in order for that knowledge to be gained), so these players must consider all strategies to be possible. Thus, if a cheap talk strategy for one player never performs better (in terms of the payoffs ultimately realized in the action game, of course) than another competing strategy and does strictly worse against at least one possible strategy profile of the opponents, then the original strategy should be discarded as suboptimal. Anything that is not weakly dominated is optimal.<sup>12</sup> This is intentionally a broad definition of a strategy; it is meant to be as loose as possible and yet at least minimally capture the requirements of optimality. Theorem 2 below proves that if all players employ communication strategies that are optimal in this loose sense, then the conversation must converge to a stably efficient equilibrium of the game. This class of equilibria, defined below, is essentially those Nash equilibria for which no coalition can break away and, on their own, force the other players to follow them to some other equilibrium that is preferred by the coalition. In two person games with distinct payoffs (a property that holds generically), this result is equivalent to Pareto optimality.
