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Semantic meaning in finite populations with conflicting interests

In our daily lives we often make decisions on the basis of information that we’ve received from parties who are affected by the outcome of those decisions. In these situations, the interested party may have an incentive to manipulate our choice by strategically revealing information. Milgrom and Roberts (1986) provided a classic model of such a situation. In what they call a persuasion game, an informed party, the seller, has private information about the quality of a product. The seller makes some assertion to the buyer, who, on the basis of that assertion, purchases some quantity of the good. It is assumed that for each quality level there is a unique quantity that the buyer prefers to purchase. The seller’s payoff, on the other hand, increases in quantity purchased, and thus the seller has an incentive to utter a misleading assertion if doing so will lead the buyer to purchase a greater quantity than she would if the assertion perfectly revealed seller’s private information. Milgrom and Roberts show that if the seller’s assertions are verifiable, these games have an equilibrium in which the sender reveals the good’s quality and the buyer uses that information to purchase her preferred quantity. In other words, honest communication is stable even in the face of the sender’s temptation to deceive. The above result hinges on the assumption that the seller’s communication is verifiable; i.e., that the buyer can freely verify the seller’s claim about her product. This means that if the quality level is x, then the seller may say “the quality level is exactly x” or she may say something less specific but still true, such as “the quality level is at least x” or “the quality level is at least x − 1.” In effect, verifiability limits the seller to saying only true things, but those truths may not be maximally specific. This is clearly a very strong assumption, and it is crucial for the main result of their paper. It is perhaps best interpreted as a quick way to idealize the notion that telling a falsehood might come with a severe cost. Maybe, for example, the penalties for perjury or false advertising are so stiff that it simply never pays to run the risk of being caught telling a falsehood. This paper abandons the assumption that assertions are verifiable, and takes a very different approach to understanding the prospects of truthful communication in persuasion-like games. These persuasion-like games differ from Milgrom and Robert’s original game in that the buyer’s action space here is finite, whereas in the original persuasion game buyers selected quantities from the non-negative real numbers. These persuasion-like games–called seduction games by Franke et al. (2012)–have been suggested as a model of a linguistic phenomenon known as pragmatic implicature. Here we investigate the emergence of communication from the perspective of evolutionary dynamics by considering a finite population of agents who update their choice of strategy by a revision protocol known as the Moran process. In this process one agent adjusts her strategy at a time, and she does so by picking an agent to imitate with probability proportional to each agent’s expected payoff in the signaling game. If we assume that there is a small noise in this imitation process, so that with small probability an updating agent just imitates at random, then it is possible to use Fudenberg and Imhof (2006)’s methods for computing the