Credible neologisms in games of communication

Abstract

1.trust another (as is the case with two-person zero-sum games).But an important role of language is coordination of actions, and game theory is the natural discipline in which to study language from that point of view, as Lewis [9] showed.In [3], we defined communication equilibrium in games, and showed that it is a strict and useful extension of Bayesian equilibrium.However, that notion, like that used by Crawford and Sobel [2] and Green and Stokey [4], requires only that given the way language is interpreted, there is no incentive to lie .In the present paper we extend the analysis by examining incentives to introduce credible neologisms.This can be viewed in two ways.Our principal interpretation is that it is an equilibrium condition that nobody should have an incentive to introduce a credible neologism: for, if someone has, then the purported equilibrium does not describe how the game will be played.We elaborate on this below, when we define a neologism-proof equilibrium.Another related approach, which we only hint at here in Section 5, is to ask when an evolving language will develop neologisms that will not immediately be discredited by lies: a self-signaling neologism will be used truthfully, at least at first.More technically, the paper can be viewed as examining the This restriction is similar to that independently proposed by Grossman and Perry [5].However, they, like the literature on stability of signaling equilibrium (Banks and Sobel [1], Kohlberg and Mertens [6], Kreps [7]) consider signals that directly affect payoffs.Their criteria, being based on dominance arguments, have no force if, as here, signals do not directly affect payoffs.The paper is organized as follows.In Section 2, we define a simple communication game, and a sequential equilibrium in such a game.We introduce the notion of a self-signaling set of types (an X with the properties just described), and argue that if there is such a set then the equilibrium is unpersuas ive , as it is vulnerable to the introduction of a credible neologism.An equilibrium not subject to such an objection we call neologism- proof .In Section 3, we give four examples to illustrate our concept, and relate it to Myerson's notion [10] of core mechanism.We show by example that neologism-proof equilibrium need not existIn Section 4, we prove some existence results for neologismproof equilibrium.In Section 5, we consider what happens if agents repeatedly play (adjusting towards best responses) a game that has no neologism-proof equilibrium.In our example, the average result depends on relative speeds of adjustment, and we suggest that the usual game-theoret ic paradigm (in which all those speeds are assumed infinite) may be misleading.Section 6 concludes 2 .Assumptions and Definitions Simple Communication GamesFirst, we define our object of analysis: a s imple This condition says that the probability distributions on T*M described by -n ( ) and s(), and by p() and s(), must be the same.In other words, given s(), R's posteriors must be is not self-signaling, we know max (a, b) > 0: say a > 0."By definition of a*(T), we know 1 + a + b < 0.Now write c = u (a*({t , t }), t ) d = u S (a*({t r t 2 }), t 2 )