A Tree Formulation for Signaling Games

The paper has as a starting point the work of the philosopher Professor D. Lewis. We provide a detailed presentation and complete analysis of the sender/receiver Lewis signaling game using a game theory extensive form, decision tree formulation. It is shown that there are a number of Bayesian equilibria. We explain which equilibrium is the most likely to prevail. Our explanation provides an essential step for understanding the formation of a language convention. The informational content of signals is discussed and it is shown that a correct action is not always the result of a truthful signal. We allow for this to be reflected in the payoff of the sender. Further, concepts and approaches from neighbouring disciplines, notably economics, suggest themselves immediately for interpreting the results of our analysis (rational expectations, self-fulfilling prophesies). 1. Introduction The philosopher Professor Lewis [1], writing on the origins and process of formation of language, discusses signaling games between a sender, who sends a signal, and its receiver. In the Lewis formulation the sender is aware of the state of the world, but the receiver is not. There are a number of alternative states and nature chooses one at random, that is, with a certain probability. Once the sender knows the state chosen, there are various signals that he can use. There is a number of alternative actions that a receiver can take in response to the signal received. Following the specific actions of the sender and the receiver there are payoffs awarded to both of them. These rewards express, for example, the utilities or money, obtained from the combination of their actions. We note in particular the games of common interest in which a resolution leads to optimal payoffs for both actors as noted by Skyrms [2]. However this type of analysis is not always complete. Notably there is little discussion of the case where the action of the receiver may be appropriate to the state of nature even if the signal sent is not. There is no discussion of what happens to the payoffs of the two agents when this is the case. Lewis makes an attempt to discuss what constitutes “true” and “untrue” signals and responses in a signaling system. We discuss these issues below. In terms of informal signaling conventions, Lewis offers the simple but illuminating example of a helper, ( ), referred to also as he, standing behind a truck gesturing to the driver, ( ), referred to also as she, to help her steer the truck into a narrow parking space. We can assume that Nature consists of the particular