Common Knowledge of Language and Iterative Admissibility in a Sender-Receiver Game

Abstract

This paper investigates the implications of common knowledge of language on cheap talk games. A general framework is proposed where language is modeled as a direct restriction on playersístrategies, and the predictions under iterative admissibility (IA) are characterized. We apply this framework to sender-receiver games a la Crawford and Sobel (1982) (CS), where the Receiver takes a one-dimensional action. We incorporate two observations about natural language into the language assumption: 1) there always exists a natural expression to induce a certain action, if that action is indeed inducible by some message, 2) messages that are more different from each other induce actions that are weakly more di§erent. It is assumed to be common knowledge that the Receiver plays only strategies that belong to language. Typically, there is a severe multiplicity issue in CS games. This procedure, on the other hand, eliminates outcomes where only a small amount of information is transmitted. Under certain regularity conditions, all equilibrium outcomes are eliminated except the most informative one. However, with an example, we point out that the normal form procedure does not take care of sequential rationality. To address this issue, we propose an extensive form procedure and characterize the solution.