Information Transmission with Almost-Cheap Talk

Misrepresenting private information is often costly, for technological, legal, or psychological reasons. I develop a model of strategic information transmission based on Crawford and Sobel (1982) (CS), but with a convex cost of lying or misreporting. There are three main results. First, I prove that a sequence of monotone equilibria converges to a CS equilibrium as the cost of misreporting shrinks to 0 only if the CS equilibrium satisfies the “No Incentive to Separate” (NITS) condition. In a cheap talk game, NITS requires that the lowest type weakly prefers the action it elicits in equilibrium to what it would elicit in the complete information game. In commonly used specifications, only the “most-informative” CS equilibrium satisfies NITS. Second, I show that under a mild technical condition, the converse is also true: any CS equilibrium satisfying NITS is the limit of a sequence of monotone equilibria as the misreporting cost shrinks to 0. This simultaneously proves existence of monotone equilibrium for small costs, under the technical condition. The third result provides a complete characterization of a class of monotone equilibria for arbitrary costs of misreporting in a cheap talk extension of the model. These equilibria display language inflation, with a region of low types fully separating when costs are large.