Information Transmission with Almost-Cheap Talk

Misrepresenting private information is often costly. This paper studies a model of strategic information transmission based on Crawford and Sobel (1982)(CS), but with a signaling dimension where there is a convex cost of misreporting. I identify a simple condition, called No Incentive to Separate (NITS), that is necessary for a CS equilibrium to be the limit of monotone equilibria as the cost of misreporting shrinks to 0. A CS equilibrium is said to satisfy NITS if the lowest type weakly prefers the action it elicits in the equilibrium to what it would elicit in the complete-information game. I also prove a converse: there is a sequence of monotone equilibria satisfying a forward-induction property that converge to any CS equilibrium satisfying NITS, so long as costless signals are available in addition to the dimension of costly misreporting. It is shown that under a standard regularity condition, only the most informative CS equilibrium satisfies NITS. The results therefore provide a novel rationale for focussing on the most informative equilibrium of the pure cheap talk game, without invoking cooperative justifications such as the ex-ante Pareto criterion. The forward-induction equilibria of the costly misreporting model also possess features that are consistent with certain empirical evidence that is hard to reconcile in a pure cheap talk setting.