Dynamic Information Revelation in Cheap Talk

This paper studies a multi-stage version of Crawford and Sobel’s communication game. In every period the receiver determines a test about the unknown state whose result is privately observed by the sender. After the sender sends a costless message about an outcome of the test, the receiver selects a test in the next period. After a finite number of periods of interaction, the receiver makes a decision. The paper offers a sequence of tests that refine sender’s information step-by-step and preserve truthtelling in every period. This sequence allows the receiver to learn the state in a subinterval of the state space with an arbitrary precision and has appealing theoretical properties. It consists of simple binary tests which reveal whether the state is above a certain cutoff, where the cutoffs are monotonic across periods and independent from results of the previous tests. Finally, we show that the relative payoff efficiency of multi-stage interaction compared to a single-stage game increases without a bound as the bias in preferences tends to zero.