Cheap talk with private signal structures

The paper considers a cheap-talk model in which the receiver privately selects the signal structure of the initially uninformed sender. After the sender privately observes a signal generated by the signal structure, the players play a standard cheap-talk game. We show that by randomizing between two perfectly informative signal structures, the receiver can elicit perfect information from the sender for the bias in preferences of any magnitude, including the case when the bias is the sender's private information. We provide the precise and simple characterization of pairs of such signal structures. The key idea is that they play a dual role: shaping the sender's posterior beliefs and the receiver's reaction to the sender's messages. These factors complement each other and jointly serve as the counterbalancing force that eliminates the sender's incentives to misreport information.