Signal Exaggeration in Bayesian Persuasion

We study a model of Bayesian persuasion where the sender cannot commit a signal to the receiver so that he can lie on the signal. The sender can choose an announced signal differently from a private real signal and should bear a cost of lying. The receiver only observes the announced signal and a signal realization drawn according to the real signal. We show that signal exaggeration in the downward direction arises in equilibrium when the sender can benefit from persuasion. Surprisingly, the sender may not lie if the lying cost is too low, in which case the equilibrium signal is uninformative. As the lying cost increases, the equilibrium real and announced signals converge to a subset (or a single signal under certain conditions) of optimal signals of Kamenica and Gentzkow [2011]. Thus our approach provides a selection guide for the case of multiple optimal signals in Kamenica and Gentzkow [2011]. If the optimal signal is unique and the lying cost is sufficiently high, the equilibrium real signal is less informative than the optimal signal. When the sender's value function is strictly convex with a sufficiently high lying cost, the sender chooses the unique optimal signal of Kamenica and Gentzkow [2011] as the announced signal