Noisy persuasion

We study the effect of noise due to exogenous information distortions in the context of Bayesian persuasion. We first characterize the optimal signal in the prosecutor-judge game from Kamenica and Gentzkow (2011) with a noisy and strongly symmetric communication channel and show that the sender's payoff increases in the number of messages. This implies that, with exogenous noise, the sender prefers to complicate communication. Then, we establish necessary and sufficient conditions for the sender's payoff to weakly increase in the Blackwell-informativeness of the noise channel when the message space and the channel are binary. The reason why a sender may benefit from additional noise is that a garbling may alter the noise structure. Subsequently, we provide sufficient conditions that extend this result to channels of arbitrary cardinality. Finally, we introduce a procedure of making a communication channel more complex and prove that increased complexity benefits the sender.