Noisy persuasion

We study the effect of noise due to exogenous information distortions in the context of Bayesian persuasion. In particular, we first provide a full characterization of the optimal signal in a standard special case that has attracted a lot of attention in the literature. Then, we ask whether more noise (a la Blackwell) is always harmful for the information designer, i.e., the sender. We show that in general this is not the case. We provide a necessary and sufficient condition for the sender to always be worse off when noise increases in a binary noisy channel. There are two ways to read our result: (a) the sender always dislikes additional noise if and only if we start with little noise in the first place, (b) the sender always dislikes additional noise if and only if this additional noise is modelled by a sufficiently symmetric channel. Then, we provide sufficient conditions that extend this result to channels of arbitrary cardinality. Finally, we show that in every noisy persuasion game, increased complexity of the message space makes the sender weakly better off, while for a rather rich class of games the improvement is strict. This is in contrast to the noiseless case, where the sender’s maximum expected utility can always be achieved with a bounded number of messages.