Persuading a Behavioral Agent: Approximately Best Responding and Learning

The classic Bayesian persuasion model assumes a Bayesian and best-responding receiver. We study a relaxation of the Bayesian persuasion model where the receiver can approximately best respond to the sender's signaling scheme. We show that, under natural assumptions, (1) the sender can find a signaling scheme that guarantees itself an expected utility almost as good as its optimal utility in the classic model, no matter what approximately best-responding strategy the receiver uses; (2) on the other hand, there is no signaling scheme that gives the sender much more utility than its optimal utility in the classic model, even if the receiver uses the approximately best-responding strategy that is best for the sender. Together, (1) and (2) imply that the approximately best-responding behavior of the receiver does not affect the sender's maximal achievable utility a lot in the Bayesian persuasion problem. The proofs of both results rely on the idea of robustification of a Bayesian persuasion scheme: given a pair of the sender's signaling scheme and the receiver's strategy, we can construct another signaling scheme such that the receiver prefers to use that strategy in the new scheme more than in the original scheme, and the two schemes give the sender similar utilities. As an application of our main result (1), we show that, in a repeated Bayesian persuasion model where the receiver learns to respond to the sender by some algorithms, the sender can do almost as well as in the classic model. Interestingly, unlike (2), with a learning receiver the sender can sometimes do much better than in the classic model.