  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.
