Regret-minimizing Bayesian persuasion

We revisit the celebrated Bayesian persuasion setting and examine how well the Sender can perform when ignorant of the Receiver's utility. Taking an adversarial approach, we measure the signaling scheme performance via (additive) regret over a single persuasion instance. We focus on Receiver with two actions: adoption and rejection, assuming that Sender aims to persuade Receiver to adopt. We show that while Sender's total ignorance of Receiver's utility is extremely harmful to her, assuming that Sender just knows Receiver's ordinal preferences over the states of nature upon adoption suffices to guarantee a surprisingly low regret even when the number of states tends to infinity. Moreover, we exactly pin down the minimum regret that Sender can guarantee in this case. We further show that such a positive result is impossible under the alternative performance measure of a multiplicative approximation ratio.