Learning to Persuade a Biased Receiver

We study a repeated information design setting in which the receiver, who is also the decision-maker, updates beliefs in a systematically biased way. More specifically, a distorted posterior in our model can be written as a convex combination of the prior and the Bayesian posterior, governed by a fixed but unknown parameter. Over repeated interactions, the sender chooses persuasive signaling schemes, observes only the receiver’s realized actions, and seeks to minimize regret relative to a full-information oracle that knows the receiver’s biased updating rule. We propose a safe exploration algorithm for learning the receiver’s bias while maintaining high persuasion value. The algorithm exploits the asymmetric cost of probing: conservative probes incur only local loss, whereas overly aggressive probes may lose the persuasive opportunity entirely. For general finite state and action spaces and arbitrary bounded utilities, our method achieves $O(\log\log T)$ regret. A matching $\Omega(\log\log T)$ lower bound shows that this rate is optimal. We further discuss the influence on receiver welfare, as well as extensions to jointly unknown prior and bias, and contextual settings with time-varying priors and utilities.