Data-driven persuasion

This paper develops a data-driven approach to Bayesian persuasion. The receiver knows the prior distribution of the state variable, the sender knows the receiver's preferences but does not know the prior, and the sender's payoffs depend on the receiver's action but not on the state. The sender observes the unconditional distribution of best responses made by a population of receivers to an unknown distribution of messages generated by an unknown and potentially heterogeneous signal. Desiring robustness, the sender seeks a signal that maximizes her worstcase payoff against the set of priors that rationalize the action data. We show positively that the two-state many-action problem has a saddle point and negatively that the two-action many-state problem does not. In the latter case, we characterize the set of optimal Blackwell experiments that are guaranteed to reveal a sufficient amount of information about the state of the world to induce a saddle point in the sender's problem.