Almost-Bayesian Quadratic Persuasion

Abstract

In this article, we relax the Bayesianity assumption in the now-traditional model of Bayesian persuasion introduced by Kamenica and Gentzkow. Unlike preexisting approaches—which have tackled the possibility of the receiver (Bob) being non-Bayesian by considering that his thought process is not Bayesian yet known to the sender (Alice), possibly up to a parameter—we let Alice merely assume that Bob behaves “almost like” a Bayesian agent, in some sense, without resorting to any specific model. Under this assumption, we study Alice's strategy when both utilities are quadratic and the prior is isotropic. We show that, contrary to the Bayesian case, Alice's optimal response may not be linear anymore. This fact is unfortunate as linear policies remain the only ones for which the induced belief distribution is known. What is more, evaluating linear policies proves difficult except in particular cases, let alone finding an optimal one. Nonetheless, we derive bounds that prove linear policies are near-optimal and allow Alice to compute a near-optimal linear policy numerically. With this solution in hand, we show that Alice shares less information with Bob as he departs more from Bayesianity, much to his detriment.