Almost-Bayesian Quadratic Persuasion with a Scalar Prior

In this article, we consider a problem of strategic communication between a sender (Alice) and a receiver (Bob) akin to the now-traditional model of Bayesian Persuasion introduced by Kamenica & Gentzkow, with the crucial difference that Bob is not assumed Bayesian. In lieu of the Bayesian assumption, Alice assumes 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 scalar. We show that, contrary to the Bayesian case, Alice's optimal response may be more subtle than revealing 'all or nothing.' More precisely, Alice reveals the state of the world when it lies outside a specific interval, and nothing otherwise. This interval increases (and the amount of information shared decreases) as Bob further departs from Bayesianity, much to his detriment.