Conjugate Persuasion

We consider a class of persuasion games in which the sender has rank-dependent (Yaari (1987)) preferences. Like much of the recent Bayesian persuasion literature, we allow the sender to choose from a rich set of information structures and assume the receiver’s action depends only on her posterior expectation of a scalar state variable. Conjugate to the standard problem, our sender’s utility is linear in posterior the mean, but may be nonlinear in probabilities. We geometrically characterize the sender’s optimal commitment payoff and identify the corresponding optimal information structure. When the state is continuously distributed, communication takes a monotone partitional form. Our characterization admits a simple analysis of comparative statics—for instance, we find that “grading on a curve” is a feature of optimal design. Finally, we apply our analysis to several problems of economic interest including information design in auctions and elections, as well as the design of equilibrium insurance contracts in the face of the ‘favorite-longshot’ bias.