Optimal monotone signals in Bayesian persuasion mechanisms

This paper develops a new approach—based on the majorization theory—to the information design problem in Bayesian persuasion mechanisms, i.e., models in which the sender selects the signal structure of the agent(s) who then reports it to the non-strategic receiver. We consider a class of mechanisms in which the posterior payoff of the sender depends on the value of a realized posterior mean of the state, its order in the sequence of possible means, and the marginal distribution of signals. We provide a simple characterization of mechanisms in which optimal signal structures are monotone partitional. Our approach has two economic implications: it is invariant to monotone transformations of the state and allows to decompose setups with multiple agents into independent Bayesian persuasion mechanisms with a single agent. As the main application of our characterization, we show the optimality of monotone partitional signal structures in all selling mechanisms with independent private values and quasi-linear preferences. We also provide sharp characterization of optimal signal structures for second-price auctions and posted-price sales.