Optimal information disclosure: A linear programming approach

Abstract

An uninformed sender designs a mechanism that discloses information about her type to a privately informed receiver, who then decides whether to act. I impose a single-crossing assumption, so that the receiver with a higher type is more willing to act. Using a linear programming approach, I characterize optimal information disclosure and provide conditions under which full and no revelation are optimal. Assuming further that the sender's utility depends only on the sender's expected type, I provide conditions under which interval revelation is optimal. Finally, I show that the expected utilities are not monotonic in the precision of the receiver's private information. negative|peach) = Pr(positive|lemon) = 1 − p. The employer is positive (negative) if the interview signal is positive (negative). Finally, the employer makes a hiring decision.I restrict attention to grading policies that generate three possible grades, A, B, or C, that convince both the positive and negative, only the positive, or no employer to hire. This is without loss of generality because convincing the negative employer also convinces the positive employer.The school chooses to maximize the probability of hire, 1 · Pr (A) + Pr (positive|B) · Pr (B) + 0 · Pr (C) subject to the constraint imposed by the prior distribution of the student's ability, m Pr (peach|m) · Pr (m) = Pr(peach) = 0 2Under the optimal grading policy, grades A and B barely persuade the negative and positive employers to hire, whereas grade C makes the employer certain that the student is a lemon; so, after some algebra, Pr (peach|A) = p, Pr (peach|B) = 1 − p, and Pr (peach|C) = 0. Using these conditions, it is easy to show that Pr (positive|B) = 2p(1 − p). The school's problem is thus a linear program: to maximize the utility function Pr(A) + 2p(1 − p) Pr(B) over probabilities Pr(A), Pr(B), and Pr(C), subject to the Bayesian budget constraint p Pr(A) + (1 − p) Pr(B) = 0 23 The Bayesian persuasion problem of this paper is mathematically similar to the delegation problem initiated by Holmström (1984). Alonso and Matouschek (2008) and Amador and Bagwell (2013) characterize necessary and sufficient conditions under which interval delegation is optimal. Their conditions resemble my conditions under which interval revelation is optimal, but their proofs are more involved.

The Interval Structure of Optimal Disclosure

“…In this section, we explain how our results extend to cases in which Sender's and Receiver's payoffs depend on both s and t , and there is no common prior. Our goal is not the extension per se, but to use the more general setup in order to compare our results to those in Kolotilin () and Kolotilin et al (). For simplicity, in this section we use examples in which Sender's type space is atomic.…”

“…Moreover, for every such that , the ratio weakly increases in s . Assumption captures the idea that Receiver's types are ranked. It implies the more general ranking assumption in Kolotilin (). For the sake of comparison with Kolotilin () and later with Kolotilin et al (), we repeat this assumption: Assumption For every such that , and every probability measure Q over , if , then . Except for some minor technical detail, this is Assumption 1 in Kolotilin ().…”

“…It implies the more general ranking assumption in Kolotilin (). For the sake of comparison with Kolotilin () and later with Kolotilin et al (), we repeat this assumption: Assumption For every such that , and every probability measure Q over , if , then . Except for some minor technical detail, this is Assumption 1 in Kolotilin (). It says that given any Sender's signal, if type is willing to accept, then a higher type t is also willing to accept.…”

“…Kolotilin () also examined optimal persuasion when the receiver has private information. He provided a linear programming approach and established the conditions under which either full or no revelation is optimal.…”

Network Effects, Bargaining Power, and Product Review Bias: Theory and Evidence

“…The theoretical framework contributes to a broad class of information transmission games first popularized by Crawford and Sobel [1982] and later revitalized by Gentzkow and Kamenica [2011]. This literature is too vast to cover in its entirety; the model used in this article most nearly resembles that of Kolotilin [2018]. In that model, the principal chooses ex ante the structure of the information transmission mechanism.…”