[Paper Review] Persuasion and Matching: Optimal Productive Transport

Anton Kolotilin, Roberto Corrao+1 more|arXiv (Cornell University)|Nov 6, 2023

Auction Theory and Applications4 citations

TL;DR

This paper introduces optimal productive transport as a general framework for information design in Bayesian persuasion with non-linear preferences. It proves that pairwise signals—where each posterior has at most two states—are always optimal, and under the twist condition, they are the only optimal solutions. The key contribution is a duality-based characterization of when disclosure is single-dipped or single-peaked, with applications to club economies, option pricing, and gerrymandering.

ABSTRACT

We consider general Bayesian persuasion problems where the receiver's utility is single-peaked in a one-dimensional action. We show that a signal that pools at most two states in each realization is always optimal, and that such pairwise signals are the only solutions under a non-singularity condition (the twist condition). Our core results provide conditions under which riskier prospects induce higher or lower actions, so that the induced action is single-dipped or single-peaked on each set of nested prospects. We also provide conditions for the optimality of either full disclosure or negative assortative disclosure, where all prospects are nested. Methodologically, our results rely on novel duality and complementary slackness theorems. Our analysis extends to a general problem of assigning one-dimensional inputs to productive units, which we call optimal productive transport. This problem covers additional applications including club economies (assigning workers to firms, or students to schools), robust option pricing (assigning future asset prices to price distributions), and partisan gerrymandering (assigning voters to districts).

Motivation & Objective

• To develop a general theory of information design for Bayesian persuasion with non-linear, single-peaked receiver preferences.
• To identify conditions under which optimal signals are pairwise, and when they induce single-dipped or single-peaked actions.
• To extend the analysis beyond persuasion to broader economic problems such as matching in club economies, robust option pricing, and partisan gerrymandering.
• To establish duality and complementary slackness theorems as core tools for characterizing optimal information structures.
• To show that under the twist condition, all optimal signals are pairwise, and to identify conditions for full disclosure or negative assortative disclosure.

Proposed method

• Formalize the persuasion problem as a first-order approach where the receiver’s optimal action satisfies a zero marginal utility condition.
• Use duality and complementary slackness theorems to derive necessary and sufficient conditions for optimality of signals.
• Define the twist condition as a non-singularity requirement ensuring that only pairwise signals are optimal.
• Analyze riskiness of prospects by comparing induced actions across pooled states, using the concept of single-dipped or single-peaked disclosure.
• Characterize optimal negative assortative disclosure as the solution to a system of ordinary differential equations.
• Apply the framework to general productive transport problems, where one-dimensional inputs are assigned to productive units to maximize output.

Experimental results

Research questions

• RQ1 Under what conditions is a pairwise signal optimal in Bayesian persuasion with non-linear preferences?
• RQ2 When does riskier disclosure (pooling extreme states) induce higher or lower actions than safer disclosure (pooling moderate states)?
• RQ3 What conditions ensure that optimal disclosure is single-dipped or single-peaked on a set of nested prospects?
• RQ4 When is full disclosure or negative assortative disclosure optimal in the absence of full information?
• RQ5 How can the optimal information structure be characterized via duality and complementary slackness in general productive transport problems?

Key findings

• A signal that pools at most two states in each realization is always optimal in Bayesian persuasion with single-peaked receiver utility.
• Under the twist condition, every optimal signal is pairwise, meaning no signal with more than two states per posterior is optimal.
• Optimal disclosure is single-dipped if and only if any single-peaked triple can be perturbed to increase sender utility by shifting weight toward the middle state.
• The optimal negative assortative disclosure pattern solves a system of ordinary differential equations, which admits explicit solutions in specific cases.
• Full disclosure is optimal when the receiver’s marginal utility is strictly increasing in the state, while negative assortative disclosure arises when the utility curvature supports riskier, non-symmetric pooling.
• The framework extends to productive transport problems, including optimal assignment in club economies, robust option pricing, and gerrymandering, where inputs are matched to productive units to maximize output.