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Persuading Stable Matching

Jonathan Shaki, Jiarui Gan, Sarit Kraus

TL;DR

The paper investigates how a fully informed principal can influence stable matching outcomes under world uncertainty via Bayesian persuasion. It develops a structural revelation principle for matching, and identifies two tractable regimes—a small number of agent types and a small number of world states—with LP- and prototype-based algorithms that achieve optimal signaling in polynomial time when the respective parameters are fixed. It also characterizes the hardness of private persuasion, showing NP-hardness even when the world set is small, thereby mapping a comprehensive complexity landscape for stable matching under information design. The results offer practical guidance for centralized matching platforms that can leverage information asymmetry to align stable matchings with policy goals while delineating when optimal persuasion is computationally feasible. The framework advances understanding of how to coordinate complex matching markets through carefully designed public and private signals in the presence of informational asymmetries.

Abstract

In bipartite matching problems, agents on two sides of a graph want to be paired according to their preferences. The stability of a matching depends on these preferences, which in uncertain environments also reflect agents' beliefs about the underlying state of the world. We investigate how a principal -- who observes the true state of the world -- can strategically shape these beliefs through Bayesian persuasion to induce stable matching that maximizes a desired utility. Due to the general intractability of the underlying matching optimization problem as well as the multi-receiver persuasion problem, our main considerations are two important special cases: (1) when agents can be categorized into a small number of types based on their value functions, and (2) when the number of possible world states is small. For each case, we study both public and private signaling settings. Our results draw a complete complexity landscape: we show that private persuasion remains intractable even when the number of worlds is small, while all other settings admit polynomial-time algorithms. We present efficient algorithms for each tractable case and prove NP-hardness for the intractable ones. These results illuminate the algorithmic frontier of stable matching under information design and clarify when optimal persuasion is computationally feasible.

Persuading Stable Matching

TL;DR

The paper investigates how a fully informed principal can influence stable matching outcomes under world uncertainty via Bayesian persuasion. It develops a structural revelation principle for matching, and identifies two tractable regimes—a small number of agent types and a small number of world states—with LP- and prototype-based algorithms that achieve optimal signaling in polynomial time when the respective parameters are fixed. It also characterizes the hardness of private persuasion, showing NP-hardness even when the world set is small, thereby mapping a comprehensive complexity landscape for stable matching under information design. The results offer practical guidance for centralized matching platforms that can leverage information asymmetry to align stable matchings with policy goals while delineating when optimal persuasion is computationally feasible. The framework advances understanding of how to coordinate complex matching markets through carefully designed public and private signals in the presence of informational asymmetries.

Abstract

In bipartite matching problems, agents on two sides of a graph want to be paired according to their preferences. The stability of a matching depends on these preferences, which in uncertain environments also reflect agents' beliefs about the underlying state of the world. We investigate how a principal -- who observes the true state of the world -- can strategically shape these beliefs through Bayesian persuasion to induce stable matching that maximizes a desired utility. Due to the general intractability of the underlying matching optimization problem as well as the multi-receiver persuasion problem, our main considerations are two important special cases: (1) when agents can be categorized into a small number of types based on their value functions, and (2) when the number of possible world states is small. For each case, we study both public and private signaling settings. Our results draw a complete complexity landscape: we show that private persuasion remains intractable even when the number of worlds is small, while all other settings admit polynomial-time algorithms. We present efficient algorithms for each tractable case and prove NP-hardness for the intractable ones. These results illuminate the algorithmic frontier of stable matching under information design and clarify when optimal persuasion is computationally feasible.

Paper Structure

This paper contains 37 sections, 20 theorems, 53 equations, 1 figure, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Challenges and Contributions
  4. Related Work
  5. Preliminaries
  6. Stable Matching in an Uncertain World
  7. Public Persuasion
  8. Private Persuasion
  9. Extension to Many-to-One Matchings
  10. An Example
  11. Structure of Optimal Policies
  12. Revelation Principle for Matching Persuasion
  13. Computing Optimal Policies
  14. Small Number of Agent Types
  15. A Type-Based Model
  16. ...and 22 more sections

Key Result

Theorem 5

There exists an optimal policy ${\sigma}: \Omega \to \Delta(\mathcal{L} \times \mathcal{M})$. Moreover, ${\sigma}$ is indicative.

Figures (1)

  • Figure 1: Agents' values in the proof of \ref{['thm:nph-small-worlds-private']}. The x-axis represent $p(\omega_2)$. The values may not be precisely the ones defined in the proof; only the relative orders at $p(\omega_1) = 0$, $0.2$, $q$, and $1$ within each group ($v_{a_i}$ or $v_{b_i}$) are important.

Theorems & Definitions (47)

  • Definition 1: Stable matching
  • Definition 2: Stable policy
  • Example 1
  • Definition 3: Cell
  • Definition 5: Indicative policy
  • Theorem 5
  • proof
  • Theorem 6: iwama1999stable
  • Definition 7: Prototype
  • Lemma 4.0
  • ...and 37 more