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Decentralized and Uncoordinated Learning of Stable Matchings: A Game-Theoretic Approach

S. Rasoul Etesami, R. Srikant

TL;DR

The paper addresses learning stable matchings in two-sided markets with unknown preferences under fully decentralized, uncoordinated interaction. It casts the problem as a complete-information stable matching game and shows that pure NE correspond to stable matchings, with mixed NE rounding to stable outcomes. It then provides two scalable learning frameworks: (i) EXP achieves logarithmic regret in hierarchical markets and local exponential convergence in general markets, and (ii) a globally convergent, decentralized alternative that leverages weak acyclicity for general markets. Together, these results connect stable matching learning to Nash-equilibrium learning in continuous-action games and offer insight into feedback design for faster convergence. The work advances principled, decentralized approaches to stable matching under uncertainty, with implications for online labor/dating platforms and other adaptive matching systems.

Abstract

We consider the problem of learning stable matchings with unknown preferences in a decentralized and uncoordinated manner, where "decentralized" means that players make decisions individually without the influence of a central platform, and "uncoordinated" means that players do not need to synchronize their decisions using pre-specified rules. First, we provide a game formulation for this problem with known preferences, where the set of pure Nash equilibria (NE) coincides with the set of stable matchings, and mixed NE can be rounded to a stable matching. Then, we show that for hierarchical markets, applying the exponential weight (EXP) learning algorithm to the stable matching game achieves logarithmic regret in a fully decentralized and uncoordinated fashion. Moreover, we show that EXP converges locally and exponentially fast to a stable matching in general markets. We also introduce another decentralized and uncoordinated learning algorithm that globally converges to a stable matching with arbitrarily high probability. Finally, we provide stronger feedback conditions under which it is possible to drive the market faster toward an approximate stable matching. Our proposed game-theoretic framework bridges the discrete problem of learning stable matchings with the problem of learning NE in continuous-action games.

Decentralized and Uncoordinated Learning of Stable Matchings: A Game-Theoretic Approach

TL;DR

The paper addresses learning stable matchings in two-sided markets with unknown preferences under fully decentralized, uncoordinated interaction. It casts the problem as a complete-information stable matching game and shows that pure NE correspond to stable matchings, with mixed NE rounding to stable outcomes. It then provides two scalable learning frameworks: (i) EXP achieves logarithmic regret in hierarchical markets and local exponential convergence in general markets, and (ii) a globally convergent, decentralized alternative that leverages weak acyclicity for general markets. Together, these results connect stable matching learning to Nash-equilibrium learning in continuous-action games and offer insight into feedback design for faster convergence. The work advances principled, decentralized approaches to stable matching under uncertainty, with implications for online labor/dating platforms and other adaptive matching systems.

Abstract

We consider the problem of learning stable matchings with unknown preferences in a decentralized and uncoordinated manner, where "decentralized" means that players make decisions individually without the influence of a central platform, and "uncoordinated" means that players do not need to synchronize their decisions using pre-specified rules. First, we provide a game formulation for this problem with known preferences, where the set of pure Nash equilibria (NE) coincides with the set of stable matchings, and mixed NE can be rounded to a stable matching. Then, we show that for hierarchical markets, applying the exponential weight (EXP) learning algorithm to the stable matching game achieves logarithmic regret in a fully decentralized and uncoordinated fashion. Moreover, we show that EXP converges locally and exponentially fast to a stable matching in general markets. We also introduce another decentralized and uncoordinated learning algorithm that globally converges to a stable matching with arbitrarily high probability. Finally, we provide stronger feedback conditions under which it is possible to drive the market faster toward an approximate stable matching. Our proposed game-theoretic framework bridges the discrete problem of learning stable matchings with the problem of learning NE in continuous-action games.
Paper Structure (22 sections, 15 theorems, 78 equations, 3 algorithms)

This paper contains 22 sections, 15 theorems, 78 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions and Organization
  4. Problem Formulation
  5. Stable Matchings with Known Preferences
  6. Stable Matchings with Unknown Preferences
  7. A Game-Theoretic Formulation and Nash Equilibrium Characterization
  8. Stable Matching Game
  9. Nash Equilibrium Characterization
  10. Logarithmic Regret for Hierarchical Markets
  11. Algorithm Design and Preliminaries
  12. Performance of Algorithm \ref{['alg:main']} for Hierarchical Markets
  13. Exponential Local Convergence for General Matching Markets
  14. Global Learning of Stable Matchings in General Markets
  15. Conclusions
  16. ...and 7 more sections

Key Result

Theorem 1

A pure strategy profile $x^*\in \{0,1\}^{n^2}$ corresponds to the characteristic vector of a stable matching if and only if it is a pure NE for the stable matching game.

Theorems & Definitions (43)

  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 1
  • proof
  • Example 1
  • Theorem 2
  • proof
  • Remark 2
  • Remark 3
  • ...and 33 more