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Stable Matching Games

Felipe Garrido-Lucero, Rida Laraki

TL;DR

The paper extends the stable matching framework by endogenizing matched payoffs as outcomes of two-player strategic games, analyzing both non-commitment (pairwise-Nash stability) and commitment (pairwise stability) regimes. It introduces a deferred-acceptance with competitions algorithm to guarantee existence and computability under mild regularity, and then strengthens stability with renegotiation proofness via constrained Nash equilibria. A broad class of feasible games (including zero-sum with value, strictly competitive, potential, and infinitely repeated games) is shown to admit CNE, and a renegotiation process is provided with convergence guarantees, linking to known models like Shapley–Shubik and Demange–Gale. The results yield robust stability notions in dynamic, strategic matching settings and open avenues for future work on broader market structures and dynamic environments.

Abstract

Gale and Shapley introduced a matching problem between two sets of agents where each agent on one side has an exogenous preference ordering over the agents on the other side. They defined a matching as stable if no unmatched pair can both improve their utility by forming a new pair. They proved, algorithmically, the existence of a stable matching. Shapley and Shubik, Demange and Gale, and many others extended the model by allowing monetary transfers. We offer a further extension by assuming that matched couples obtain their payoff endogenously as the outcome of a strategic game they have to play in a usual non-cooperative sense (without commitment) or in a semi-cooperative way (with commitment, as the outcome of a bilateral binding contract in which each player is responsible for her part of the contract). Depending on whether the players can commit or not, we define in each case a solution concept that combines Gale-Shapley pairwise stability with a (generalized) Nash equilibrium stability. In each case we give necessary and sufficient conditions for the set of solutions to be non-empty and provide an algorithm to compute a solution.

Stable Matching Games

TL;DR

The paper extends the stable matching framework by endogenizing matched payoffs as outcomes of two-player strategic games, analyzing both non-commitment (pairwise-Nash stability) and commitment (pairwise stability) regimes. It introduces a deferred-acceptance with competitions algorithm to guarantee existence and computability under mild regularity, and then strengthens stability with renegotiation proofness via constrained Nash equilibria. A broad class of feasible games (including zero-sum with value, strictly competitive, potential, and infinitely repeated games) is shown to admit CNE, and a renegotiation process is provided with convergence guarantees, linking to known models like Shapley–Shubik and Demange–Gale. The results yield robust stability notions in dynamic, strategic matching settings and open avenues for future work on broader market structures and dynamic environments.

Abstract

Gale and Shapley introduced a matching problem between two sets of agents where each agent on one side has an exogenous preference ordering over the agents on the other side. They defined a matching as stable if no unmatched pair can both improve their utility by forming a new pair. They proved, algorithmically, the existence of a stable matching. Shapley and Shubik, Demange and Gale, and many others extended the model by allowing monetary transfers. We offer a further extension by assuming that matched couples obtain their payoff endogenously as the outcome of a strategic game they have to play in a usual non-cooperative sense (without commitment) or in a semi-cooperative way (with commitment, as the outcome of a bilateral binding contract in which each player is responsible for her part of the contract). Depending on whether the players can commit or not, we define in each case a solution concept that combines Gale-Shapley pairwise stability with a (generalized) Nash equilibrium stability. In each case we give necessary and sufficient conditions for the set of solutions to be non-empty and provide an algorithm to compute a solution.

Paper Structure

This paper contains 20 sections, 19 theorems, 61 equations, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. Contributions
  4. Matching games
  5. No commitment
  6. Pairwise-Nash stability
  7. Commitment
  8. Pairwise stability
  9. Shapley-Shubik and Demange-Gale models
  10. Commitment with renegotiation Proofness
  11. Renegotiation proofness
  12. Constrained Nash equilibria
  13. Feasible games
  14. Zero-sum games with a value are feasible
  15. Potential games are feasible
  16. ...and 5 more sections

Key Result

Theorem 1

Suppose that all the sets in the family $\mathcal{C}$ are non-empty and compact, and that agents' payoff functions are continuous. Then, there exists a pairwise stable allocation with respect to the family $\mathcal{C}$.

Figures (5)

  • Figure 1: Stable allocations \ref{['ex:prisoners_dilemma_example']}
  • Figure 2: Feasible payoffs region
  • Figure 3: Matching pennies
  • Figure 4: Uniform equilibrium payoffs
  • Figure 5: Feasibility proof infinitely repeated games

Theorems & Definitions (69)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1
  • Example 2
  • Example 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • Remark 1
  • ...and 59 more