Stable Matching Games

TL;DR

This paper extends the classical stable matching model by incorporating endogenous payoffs derived from strategic games played by matched pairs, analyzing stability under both non-cooperative and semi-cooperative settings.

Contribution

It introduces a new framework combining Gale-Shapley stability with Nash equilibrium concepts, providing conditions for solution existence and algorithms for computation.

Findings

Necessary and sufficient conditions for solution existence.

Algorithms to compute stable matchings with strategic payoffs.

Analysis of stability under both non-cooperative and semi-cooperative scenarios.

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.