Stable Matching Games.

In 1962, Gale and Shapley introduced a matching problem between two sets of agents $M$ and $W$ (men/women, students/universities, doctors/hospitals), who need to be matched by taking into account that each agent on one side of the market has an $\textit{exogenous}$ preference order over the agents of the other side. They defined a matching as stable if no unmatched pair can Pareto improve by matching together. They proved the existence of a stable matching using a "deferred-acceptance" algorithm. Shapley and Shubik in 1971, extended the model by allowing monetary transfers (buyers/sellers, workers/firms). Our article offers a further extension by assuming that matched couples obtain their payoff $\textit{endogenously}$ as the outcome of a strategic-form game they have to play. A matching, together with a strategy profile, is $\textit{externally stable}$ if no unmatched pair can form a couple and play a strategy profile in their game that Pareto improves their previous payoffs. It is $\textit{internally stable}$ if no agent, by individually changing his/her strategy inside his/her couple, can increase his/her payoff without breaking the external stability of his/her couple (e.g. the partner's payoff decreases below his/her current market outside option). By combining a deferred acceptance algorithm with a new algorithm, we prove the existence of externally and internally stable matchings when couples play strictly competitive games, potential games, or infinitely repeated games. Our model encompasses and refines matching with monetary transfers (Shapley-Shubik 1971, Kelso-Crawford 1982, Demange-Gale 1986, Demange-Gale-Sotomayor 1986) as well as matching with contracts (Blaire 1988, Hatfield-Milgrom 2005).