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.

翻译：Gale和Shapley提出了一个涉及两组代理的匹配问题，其中每组代理对另一组代理具有外生偏好排序。他们将匹配定义为稳定的，如果不存在一个未配对组可以通过组成新配对同时提升双方效用。他们通过算法证明了稳定匹配的存在性。Shapley与Shubik、Demange与Gale以及众多学者通过引入货币转移对该模型进行了扩展。我们提出进一步扩展，假设配对后的个体其收益内生于他们必须进行的策略博弈——这种博弈可以是通常的非合作形式（无承诺），也可以是半合作形式（有承诺，即双方通过具有约束力的双边契约达成结果，其中每个契约方对其负责部分承担义务）。根据参与者是否具有承诺能力，我们分别定义了结合Gale-Shapley成对稳定性与（广义）纳什均衡稳定性的解概念。针对每种情形，我们给出了解集合非空的充要条件，并提供了求解算法。