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Gale-Shapley introduced a matching problem between two sets of agents where each agent on one side has a preference over the agents of the other side and proved algorithmically the existence of a pairwise stable matching (i.e. no uncoupled pair can be better off by matching). Shapley-Shubik, Demange-Gale, and many others extended the model by allowing monetary transfers. In this paper, we study an extension where matched couples obtain their payoffs as the outcome of a strategic game and morearXiv:2107.07440v2 fatcat:3qpxqfmqmvhcjjsmwqi23s6pzu