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The authors define the median stable matching for two-sided matching markets with side payments and prove constructively that it exists. They study one-to-one matching problems with endogenous side transfers as in Shapley and Shubik (1971), Crawford and Knoer (1981) and Kelso and Crawford (1982). In this problem, there are two sides of the market, firms and workers. Each agent on one side of the market has cardinal preferences over the agents on the other side. A matching in the setup not only assigns who is going to be matched with whom but also the monetary transfers to be made.
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