Approximately Stable Matchings with Budget Constraints

This paper considers two-sided matching with budget constraints where one side (firm or hospital) can make monetary transfers (offer wages) to the other (worker or doctor). In a standard model, while multiple doctors can be matched to a single hospital, a hospital has a maximum quota: the number of doctors assigned to a hospital cannot exceed a certain limit. In our model, a hospital instead has a fixed budget: the total amount of wages allocated by each hospital to doctors is constrained. With budget constraints, stable matchings may fail to exist and checking for the existence is hard. To deal with the nonexistence of stable matchings, we extend the "matching with contracts" model of Hatfield and Milgrom, so that it handles approximately stable matchings where each of the hospitals' utilities after deviation can increase by factor up to a certain amount. We then propose two novel mechanisms that efficiently return such a stable matching that exactly satisfies the budget constraints. In particular, by sacrificing strategy-proofness, our first mechanism achieves the best possible bound. Furthermore, we find a special case such that a simple mechanism is strategy-proof for doctors, keeping the best possible bound of the general case.