We introduce the Stable Matching Problem with Minimum Utility Gap, which seeks a stable matching in which the utilities received by individual agents are as balanced as possible. Our framework can handle many-to-many matchings and general utility functions on partner sets that are consistent with the agents' preferences. We consider two measures for comparing agents' utilities: the difference between the maximum and minimum utilities, and their ratio. We provide a polynomial-time algorithm for both versions. The algorithm exploits the rotation-poset representation of the set of stable matchings and, in particular, the fact that the rotations affecting each agent form a chain in this poset. To position our result, we also clarify its relation to existing frameworks: we show that our objectives are not captured by the recent minimum-cut representability framework, while identifying a special case that admits a submodular function minimization interpretation.

翻译：暂无翻译