On Achieving Leximin Fairness and Stability in Many-to-One Matchings

The past few years have seen a surge of work on fairness in allocation problems where items must be fairly divided among agents having individual preferences. In comparison, fairness in settings with preferences on both sides, that is, where agents have to be matched to other agents, has received much less attention. Moreover, two-sided matching literature has largely focused on ordinal preferences. This paper initiates the study of fairness in stable many-to-one matchings under cardinal valuations. Motivated by real-world settings, we study leximin optimality over stable many-to-one matchings. We first investigate matching problems with ranked valuations where all agents on each side have the same preference orders or rankings over the agents on the other side (but not necessarily the same valuations). Here, we provide a complete characterisation of the space of stable matchings. This leads to FaSt, a novel and efficient algorithm to compute a leximin optimal stable matching under ranked isometric valuations (where, for each pair of agents, the valuation of one agent for the other is the same). Building upon FaSt, we present an efficient algorithm, FaSt-Gen, that finds the leximin optimal stable matching for a more general ranked setting. When there are exactly two agents on one side who may be matched to many agents on the other, strict preferences are enough to guarantee an efficient algorithm. We next establish that, in the absence of rankings and under strict preferences (with no restriction on the number of agents on either side), finding a leximin optimal stable matching is NP-Hard. Further, with weak rankings, the problem is strongly NP-Hard, even under isometric valuations. In fact, when additivity and non-negativity are the only assumptions, we show that, unless P=NP, no efficient polynomial factor approximation is possible.