On Achieving Leximin Fairness and Stability in Many-to-One Matchings (2009.05823v4)

Abstract: 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.

• The paper characterizes stable matchings in ranked valuations, providing a structural basis for algorithm design in many-to-one matching scenarios.
• The paper develops the FaSt and FaSt-Gen algorithms, which compute leximin optimal stable matchings under different valuation settings with provable runtime bounds.
• The paper establishes NP-Hardness and inapproximability results, highlighting the intrinsic computational challenges in balancing fairness and stability.

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

This paper focuses on addressing fairness within many-to-one matching problems under cardinal valuations, a scenario less explored compared to allocation problems where items are divided among agents with individual preferences. The novelty of this work lies in the intersection of fairness and stability, particularly under cardinal valuation settings, deviating from the ordinal preferences that have traditionally dominated two-sided matching literature.

Core Contributions

1. Characterization of Stable Matchings in Ranked Valuations: The paper introduces a complete characterization of stable matchings when all agents on each side follow consistent rankings over agents on the opposing side, although they may differ in their actual valuations. This structural property provides a pivotal basis for designing efficient algorithms.
2. Algorithm for Ranked Isometric Valuations - FaSt: A critical achievement in this paper is the development of the FaSt algorithm, which efficiently finds a leximin optimal stable matching in the ranked isometric valuation setting. Conditions include each pair of agents receiving identical valuation from each other, alongside strict preferences that maintain agent rankings. The algorithm operates in $O(mn)$ time, where $m$ represents the number of colleges and $n$ denotes the number of students.
3. Extension to General Ranked Valuations - FaSt-Gen: Building on FaSt, the research presents FaSt-Gen, an algorithm extending to environments with general ranked preferences without the isometric valuation constraint. The runtime complexity here increases to $O(m^2n^2)$, but it reflects significant progress in handling more generalized ranking-based scenarios.
4. Complexity Results:
   • It establishes the NP-Hardness of identifying a leximin optimal stable matching under strict preferences when agents do not share a consistent ranking. This underscores the computational challenges in scenarios lacking structured valuation patterns.
   • The paper further exposes the problem as strongly NP-Hard under isometric valuations with weak rankings, broadening the understanding of problem difficulty even under specific constraints.
   • A significant theoretical contribution is demonstrating the impossibility of a polynomial-factor approximation for leximin optimal matchings with general additive valuations unless P = NP. This result delineates the intrinsic complexity barriers in optimizing fairness over stability for general two-sided environments.

Practical and Theoretical Implications

• Practical Settings: The findings have direct implications for various real-world applications, such as college admissions and labor market placements, where fairness in institutional allocations profoundly affects both individuals and organizations.
• Algorithmic Paradigm: The framework and techniques developed, particularly under structured settings such as ranked valuations, inform a promising direction for further explorations into matching problems that balance multiple criteria like stability and leximin fairness. These algorithms provide a template for integrating fairness standards into matching processes with tractability.
• Research Trajectories: By elucidating settings (like those with ranked isometric valuations) that yield efficient solutions, alongside demarcating the boundaries of intractability, this paper sets a foundational benchmark. Future research could investigate refined subclasses of preferences that admit efficient or approximate leximin solutions or explore leximin-based formulations extending beyond the traditional matching paradigms to other bipartite resource allocations.

In summary, the paper investigates an underserved area in matching theory by focusing on leximin fairness under stability constraints, providing both efficient solutions and outlining the inherent complexity challenges. Its insights contribute significantly to the theoretical and practical landscapes of economic and computational matchings, especially under cardinal evaluation contexts.