On Achieving Fairness and Stability in Many-to-One Matchings Shivika Narang1, Arpita Biswas2, and Y Narahari1 1Indian Institute of Science (shivika, narahari @iisc.ac.in) 2Harvard University ([email protected]) Abstract The past few years have seen a surge of work on fairness in social choice literature. A major portion of this work has focused on allocation settings where items must be allocated in a fair manner to agents having their own individual preferences. In comparison, fairness in two- sided settings where agents have to be matched to other agents, with preferences on both sides, has received much less attention. Moreover, two-sided matching literature has mostly focused on ordinal preferences. This paper initiates the study of finding a stable many-to-one matching, under cardinal valuations, while satisfying fairness among the agents on either side. Specifically, motivated by several real-world settings, we focus on leximin optimal fairness and seek leximin optimality over many-to-one stable matchings. We first consider the special case of 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). For this special case, 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). The running time of FaSt is linear in the number of edges. Building upon FaSt, we present an efficient algorithm, FaSt-Gen, that finds the leximin optimal stable matching for ranked but otherwise unconstrained valuations. The running time of FaSt-Gen is quadratic in the number of edges. We next establish that, in the absence of rankings, finding a leximin optimal stable matching is NP-Hard, even under isometric valuations. In fact, when additivity and non- negativity are the only assumptions on the valuations, we show that, unless P=NP, no efficient polynomial factor approximation is possible. When additivity is relaxed to submodularity, we find that not even an exponential approximation is possible. arXiv:2009.05823v3 [cs.GT] 15 Jul 2021 1 Introduction Fairness is a very natural objective in most practical situations. In the past decade, the computa- tional problem of achieving fairness has been receiving intense attention [13, 12, 30, 15]. Several fairness notions have been studied for fair allocation problems, where agents have preferences over a set of items which must be allocated among the agents in a fair manner. In these settings, the items do not have any preferences over the agent to whom they are allocated. In contrast, two- sided (bipartite) matching problems [18, 11, 14, 28] assume two groups of agents where agents in each group have preferences over the agents belonging to the other group. The task is to match the agents of one group to the agents of another group. In such a setting, stability and fairness are clearly two very natural and desirable objectives. The fairness notions studied in the fair allocation literature have largely been unexplored for two-sided matching settings, with the exception of some very recent work [17, 21]. One reason 1 could be that most of this literature considers cardinal valuations, whereas the literature on match- ings, almost exclusively, works with ordinal valuations. We borrow the fairness notions from the fair allocation literature to the context of matchings with cardinal valuations. In particular, we focus on many-to-one matchings which are widespread in practice. Many of these settings correspond to the ranked valuation setting where there are inherent rankings1 across the agents on each side. That is, 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). For example, in labour market settings, workers are ranked based on their experience, while employers may be ranked on the wages they offer. In this paper, we use the college admissions example for ease of presentation, but our results remain applicable to other scenarios. In India, for example, admissions to undergraduate engineering degrees are based on a central- ized annual examination called the Joint Entrance Exam (JEE). Based on their performance in the JEE, the students are given an All India Rank (AIR). The AIR determines (the order in which they get to choose) the college they want to get admitted to in a centralized admissions process. The colleges in turn are ranked by their reputation which is determined by factors such as employment secured by the alumni, research opportunities that the college opens up, etc. For students, the reputation is an indicator of how much a degree from this college will help their future prospects. This ranking of colleges is the same for nearly all students (except for a small fraction of students who have a strong location preference), and is often publicly available through rankings published from surveys commissioned by national newspapers and magazines. Clearly, stability is a key criterion for college admissions. Informally, stability in this context requires that there should not be a college - student pair where both can benefit by deviating from the given matching by matching with each other. This is clearly a very logical criteria and has been shown to be efficient. In fact, it was found that using a stable matching mechanism in the context of IIT admissions was found to eliminate some of the inefficiencies that were observed when stability was not considered [8]. While stability is of crucial importance, there is another important issue that needs attention, namely fairness. Newer colleges are invariably ranked lower than well established colleges. As a result, even though the newer colleges have adequate capacity and competitive quality, they are ignored by (especially the top ranking) students. Observe that matchings where the majority of students are matched to older (higher ranked) colleges may be stable but are clearly unfair to newer colleges. This results in a poor reputation and a loss of opportunity for the newer colleges. Consequently, these colleges continue to be poorly ranked in future years as well, having similar quality. To offset this discrepancy, a reasonable notion of fairness could be implemented in the centralized admission process to these colleges. If we can ensure that newer colleges get more students, this will provide more opportunities for them to excel and improve their rankings. Of course, in our attempt to help the colleges, we must not ignore the interests of the students. Fairness must be maintained for the students and the colleges, while also being efficient. While there are a variety of fairness notions prevalent in the literature, given that they focus on allocation settings, they are not always helpful in maintaining fairness across the two sides of the bipartite graph. The question of whether agents on both sides must be treated equally is up for debate, but in this paper we assume that agents from either side are equally important. To this end, we have picked leximin optimality [10, 30] as the notion of fairness to explore. In conjunction with stability, this also ensures some degree of efficiency, as stable matchings are Pareto optimal. 1The class of ranked valuations has been well studied in the context of fair division (under various names) [1, 4, 12]. This class is, in fact, a generalization of identical valuations, which is also a very well studied class of valuation functions [30, 5, 6, 15]. 2 Informally, a leximin optimal (fair) matching is the one that maximizes the valuation of the worst- off agent (i.e. satisfies the maximin criteria), and out of those matchings that achieve this, maxi- mizes the valuation of the second worst-off agent, and so on. This would essentially minimize the discrepancy in the valuations achieved by all the students and colleges. Taking the leximin optimal outcome over the valuations of all students and colleges ensures a balance in the interests on both sides. This notion applies very well to the college admissions context since it tries to improve the allocations to lower ranked colleges, without disregarding the interests of the students. This is because any matching where a student has lower value being sent to a college than the college without this student would never be the leximin optimal. Another appealing merit of a leximin notion is that it is an optimization based fairness notion with a guaranteed optimal outcome over the space of stable matchings. While maximizers of notions like egalitarian welfare and Nash social welfare are also guaranteed to exist, we find that they are not sufficient for guaranteeing fairness for agents on either side over stable matchings. Other popular fairness notions like envy, equitability, and MMS need not coexist with stability. As a result, leximin seems to be an excellent fairness notion to explore for two-sided matchings. Due to space considerations a comprehensive discussion on how these various fairness notions interact with stability is deferred to Appendix A. We have also investigated the specific case of isometric valuations where the valuation of a student for a college is the same as the valuation of the college for the student in return. While this may appear very restrictive, it is not separated from reality. Students in India choose to pursue technical degrees in large part due to the expectation of a well paying job after graduation. The jobs that these students secure, in turn, determine the reputation of their colleges. Consequently, in such a setting, the value that a student and a college gain from each other is the expected quality of the future opportunities awaiting the student at the time of graduation.