Matchings, in particular stable matchings, have been studied for several decades, from both theoretical and applied perspectives [gale1962college, roth1982economics, teo1998geometry, teo2001gale, ashlagi2018stable, yahiro2018strategyproof, bogomolnaia2004random, caragiannis2019stable, narang2020study]. Likewise, the concept of fairness has recently been receiving intense attention, especially in social choice literature [budish2011combinatorial, barman2017groupwise, barman2019fair, garg2020approximating, plaut2020almost, freeman2019equitable, chen2020fairness]. Several fairness notions have been studied for allocation problems, where agents have preferences over a set of items and these items have to be allocated among the agents in a fair manner. However, the items do not have any preferences for the agent to whom they are allocated. In contrast, matching problems assume two groups of agents; each agent in a group has preferences for all the agents belonging to the other group. The task is to match agents of one group to the agents of another group. Stability and fairness are two very natural and desirable properties for agents on both sides in matching problems.
Surprisingly, the fairness notions studied in fair allocation literature have been largely unexplored for matching settings. One possible reason could be the fact that the majority of fair allocation literature considers cardinal utilities whereas matchings literature almost exclusively considers ordinal utilities. We are, thus, motivated to study a matching setting with cardinal utilities. In particular, we focus on many-to-one matchings which are widespread, ranging from the well-studied school choice problem [abdulkadirouglu2003school, ashlagi2018stable] to labour markets [marti2001lattice, caragiannis2019stable].
The primary motivation for our work is somewhat different from the examples prevalent in prior work. We consider a problem that has become crucial in recent months: matching COVID-19 patients to appropriate healthcare facilities available in a given city or region. Imagine a city that provides universal healthcare to all its residents. In the face of a COVID-19 outbreak, they would like to allocate their infected citizens to the available facilities. This results in a matching problem with infected citizens (patients) on one side and healthcare services (facilities) on the other side. Different healthcare facilities, such as intensive care units, hospital wards, and outpatient consultancy, would typically be capable of different types of care facilities. On the other hand, each patient carries a different risk factor due to their preexisting conditions and comorbidities. For example, a young person in their 20s, presenting mild symptoms, with no other health issues is a low risk individual. On the other hand, a person in their 80s with diabetes is a high risk individual. We assume that there may be patients ranging from very high risk factor to very low risk factor and we wish to provide care to all individuals.
As the city provides universal healthcare, patients do not pay for their treatment. Hence, their utility depends entirely on how much of an improvement they see in their health and how soon it is. In other words, a patient’s utility is a combination of their recovery rate and the magnitude of improvement in their symptoms. Clearly, this depends on the risk factor of the patient as well as the facility to which the patient is allocated. The lower the risk factor, the lesser will be the extent of improvement, and thus, lower the utility. Similarly, the better the quality of care available, the faster the patient improves. The facilities, in turn, get their funding based on the impact they have, that is, the total utility they provide to the patients matched to them. Hence, each facility wishes to maximize the total utility derived from treating the patients assigned to them.
We make two important observations about the example. Firstly, the utility that a patient and a facility obtain from being matched to each other is the same. We refer to such utilities as isometric utilities. The second is that there is a ranking over the facilities which is consistent across the patients (for instance, for any patient, the utility from a hospital ward is greater than that from a consultant). Furthermore, the patients can be ranked in accordance with their risk factors assuming that the more the health risk the more utility is derived from any facility. These two observations lead us to an interesting utility space with isometric utilities and consistent rankings. We succinctly call these as ranked isometric utilities
. Other practical settings where ranked isometric utilities apply include recommending online ads (utility is the probability of purchase), matching student volunteers to charities (utility is the quality of work done by the student), and school choice without monetary considerations (utility is the score of the student on a standardised test).
Clearly, we would like these matching solutions to be stable in the sense that low-risk patients should not be under intensive care while high-risk patients are confined to consultancy service only. However, stability alone is not enough because it may not guarantee fairness. Sending all patients to the same facility would preserve stability but is clearly unfair to all other facilities as they get utility 0. One popular fairness notion is that of envy-freeness (EF) [foley1967resource, varian1974equity, stromquist1980cut], studied in fair allocation literature. Informally, an EF allocation guarantees that every agent would prefer its own allocation over any other agent’s allocation. Since it is not possible to find such a solution in the matching setting, we consider a relaxation of EF, namely, envy-freeness up to one item (EF1) [budish2011combinatorial]. We show that it may not be possible to find a stable matching that satisfies EF1. We get into further details on this in Section 4.1.
Next, we study another fairness notion called leximin optimality [bezakova2005allocating, plaut2020almost] and show that it is a relevant solution concept in the matching setting (detailed in Section 3). Informally, a leximin optimal (fair) matching is one that maximizes the utility of the worst off agent, and out of those matchings that achieve this, maximizes the utility of the second worst off agent, and so on. This would essentially minimize the discrepancy in the utilities achieved by all the patients and facilities. Taking the leximin optimal, over the utilities of all patients and facilities, ensures a balance in the interests of both patients and facilities. We show that leximin optimality does not suffer from any of the shortcomings that occur with envy considerations.
The majority of this paper is dedicated to finding a leximin optimal stable matching under ranked isometric utilities. The problem of finding a leximin optimal solution for the allocation problem is NP-Hard [bezakova2005allocating, plaut2020almost]. We show in Lemma 1 that it is also NP-Hard to find a leximin optimal stable matching under isometric utilities. However, for the subspace of matching problems with ranked isometric utilities, we devise a polynomial time algorithm for finding a leximin optimal stable matching. In particular, we present an algorithm, which we call FaSt (Fair and Stable) that outputs a leximin optimal stable matching in time where is the number of facilities and is the number of patients. Further, we show that this algorithm finds the leximin optimal matching in settings more general than ranked isometric utilities.
We explore the twin objectives of stability and fairness for many-to-one matchings under cardinal utilities. In particular, we consider the well motivated special case of isometric utilities. We first show that a stable matching satisfying EF1 need not exist. Then, we primarily focus on leximin optimal stable matching. The main contributions of this paper are as follows:
We show that finding a leximin optimal stable matching is NP-Hard under isometric utilities, and as an obvious consequence, in more general settings as well (Section 4).
We obtain a characterisation for the space of stable matchings in the case of ranked isometric utilities (Section 5.1).
We present a novel polynomial time algorithm, which we call FaSt, that finds the leximin optimal stable matching under ranked isometric utilities (Section 5.2). On a high level, our algorithm uses the structure on the space of stable matchings to start with a patient optimal stable matching, and then iteratively improves upon the leximin value by increasing the utilities of the facilities, while maintaining stability.
Finally, we discuss a general space of utility functions (beyond ranked isometric utilities) for which FaSt correctly finds the leximin optimal stable matching.
We believe we are the first to explore matching problems with ranked isometric utilities and study the objective of finding a leximin optimal stable matching.
2 Related Work
Stable matchings and fairness have been studied almost independently for decades in social choice literature. The formal study of the stable matching problem began with Gale and Shapley’s seminal paper [gale1962college] where they first introduced the notion of stability. Their work initiated decades of research on both the theory and applications of the stable matching problem [roth1982economics, teo2001gale, caragiannis2019stable, kamiyama2019many]. We focus on a many-to-one matching setting in bipartite graphs. Many-to-one matchings, having immense practical relevance, have been studied from a variety of angles. Both theoretical [sethuraman2006many, wu2018lattice, alkan2001preferences, klaus2005stable] and practical aspects [abdulkadirouglu2009strategy, Correa2019chile, gonczarowski2019matching, Baswana2019india] of stable many-to-one matchings have been well studied. Important applications of many-to-one matchings have initiated large bodies of work including school choice [abdulkadirouglu2009strategy, abdulkadirouglu2003school, ashlagi2018stable, roth1989college, biro2010college], matching residents to hospitals [kavitha2004strongly, aldershof1996stable, klaus2005stable, irving2000hospitals, irving2003strong] and kidney exchange[roth2005kidney, roth2007efficient, biro2008three, irving2007cycle]. Our work builds on this space by focusing on fairness along with stability. While this work is inspired from the setting described in the introduction, it is relevant across the various applications of many-to-one matchings.
2.1 Fairness Concepts in Matching Problem
Fairness is extremely desirable property for many matching scenarios. However, fairness in matching settings has often been defined from context-specific angles, such as college admissions [zhang2018strategyproof, yahiro2018strategyproof, nguyen2019stable, lien2017ex]. Our work looks at fairness from a more universal angle. Some prior literature has focused on combating the inherent bias towards the proposing side in the Gale Shapley algorithm [sethuraman2006many, klaus2009fair, huang2016fair, tziavelis2019equitable]. There is also some work on procedural fairness of the matching algorithms [klaus2006procedurally, tziavelis2020fair]. However, this literature has almost exclusively considered settings with ordinal preferences. In this paper, we consider cardinal utilities and show that it is possible to adopt various fairness definitions from fair allocation literature, like, envy-freeness and leximin optimality.
As a slight relaxation of stability, prior work [wu2018lattice, aziz2019matching, yokoi2020envy] has studied envy-free matchings, where “envy-free” means the absence of blocking pairs. However, many-to-one matching literature often defines stability as no blocking pairs. To avoid confusion with the fair allocation concept of envy-freeness we define stability as the absence of blocking pairs. Moreover, our results extend well even when there are some capacity constraints (see Section 6).
2.2 Fairness Concepts in Allocation Problem
Fair allocation refers to the problem of fairly allocating a set of items among a set of agents when their cardinal utilities are known. The fairness notions in allocation problems, unlike matching problems, are defined with respect to the agents only (and not the items). In this work, we adopt these fairness notions to define two-sided fairness in the matching scenario. We now discuss some of the popular fairness notions. One popular fairness notion is called envy-freeness (EF) [foley1967resource, varian1974equity, stromquist1980cut], which ensures that every agent values her allocated bundle at least as much she values the bundles allocated to any other agent. However, an EF allocation is often not achievable when the items are indivisible. For instance, if there is only one indivisible item, positively valued by two agents. Thus, weaker fairness notions like envy-freeness up to one item (EF1) [budish2011combinatorial] have been studied for such settings. When the utilities satisfy monotonicity property, an EF1 allocation always exists and can be efficiently obtained [lipton-envy-graph]. We adopt the EF1 definition to the many-to-one matching problem and show that there are matching instances where no solution satisfies both EF1 fairness and stability.
Another fairness notion called leximin optimality has gained attention because of mainly two reasons: ) it always exists and ) whenever the marginal utilities are strictly positive, is Pareto Optimal. The hardness of finding the leximin optimal allocation was established in [bezakova2005allocating, plaut2020almost]. In special cases, when the utilities are dichotomous (or binary) [bogomolnaia2004random, kurokawa2015leximin], leximin optimality can be achieved in polynomial time. Dichotomous preferences enable us to surpass a lot of impossibilities as the setting is relatively restrictive. Bogomolnaia and Moulin show that the maximum weight matching is stable under dichotomous preferences and satisfies a variety of properties including leximin optimality. The main objective of the paper is not leximin fairness however, rather it is simply a consequence of their other work. Recent results by Benabbou et al. [benabbou2020finding] and Chen and Liu [chen2020fairness] study the properties of a leximin optimal allocation for restricted settings. Of course, it must be taken into account that properties and algorithms that hold in allocation settings do not necessarily hold in matching settings. Hence, the existing algorithms cannot be used as is. To the best of our knowledge, we are the first to consider leximin optimality for the matching problem, without restricting to dichotomous valuations.
3 Preliminaries and Main Results
We now setup our model and give the necessary definitions in order to state our main results.
3.1 Definitions and Notations
Let be a non-empty, finite, and ordered set of patients and be a non-empty, finite, and ordered set of facilities. We assume that there are at least as many patients as facilities, that is, . Note that, when , the best we can do is match the patients to facilities as in a maximum weight matching.
Let be the utility function of agent . We assume that the utilities are additive, in that the utility of a facility is the sum of its utilities for each of its matched patients. Thus, each isometric utilities instance can be uniquely identified by the tuple . Here is an matrix where is the utility of for matching with , or . Whenever we speak of rankings or ranked isometric utilities we shall assume that for all and for all . Hence, each row and column of the matrix are sorted in decreasing order.
The goal is to find a many-to-one matching of the bipartite graph such that satisfies stability as well as fairness properties. A matching is a subset of the edge set such that each patient has at most one incident edge present in the matching, whereas a facility may have multiple incident edges. Alternatively, can be defined as a function that maps an element to a set of elements, such that for each element , the function satisfies and , and for each element , the function satisfies and s.t. for each , and for each , for all . The utility of an element under a matching is defined as
Next, we define two desirable properties of a matching, namely, stability and leximin optimality.
Definition 1 (Stable Matching).
A matching of instance is said to be stable if there does not exist a blocking pair .
Definition 2 (Blocking Pair).
Given a matching , a tuple is called a blocking pair if and there exists such that and . That is, there should be no (patient, facility) pair s.t. they prefer to be matched to each other than to be matched as in .
We denote the space of stable matchings111Note that our definition of stability does not assume or imply any minimum/maximum capacity on the number of patients matched to a facility unlike in [gale1962college, roth1985college]. which do not leave any agent unmatched as .
Our work is focused on finding a leximin optimal matching. The leximin tuple of any matching is simply the tuple containing the utilities of all the agents under this matching listed in non-decreasing order. Note that the position of a particular agent’s utility in the leximin tuple may change under different matchings. The leximin tuple of a matching will be denoted by
Definition 3 (Leximin Superior).
We say that matching is leximin superior to if there exists a valid index such that for all and .
We shall say that the leximin value of is greater than that of if is leximin superior to .
Definition 4 (Leximin Optimal).
A leximin optimal matching is the matching that is leximin superior to all other possible matchings. In other words, maximizes the utility of the worst-off agent, among those that do this, maximizes the utility of the second worst-off agent and so on.
Essentially, we wish to find the matching that maximizes the left most value in the leximin tuple, of those that do, find the one that maximizes the second value and so on. In general, this problem is NP-Hard. Our goal is to find a leximin optimal matching over the space of stable matchings.
3.2 Main Results
The main focus of this paper is to find a fair and stable many-to-one matching. We first show that envy and stability are not always compatible. As a result, we turn to leximin optimality over the space of stable matchings. To this end, we study an interesting class of many-to-one matching instances called isometric utilities. This setting has a large number of practical applications, well beyond matching patients to facilities setting. The fact that the utilities are isometric does not restrict the agents from having heterogeneous preference ordering over the agents on the other side. We use this to establish the hardness of finding the leximin optimal stable matching under isometric utilities, by a reduction from the balanced partition problem.
A set of integers such that admits a balanced partition if and only if the leximin optimal stable matching of the instance allocates utility of to both and .
Consequently, if we wish to ensure that computation of the leximin optimiser is efficient, we would like to ensure that we have some structure over the space of stable matchings which we might hope to iterate over. Subsequently, we study matching instances under identical rankings where we do find structure.
Given an instance of ranked isometric utilities, a matching is stable if and only if, for all , where and
Lemma 2 ensures that a stable solution for a ranked isometric instance would necessarily allocate contiguous patientsto each . We exploit this structure in the algorithm, FaSt (Algorithm 2). We establish the correctness of our algorithm in the proof of the following theorem.
FaSt (Algorithm 2) finds a leximin optimal and stable matching for ranked isometric utilities in time .
Further, in Section 6, we list various spaces of utility functions where this algorithm can be extended. Before we prove the algorithmic results, we first demonstrate some hurdles to achieving fairness and stability in the isometric utilities space.
4 Impediments to Fairness in the Stable Matching Problem
Before we begin our discussion on leximin, let us first motivate the need for leximin optimality over the space of stable matchings. In particular, we show that envy, a well-defined and popular fairness notion, may not be compatible with stability, even under ranked isometric utilities.
4.1 Envy and Stability Don’t Mix
Often the first fairness notion that comes to mind when we think of fair allocations is envy-freeness(EF). Envy-free allocations or matchings are those where no agent has a strictly higher utility for another agent’s allocation than they have for their own. Clearly, EF allocations/matchings need not exist in indivisible settings. Consider and with utilities , , , and . Let be a matching such that and . Since , the agent envies , similarly, agent envies . It is easy to see that in every possible matching , there will always be at least one agent who will envy the other agent.
Envy-freeness upto one item (EF1) is a popular notion of fairness for allocations of indivisible items. An allocation is said to be EF1 if for every distinct pair of agents and , there exists such that . An allocation is envy-free up to any good (EFX) if for all , . In one-to-one matchings, EF1 and EFX are achieved trivially.
In the case of many-to-one matchings, finding EF1 matchings, in particular, matchings that are EF1 for the facilities, can be done by using a round robin procedure [caragiannis2019unreasonable, barman2017groupwise]. However, such a matching need not be stable, even under ranked isometric utilities. In fact, there exist ranked isometric utilities instances where no matching simultaneously satisfies stability and EF1. Consider the following example: let and . The utility matrix is as in Table 2.
Now it is easy to verify that the only EF1 matching is . This however is not stable as form a blocking pair. Hence, when stability is non-negotiable, EF1 cannot be the fairness notion of choice for isometric utilities. Consequently, neither can EFX.
It has been shown that the space of stable matchings of any given instance can be captured as the extreme points of a linear polytope [roth1993stable, teo1998geometry]. Thus, we can optimize any linear function over this space. Consequently, our next idea may be to look for a stable matching that minimizes average envy, or equivalently, total envy. But that too can often lead to matchings that are inherently unfair. Consider the example given in Table 2. The stable matchings in this example and the envy they induce is as in Table 2.
Clearly, in this example, matching all the patients to reduces the total/average envy but this is obviously unfair to . This is happening despite the fact that there are more patients than there are facilities. Note that in this example, there is in fact a ranking and yet envy doesn’t work well. In order to avoid such outcomes, we study leximin optimal fairness.
4.2 Hardness of Leximin Optimality in No-Rank Settings
While envy is not a feasible fairness notion for many-to-one matchings, leximin avoids all the downfalls of envy. It is guaranteed to exist, and it ensures that each facility will be matched to at least one patient. In some sense, it aims to bridge the disparity between the agent with the lowest utility and one with the highest. Despite its many merits, leximin too has its disadvantages, namely its intractability, even under isometric utilities. We now state our first main result, the computational hardness of finding a leximin optimal stable matching under isometric utilities. We show a reduction from the partition problem to the current problem.
Partition Problem: Given a set of integers , such that , find a -partition of which satisfies .
Given an instance of the partition problem , we create a matchings instance as follows. Set , and set . In such a setting it is easy to see that all matchings are stable as no patient has any incentive to deviate.
In any matching, patients will always get the same value, that is patient always gets value . Thus for a leximin optimal matching, it suffices to check the values that the facilities attain.
Let admit a balanced partition . Let be the matching which matches to all the patients whose values are in and matches to the rest. Here, clearly, both facilities get value . The leximin tuple of will list the values first, in non-decreasing order and the last two entries will both be . Any matching that gives any one of the facilities, say , higher utility, will naturally decrease the utility of the other facility, . Hence ’s value will either be lower in the same position in the leximin tuple, or be to the left, resulting in a lower leximin value in both cases. Hence, is a leximin optimal matching.
Conversely, if the leximin optimal matching gives value to both and then the partition created by the matching is clearly balanced. ∎
Note that the reduction actually reduces the partition problem to a setting where there are rankings, but ties are permitted. This discourages us from looking at settings other than those with strict rankings over the two partitions.
5 Leximin for Ranked Isometric Utilities
In view of the hardness of finding the leximin optimal matching under isometric utilities without any further restrictions, we now focus on a class of matching instances, called ranked isometric utilities. Ranked isometric utilities are settings where all the patients have the same strict preference order over the facilities and vice-versa. We now characterise the space of stable matchings for any matching problem with ranked isometric utilities.
5.1 Characterizing a Stable Matching
Recall that the agents are numbered in decreasing order of the utilities they induce (i.e., according to their ranks). That is and . This ranking gives structure to the space of stable matchings. We find that for a matching to be stable, it must be in accordance with the rankings. See 2
We prove the forward implication by assuming to be a stable matching that matches each facility to patients. We inductively prove that the required property holds. We first show that is the set of first patients222We can assume without lose of generality that ., that is, . Suppose not, then there exists some such that and that for some . Consequently, there must be an such that . However, and , by assumption. Thus, form a blocking pair, which contradicts the fact that is a stable matching. We now assume that, for the stable matching , the lemma holds for the first facilities, i.e., where for all
We now show that the lemma is true for . Suppose not, then there exists such that and . This implies that, there must exist some such that . Let . Now, as the instances are ranked, and . This implies that form a blocking pair which contradicts the stability of .
This proves that if is a stable matching that matches to patients, then must be equal to .
We now prove the reverse implication. Let be a matching which matches to patients and where for all .
Fix , and let . If , clearly does not form a blocking pair with any facility. If , then prefers to . But, all these facilities prefer each of the patients matched to them to , because of the ranking. Consequently, does not form any blocking pairs. As a result, there are no blocking pairs in and it is a stable matching. ∎
Note that the number of stable matchings under ranked instances is thus . Nonetheless, this seemingly simple fact provides a structure over the leximin values of stable matchings under ranked isometric utilities. Consequently, it enables us to find the leximin optimal stable matching. We now list some observations about the structure of a leximin optimal stable matching over the set of stable matchings under ranked isometric utilities.
A leximin optimal stable matching matches each facility to at least one patient, i.e. for all .
The top-ranked patient is always matched to the top-ranked facility, and the least-ranked patient is matched to the least-ranked facility. That is, and .
If is matched to then must be matched to either or . That is, .
For any stable matching , under ranked isometric utilities, the utilities of the patients will appear in accordance with their rank in the leximin tuple, that is, for any , , for all .
For any matching under isometric utilities, the utility of a facility will always be greater than or equal to the utility of any patient matched to it. That is, for each facility and for all .
Note that Observations , and do not rely on the isometric utilities property and will hold for all ranked instances with positive utilities. These observations are critical for the design of the algorithm and its proof of correctness in Theorem 1, which states that FaSt (Algorithm 2) outputs a leximin optimal fair and stable matching under ranked isometric utilities.
5.2 FaSt: An Algorithm to find a Fair and Stable Matching
In this section, we present an time algorithm, called FaSt, to find a leximin optimal (fair) and stable matching under ranked isometric utilities. The algorithm starts with a stable matching and gradually finds leximin optimal matching by improving the utilities of the facilities according to lexicographic order one-by-one, keeping stability and non-zero utilities for all agents as an invariant in each update. By Observation 3 and 5 we can start with patient and iteratively decide the matchings for higher ranked patients.
The initial stable solution matches patient to facility and patient to facility (using Observation 2). Moreover, the first patients are matched to , whereas each of the last patients are matched to the last facilities, i.e., for each .
After the initialization, the algorithm systematically increases the number of patients matched to the lowest ranked facility . A patient is added to if . To increase the number of patients matched to facility, the procedure Demote (Algorithm 1) is used, which also ensures that each facility is matched to at least one patient. We describe this subsequently. In the unlikely event when , a carefully designed look-ahead condition decides whether to include to or to stop adding anymore patients to . Note that increasing the set results in decrease of utilities of the patients matched to (since these patients are demoted from higher ranked facilities). Our algorithm balances the utilities of patients and facilities by ensuring that whenever for some , no further patients are matched to . In other words, since assigning to would decrease the utility of by a large amount, the algorithm would not assign to . At that point, the matching to the facility is fixed. We shall say that the matching of a patient (or a facility ) is fixed if and only if they are matched as in a leximin optimal stable matching.
After is fixed, the process (of increasing the number matched patients) is repeated for , , and so on. The algorithm stops when one of the two happens: either the bottom facilities get fixed, or the is matched to top-ranked patient only. Note that, every time a patient is demoted, no patient sees an increase in their utility and no facility sees a decrease in their utility. As a result, whenever a patient is demoted, their position in the leximin representation either moves to the left or stays in the same position, and the position of the corresponding facility moves to the right. Since, the utilities of the previously matched patients are unaffected, due to Observation 4 their positions stay the same. Hence, the algorithm optimizes for one leximin position at a time, before moving on to the next one.
Demote (Algorithm 1) is critical to maintaining the invariant of a stable matching where all agents have non-zero utility. If we decide to send patient to facility when they are currently matched to , it is not enough to simply do this one step. Doing this alone will make ’s utility which violates our invariant. As a result, we need to bring the patient and match her to , this must continue till we send ’s lowest ranked patient to . For this to be feasible, must be matched to at least patients. If not, no transfers are possible and as a result, no further improvement can be made to the leximin tuple. We ensure this feasibility in the while loop condition in Step 7 of FaSt.
We first prove that the algorithm always computes a leximin optimal stable matching.
Correctness: During initialization, FaSt matches to , and fixes the match by assigning to the set . This step indicates that remains matched to throughout the execution of the algorithm. Recall that, by Observation 4 listed in Section 5.1, for the optimal stable matching , the first entry in its leximin representation is , which is ensured by FaSt.
We shall say that the matching of a patient (or a facility ) is fixed if and only if the corresponding matching is a part of a leximin optimal matching. Let the matchings of be fixed, and let be matched to . Hence, the matchings of are also fixed.
By Observation 4, and must occur to the right of in the leximin tuple of any stable matching, in particular the leximin optimal one. Similarly, must be listed to the left of . This will not change, irrespective of the way the other agents are matched, as long as stability is maintained. Also, since ’s utility for does not depend on how any other agent is matched, it will not change once it is fixed. As a result, matches to if and only if it results in a leximin superior matching.
By sending to , now moves to the left of its current position in the leximin tuple. For an improvement in the leximin tuple, it shouldn’t move more than one position, and its value at the new position should not be lower than the current value. Thus, if is less than or equal to ’s current utility, must be fixed to , the matching of must be matched as in the current matching. Note that, adding more patientsto ’s matching will continue to have at the same position, due to Lemma 2 and as a result will remain leximin inferior to the current matching.
By the same reasoning, if is strictly lower than ’s current utility, must be fixed to , the matching of must be matched as in the current matching. On the other hand if is strictly greater than ’s current utility, must be fixed to .
In the case where these two values are equal, we must look ahead to see whether by demoting we eventually result in a leximin inferior matching. Note that simply comparing ’s new utility need not suffice. By demoting we may land in a setting where ’s new utility lies between and , resulting in a matching which is leximin inferior to the current one. However by sending to we may actually improve upon the current leximin tuple. Let the position of when matched to be . Thus, we must look ahead, comparing the new leximin values (obtained by sending to ) at position with the current leximin value at position . If the values are equal we may have to look ahead further and accordingly fix the matching.
Thus, when the matchings of and are fixed we can correctly match as in a leximin optimal matching.
It is straightforward to see that the algorithm takes time to terminate, which is linear in the number of edges of the underlying bipartite graph. We provide a proof for completeness.
Termination: The algorithm considers each patient for a particular facility at most once. Hence, the algorithm concludes in time. Note that, in the absence of ties, each patient is considered for at most facilities, thus taking time only. However, if ties occur, the tie breaking routine can take time for each facility. The update to the leximin tuple and the array can be done in time as well. Thus, the algorithm takes time in the worst case. ∎
6 Extending to Other Settings
While the proofs and discussions thus far have focused on many-to-one matchings under isometric utilities, with no constraints on the capacities, the approach behind FaSt can correctly find the leximin optimal in a variety of other settings. We now discuss some of these.
6.1 Capacity Constraints
The majority of prior work on many-to-one matchings assumes that each facility has a maximum capacity or budget on the number of patients it can accommodate. In such settings, stability is defined as the absence of any blocking pairs, with the additional constraint that no patient prefers some other facility (which has not reached its capacity) over their current match. The proof of computational hardness of finding leximin stable matching (in Section 4) holds true even with capacities—by setting the capacity of both facilities in the example as . However, under rankings, adding a capacity aspect to the definition of stability will give exactly one stable matching in each instance. This will be precisely the matching which matches to the top patients and to and so on, where is the capacity of .
Alternately, one may wish to continue to define stability only as the absence of blocking pairs, without capacity considerations. In such a case, a simple tweak to FaSt suffices. In the very first if condition (Step of Algorithm 2), we need to also check if is at its given capacity. Recall that this is the condition which checks if the current utility of is already greater than or equal to ’s current utility. It is simple to see that this will correctly find the leximin optimal stable matching, given that all facilities have capacity at least (similar to the proof of Theorem 1). Note that facilities that have capacity can be discarded at the outset. Thus, the same approach can be used to find the leximin optimizer under ranked isometric utilities over the space of matchings which have no blocking pairs.
6.2 Beyond Isometric Utilities
FaSt proceeds by increasing the number of patients allotted to the lowest ranked unfixed facility till there is a decrease in the leximin value. The success of this approach is contingent on 1) the inherent rankings 2) utility functions of the facilities being monotone increasing333that is, for all , we have that , for each , and 3) the five observations listed in Section 5.1. We do not rely directly on the fact that the utilities are isometric. Further, whenever we have these three requirements met in a matchings instance, we can use FaSt as is to find the leximin optimal stable matching. Essentially, FaSt works correctly for the space of monotone increasing utility functions with rankings where and for all and .
Rankings give the space of stable matchings some structure over which we are able to iterate. Consequently, rankings are essential to the success of our algorithm. We don’t expect that in the absence of some structure we will be able to find a leximin optimal stable matching efficiently. FaSt does not work for instances with rankings for which Observation 5 does not hold. However the underlying approach works when the number of facilities is exactly 2. Recall that the approach was to start with a matching which will match only to all the rest to . We then keep increasing the number of students matched to till we see a decrease in the leximin value.
Consider a setting where when where . When all the patients must be divided between the two facilities such that moving a patient from either one to the other will decrease the leximin value. Due to the structure of the stable matchings under rankings (Lemma 2 applies here) if a patient cannot be matched with then no patient ranked higher can be matched with . A decrease in the leximin value could happen in one of two ways: either ’s utility is already lower than ’s so removing any further patients will only cause the leximin value to decrease further. Otherwise, the utility of is already greater than that of the patient ranked just above the highest patient matched to . As a result, matching that patient to would decrease the leximin value. Note that when is so large that , for all complete matchings , finding the leximin optimal essentially reduces to finding a leximin optimal allocation under constraints. An algorithm which efficiently finds a leximin optimal matching for when , would imply that the corresponding leximin optimal allocation under constraints can be found in polynomial time.
7 Conclusion and Future Work
This work considers the problem of finding a fair and stable many-to-one matching under cardinal utilities. We study a notion of fairness which is guaranteed to exist: a leximin optimizer, which has been hitherto unexplored for stable matchings. We show that finding a leximin optimal stable matching is NP-Hard for the space of isometric utilities. However, when there is an inherent ranking over agents on both sides, we provide an algorithm, called FaSt, that finds a leximin optimal stable matching in time linear in the number of edges (of the underlying bipartite graph). Our algorithm, however, is applicable beyond ranked isometric utilities.
The key reason that we are able to efficiently compute the leximin optimal stable matching is the underlying structure on the space of stable matchings under rankings, this characterisation is also a non-trivial contribution of this work. An immediate follow up would be to resolve this question for the general space of ranked (non-isometric) utilities. We believe that this work will encourage further research on fair and stable matchings under cardinal utilities, with different definitions of fairnes and various subclasses of many-to-one matching problems.