{"title": "Equitable Stable Matchings in Quadratic Time", "book": "Advances in Neural Information Processing Systems", "page_first": 457, "page_last": 467, "abstract": "Can a stable matching that achieves high equity among the two sides of a market be reached in quadratic time? The Deferred Acceptance (DA) algorithm finds a stable matching that is biased in favor of one side; optimizing apt equity measures is strongly NP-hard. A proposed approximation algorithm offers a guarantee only with respect to the DA solutions. Recent work introduced Deferred Acceptance with Compensation Chains (DACC), a class of algorithms that can reach any stable matching in O(n^4) time, but did not propose a way to achieve good equity. In this paper, we propose an alternative that is computationally simpler and achieves high equity too. We introduce Monotonic Deferred Acceptance (MDA), a class of algorithms that progresses monotonically towards a stable matching; we couple MDA with a mechanism we call Strongly Deferred Acceptance (SDA), to build an algorithm that reaches an equitable stable matching in quadratic time; we amend this algorithm with a few low-cost local search steps to what we call Deferred Local Search (DLS), and demonstrate experimentally that it outperforms previous solutions in terms of equity measures and matches the most efficient ones in runtime.", "full_text": "Equitable Stable Matchings in Quadratic Time\n\nNikolaos Tziavelis\n\nNortheastern University\n\nIoannis Giannakopoulos\n\nNTU Athens\n\nKaterina Doka\nNTU Athens\n\nNectarios Koziris\n\nNTU Athens\n\nPanagiotis Karras\nAarhus University\n\nAbstract\n\nCan we reach a stable matching that achieves high equity among the two sides\nof a market in quadratic time? The Deferred Acceptance (DA) algorithm \ufb01nds a\nstable matching that is biased in favor of one side; optimizing apt equity measures\nis strongly NP-hard. A proposed approximation algorithm offers a guarantee only\nwith respect to the DA solutions. Recent work introduced Deferred Acceptance\nwith Compensation Chains (DACC), a class of algorithms that can reach any stable\nmatching in O(n4) time, but did not propose a way to achieve good equity. In\nthis paper, we propose an alternative that is computationally simpler and achieves\nhigh equity too. We introduce Monotonic Deferred Acceptance (MDA), a class of\nalgorithms that progresses monotonically towards a stable matching; we couple\nMDA with a mechanism we call Strongly Deferred Acceptance (SDA), to build an\nalgorithm that reaches an equitable stable matching in quadratic time; we amend\nthis algorithm with a few low-cost local search steps to build Deferred Local Search\n(DLS), which, as we demonstrate experimentally, outperforms previous solutions\nin terms of equity measures and matches the most ef\ufb01cient ones in runtime.\n\n1\n\nIntroduction\n\nA matching process on a two-sided market can determine who gets which job [40, 31], school\nplace [44], or spouse. Gale and Shapley [16] proposed1 a model, in which each agent (e.g., woman\nor man) ranks members of the other set by strict order of preference; then agents on the one side\nissue proposals (i.e., offers) to those on the other side by that order; recipients hold the best proposal\nthey have received, without commitment, until nobody wishes to propose. This O(n2) algorithm,\ncalled the Deferred Acceptance (DA) algorithm in contradistinction to immediate acceptance [41],\nleads to a stable solution; that is, no pair of agents would rather be matched with each other than\nwith their assigned partners. The DA algorithm has had a profound in\ufb02uence on market design\nand stands at the basis of a number of centralized labor market clearinghouses around the world,\nallowing failed markets to be reorganized [41]. Roth and Shapley shared the Nobel Memorial Prize\nin Economic Sciences for their work in \u201cthe theory of stable allocations and the practice of market\ndesign\u201d, re\ufb02ecting also Roth\u2019s application of these results to real-world markets [32].\nThe problem instance size may be large. In China, over 10 million students apply for admission to\nhigher education institutions annually through a centralized process [32]. Similar centralized schemes,\nin which students apply for education programs and are ranked according to their scores [8], occur in\nGermany [9], Greece, Hungary [7], Ireland, Spain [37], Turkey [6, 8, 4], and several school districts\nin the USA [1, 3, 2]. Apart from the instance size, the set of possible stable matchings is large in\nreal-world markets [20], and exponentially growing in the worst case [29, 30]. Still, the DA algorithm\nreturns a solution optimal for proposers, as each proposer gets the best match possible in any stable\n\n1The US resident matching program had used this algorithm since 1952 for junior doctor recruitment [40].\n\n33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada.\n\n\fmatching, and, in reverse, pessimal for recipients [16, 34]; thus, it \ufb01nds either the man-optimal or the\nwoman-optimal stable matching. Yet many real-world markets require stable matchings that are fair\nto both sides [19, 43, 15]. For example, in a health care market, each surgeon may have preferences\nfor which anesthetist to work with, and vice versa; an impartial allocation that eschews any favoritism\nwould arguably lead to a sense of fairness and better performance [39]. There is then a practical\nneed to \ufb01nd stable matchings that do not favor any side. Unfortunately, minimizing apt measures\nof equity or balance between the two sides is NP-hard [26, 15]. An approximation algorithm [25]\nprovides a guarantee only with respect to the biased DA solutions. Thus, there is a need for ef\ufb01cient\nand effective algorithms that produce equitable stable matchings [21, 42].\nIn this paper, we provide the \ufb01rst, to our knowledge, quadratic-time algorithms that reach stable\nmatchings of good equity measures. We \ufb01rst introduce a class of algorithms called Monotonic\nDeferred Acceptance (MDA), which exploit the growth of a monotonic state function; then, we\nintroduce a new proposal mechanism, Strongly Deferred Acceptance (SDA), by which an agent\ncannot be in a pair and issue proposals at the same time. We devise an algorithm utilizing MDA and\nSDA, POWERBALANCE, that terminates in O(n2) time, and enhance it with a few selective low-cost\nlocal search steps to produce even more equitable solutions. We call the full operation Deferred Local\nSearch (DLS). Our experimental study with simulated markets shows that DLS outperforms the state\nof the art in equity measures and matches the most ef\ufb01cient heuristics in runtime.\n\n2 Background and Related Work\nAn instance I of the stable marriage problem (SMP) comprises of a set W = {w1, w2, . . . , wn} of n\nwomen and a set M = {m1, m2, . . . , mn} of n men, and for each person (or agent) a preference\nlist, i.e., a total order of the members of the opposite side from most to least preferable. Let (cid:96)q be\nthe preference list of agent q; (cid:96)q[k] = p means that q ranks p as its kth preference, with k = 0\ndenoting the highest preference; we also write prq(p) = k. If a woman w prefers m1 to m2, i.e.,\nprw(m1) < prw(m2), we denote that as m1 (cid:31)w m2; likewise for men\u2019s preferences. A (perfect)\nmatching \u00b5 on I is a set of n disjoint man-woman pairs. If a woman w and a man m are matched in\n\u00b5, we write \u00b5(w) = m and \u00b5(m) = w. A woman w and a man m form a blocking pair for \u00b5 when:\n(i) \u00b5(m) (cid:54)= w; (ii) w (cid:31)m \u00b5(m); and (iii) m (cid:31)w \u00b5(w). A matching \u00b5 is stable if no blocking pair\nexists for \u00b5, otherwise it is unstable. The SMP calls for \ufb01nding a stable matching.\n\nThe Deferred Acceptance Algorithm In the Deferred Acceptance (DA) algorithm [16], each man\nm starts out from his \ufb01rst preference, with an index \u03bam = 0, and proposes to the woman at entry\n(cid:96)m[\u03bam], increasing \u03bam in each iteration, as long as he remains unmatched. A woman w accepts a\nproposal from a man m to form pair (w, m) if she is single or m is more preferable to her than the\ncurrent \ufb01anc\u00e9, \u00b5(w). We express this acceptance condition by the following Boolean predicate:\n\naccept(w, m) = single(w) \u2228 m (cid:31)w \u00b5(w),\n\n(1)\n\nwhere \u00b5 is the matching created so far. If the proposal is rejected, m moves to preference \u03bam + 1. The\nDA algorithm reaches a stable matching in O(n2) steps [19]; the number of pairs never decreases:\nwhen a woman breaks one pair and creates another, her preference for her \ufb01anc\u00e9 may only improve;\ncontrariwise, a man\u2019s preference for his \ufb01anc\u00e9e may only worsen.\n\nBreakmarriage and Rotations The DA algorithm is biased: it returns, out of a set of stable\nmatchings that may grow exponentially in the worst case [22], one that is most preferable to each\nproposing agent and least preferable to each recipient agent [16, 34]. For example, if men\u2019s \ufb01rst\npreferences do not con\ufb02ict, each man may obtain his \ufb01rst choice, regardless of how satisfactory\nthat is to women. The complete set of stable matchings for a problem instance forms a distributive\nlattice under a natural dominance relation2, in which the unique maximum and minimum elements\nare the two gender-optimal matchings [29, 30]. This lattice can be traversed through breakmarriage\noperations [34]: starting out from a stable matching \u00b5, we break a pair (m, w); then man m proceeds\nas per the DA algorithm, initiating a sequence of proposals that terminates either with a man being\nrejected by all women (a dead-end) or to a new stable matching \u00b5(cid:48). During this operation, there is\nexactly one single man at any time, who makes the next proposal. The resulting stable matching is\ndominated by the initial one, in the sense that all men who changed partners are worse off.\n\n2A stable matching dominates another when it is strictly preferred by one gender.\n\n2\n\n\fA breakmarriage operation [34] corresponds to one or more rotations, i.e., minimal operations\nwhereby a cyclically ordered sequence of pairs exchange partners, transforming one stable matching\nto another [22]. A precedence relation de\ufb01nes a partial order by which rotations can be performed,\nthe rotation poset. Each stable matching corresponds to a closed subset of the rotation poset [22];\napplying this subset of rotations from one lattice end, in any valid order, results to the same stable\nmatching. All rotations are found in O(n2) time via breakmarriage operations [18, 19].\n\nDe\ufb01ning Fairness The bias of the DA algorithm calls for solutions that optimize some measure of\nfairness. Knuth [29, 30] describes, with credit to Selkow, an O(n4) algorithm to \ufb01nd a stable matching\n\u00b5 that minimizes the lowest preference assigned to any agent, or regret cost r(\u00b5); others proposed an\nO(n2) algorithm [18] and another O(n4) algorithm to the same effect [39]. Still, a minimum-regret\nmatching may coincide with one of the DA outputs, even when there are many stable matchings [21].\nTo consider the big picture, we de\ufb01ne two quantities: the sums of women\u2019s and men\u2019s preferences for\n\ntheir matches in a stable matching \u00b5: \u21181 =(cid:80)\n\n(m,w)\u2208\u00b5 prm(w), \u21182 =(cid:80)\n\n(m,w)\u2208\u00b5 prw(m).\n\nGiven a pair (m, w)\u2208 \u00b5, m envies the partners of all women w(cid:48) such that w(cid:48) (cid:31)m w; the egalitarian\ncost [19] is a measure of fairness that counts the number of envy situations in the market [38].\nEg(\u00b5) = \u21181 + \u21182. A stable matching of minimum egalitarian cost is found in O(n3) [23, 14]. Still,\nby such a matching, one side may fare much better than the other. The sex equality cost [19] measures\nthe gap between the two sides\u2019 sums of preferences for their matches: SEq(\u00b5) = |\u21181 \u2212 \u21182|.\nStill, sex-equality may compromise overall happiness: by this measure, a stable matching in which the\ntwo sides are closer to each other is preferred over another matching in which both sides fare better,\nbut at an increased gap. The balance cost [15] provides an alternative: Bal(\u00b5) = max{\u21181, \u21182},\nminimizing the unhappiness of the most unhappy side [33]. Our goal is to \ufb01nd stable matchings\nof low sex equality and balance cost. Unfortunately, minimizing the sex equality cost is strongly\nalgorithm, which for some \ufb01xed \u03b5 > 0,\nreturns a matching \u00b5 such that SEq(\u00b5) \u2264 \u03b5\u2206, where \u2206 is the least sex-equality cost among the two\nDA outputs, or reports that no such matching exists. We revisit this algorithm in our experimental\nstudy. Minimizing balance is also NP-hard [15]. Manlove [32] constructs an instance, credited to\nMcDermid, in which no balanced stable matching is a sex-equal stable matching, and vice versa.\n\nNP-hard [26, 33]. Iwama et al. [25] gave an O(cid:16)\n\nn3+ 1\n\n(cid:17)\n\n\u0001\n\nDA-Extending Procedures Past research [13, 35, 17, 11] has proposed procedures that aim to\n\ufb01nd a fair stable marriage by extending the DA algorithm; they allow agents on both sides to issue\nproposals, one after another, each agent following the order of its preference list. At any time, \u03baa\ndenotes the position on the preference list of agent a where a issues a proposal when its turn comes; \u03baa\nincreases with every rejection. When a accepts a proposal from b, such that b (cid:31)a (cid:96)a[\u03baa], then it sets\n\u03baa = pra(b), i.e., it upgrades \u03baa to the position of b in its preference list, so that it resumes proposals\nin case of a divorce. Yet an agent a may not skip forward positions in its preference list. We call this\nclass of algorithms [13, 35, 17, 11] DA-extending procedures; all DA-extending procedures arrive\nat a stable matching \u00b5 iff each agent a is in a couple with its preference at \u03baa, i.e., \u00b5(a) = (cid:96)a[\u03baa].\nHowever, they may enter endless loops. Dworczak [11] suggests a variant, Deferred Acceptance with\nCompensation Chains (DACC), that immediately compensates any agent a abandoned by a partner\nthat had proposed to a, letting a issue proposals until it \ufb01nds a new partner. Dworczak [11] does not\nprove termination in the case in which two divorcees need to be compensated in the same round of the\nalgorithm, and gives no polynomial runtime bound for DACC; after communication with the author,\nwe have ascertained that DACC terminates in O(n4) [12]. Still, there is no suggestion in [11] on how\nto generate operations that quickly converge to a solution achieving high fairness. The main idea of\nDACC is reminiscent of EROM [39, 27], a regret-minimizing O(n4) procedure that lets all agents\npropose with progressive receptiveness: in round k, only preferences ranked up to k may be proposed\nto and accepted. EROM compensates every agent abandoned by its partner; at its \ufb01nal stage, when\nk = n, it enacts compensation chains that go on until a single agent accepts a proposal. Yet, contrary\nto DACC, EROM only accesses a regret-minimizing sublattice of the stable marriage lattice. In this\nrestrictive nature, EROM is akin to LOTTO [5], a random serial dictatorship mechanism that reduces\nthe space of attainable stable matchings in favor of a randomly chosen agent in each iteration.\nIn another direction, a local search algorithm, BILS [45], starts out from the two DA solutions and\nbidirectionally traverses the lattice of stable marriages via breakmarriage operations, guided by a cost\nmeasure. When the two operations meet each other in terms of cost, it outputs the one of best cost.\n\n3\n\n\f3 Enforcing Monotonicity\n\nWe aim to provide a procedurally fair [28] DA-extending procedure that converges to a stable\nmatching of high equity in quadratic time. We \ufb01rst introduce some basic concepts.\nDe\ufb01nition 3.1 (Proposal index). Given a set of agents A = {ai} in a two-sided market, the proposal\nindex \u03baa of an agent a is the index of the position in a\u2019s preference list, such that a intends to make its\nnext proposal (offer) to the agent (cid:96)a[\u03baa] (or to none, if \u03baa = n); \u03baa advances to the next position with\neach rejection, yet backtracks to the position pra(b) of an agent b who proposes to a, if b (cid:31)a (cid:96)a[\u03baa].\nDe\ufb01nition 3.2 (State). Given a set of agents A = {ai} in a two-sided market the state of A at a\ngiven time is the set {\u03baai}, where \u03baai is ai\u2019s current proposal index value.\nDe\ufb01nition 3.3 (Frontier index). Given a set of agents A = {ai} in a two-sided market the frontier\nindex \u03bba of each agent a is the largest value that a\u2019s proposal index \u03baa has assumed so far, i.e., the\nfarthest position in a\u2019s preference list to which a has ever made an offer.\nDe\ufb01nition 3.4 (Idle agent). An agent a is idle when it has proposed to all its preferences up to its\ncurrent match or the end of its preference list, i.e., \u03baa = pra(\u00b5(a)) or \u03baa = n.\nDe\ufb01nition 3.5 (Idle couple). A couple {a, b} \u2208 \u00b5 is idle when both of its members are idle, i.e.,\n\u03baa = pra(b) and \u03bab = prb(a), hence a = (cid:96)b[\u03bab] and b = (cid:96)a[\u03baa]; in other words, a and b have both\nproposed to each other and none of them is still making offers to other options.\n\nMonotonic Events A procedure of proposals issued by both sides that does not terminate must\neventually bring A back to a state where it has already been. In reverse, as long as a procedure\nbrings A to states where it has never been before, it is not in a loop. Thus, if we reach a state never\nencountered before, then an algorithm is not looping. We can determine that we reach a state never\nencountered before when a monotonically non-decreasing function of state grows to a value never\nreached before. By enforcing the growth of such functions, we ensure that the algorithm in question\ndoes not loop. We call an event of growth of such a function a monotonic event.\nDe\ufb01nition 3.6 (Monotonic event). Given a set of agents A = {ai} in a two-sided market and an\nalgorithm operating on it, a monotonic event is the increase of a function that is monotonically\nnon-decreasing during the algorithm\u2019s operation and upper-bounded by a maximum value.\n\nWe now de\ufb01ne two such functions. Each frontier index \u03bba is monotonically non-decreasing, as by\nde\ufb01nition it cannot be decreased during an algorithm\u2019s operation, and is upper-bounded by n. Thus:\nCorollary 3.1. The increase of a frontier index \u03bba is a monotonic event.\nThe number of idle couples C is also monotonically non-decreasing: if such a couple is broken by one\npartner a, then a accepts a proposal from a more preferable option b, and thereby remains idle with\n\u00b5(a) = b = (cid:96)a[\u03baa], while the proposing agent b becomes idle, as it has just proposed to a = (cid:96)b[\u03bab].\nCorollary 3.2. The increase of C = |{{a, b} \u2208 M|a = (cid:96)b[\u03bab] \u2227 b = (cid:96)a[\u03baa]}| is a monotonic event.\nWe call the class of algorithms that enforce monotonic events Monotonic Deferred Acceptance\n(MDA). The following theorem de\ufb01nes an example of an MDA procedure.\nTheorem 3.1. Assume an algorithm operates on a set of agents A = {ai} in a two-sided market,\nstarting from any state. Then continuous proposals by agents on the same side will lead, in at most\nO(n2) steps, to one of the following events: (i) a frontier index \u03bba increases, or (ii) a new, additional\nidle couple is formed, hence C increases, or (iii) all agents on the proposing side become idle.\n\nProof. As proposing-side agents do not receive offers, none of them rises to a more preferable\nposition in its preference list. Thus, each proposing-side agent a increases \u03baa. Eventually, one of\nthem may reach and exceed \u03bba, a monotonic event. Alternatively, an agent a, may issue a proposal\nand form a new idle couple (either by proposing to its current match \u00b5(a) or to a single agent) before\nit reaches \u03bba, also a monotonic event. If no such event occurs, then each proposing agent a either\n(i) is already idle, or (ii) has a proposal accepted at \u03baa \u2264 \u03bba and becomes idle without forming\nan additional idle couple, or, (iii) has \u03bba = n and \u03baa reaches that terminal position, hence again a\nbecomes idle. Therefore, eventually either a monotonic event occurs, or all agents on the proposing\nside are rendered idle; that happens in at most O(n2) steps, the amount of all possible proposals.\n\n4\n\n\fCases (i) and (ii) in Theorem 3.1 constitute monotonic events. If such an event occurs, the algorithm\nis not in a loop; we can then switch from the one side, A, to the other side, B, so as to give to agents\non both sides the opportunity to receive and issue proposals, and insist on side B until a monotonic\neven occurs; the sooner a monotonic event occurs and we switch side, the more evenly we treat\nthe two sides. However, no monotonic event occurs in Case (iii), when all agents on side A are\nrendered idle. It is tempting to think that, with all agents on side A already idle, the termination of\nthe algorithm is imminent, after a few proposals from side B. Unfortunately, this is not the case, as\nthere may exist a couple {a, b} with an idle agent a on side A and a non-idle partner b on side B, i.e.,\nwith (cid:96)a[\u03baa] = b but (cid:96)b[\u03bab] (cid:31)b a; after switching to side B, b may propose to others on side A and\nhence abandon a; thereby, a is rendered non-idle, and hence we will still need to return to proposing\nwith side A. In other words, the allowance for couples in which one partner is non-idle3 renders the\ntermination of the algorithm problematic and calls for measures like those in [11], which incur a high\ncomputational overhead. In the following, we introduce our proposal that overcomes this problem.\n\n4 Strongly Deferred Acceptance\n\nSince termination is rendered problematic by couples that contain a non-idle agent, we reason that\nwe should disallow the creation of such couples in the \ufb01rst place; in other words, every couple should\nbe an idle couple. By that precaution, once all agents on one side, A, are rendered idle, no agent a\non side A can be abandoned by its partner: if such partner b exists, it is necessarily idle, and every\nagent on side A that could propose to b is idle too. Then, as we will show, after all agents on side A\nare rendered idle, the algorithm can securely terminate by letting agents on side B propose. Yet, to\ndisallow the creation of couples with a non-idle agent, we should modify the proposal acceptance\ncriterion in Equation (1), employed by the DA and DA-extending algorithms [16, 17, 11]. By this\ncriterion, as discussed in Section 2, an unmatched agent a accepts a proposal from any agent b on\nthe other side; thereafter, it may continue issuing proposals of its own, as long as \u03baa < pra(b), i.e.,\n(cid:96)a[\u03baa] (cid:31)a b. We propose a simpler acceptance criterion that eschews this duplicity: an agent q\naccepts a proposal from another agent p if and only if p is preferable to q over its next proposal target:\n\naccept(q, p) = p (cid:31)q (cid:96)q[\u03baq]\n\n(2)\n\nIn case of acceptance, q sets \u03baq = prq(p), otherwise p moves on to preference \u03bap + 1. We call this\nmechanism Strongly Deferred Acceptance (SDA).\n\nProperties We now study the capacity of an SDA procedure using an arbitrary order of proposals\nto terminate to stable solutions from a given starting state, i.e., its stability and reachability properties.\nDe\ufb01nition 4.1. An SDA proposal procedure terminates when it brings all agents to an idle state.\nDe\ufb01nition 4.2. Given a stable matching \u00b5, we characterize the position of agent p with respect to\n\u00b5 in terms of its \u03bap as follows: (i) if \u03bap < prp(\u00b5(p)), p is \u00b5-overrated, i.e., proposing above its\nmatch in \u00b5; (ii) if \u03bap = prp(\u00b5(p)), p is \u00b5-pivotal, i.e., ready to propose to its assignee in \u00b5; (iii) if\n\u03bap > prp(\u00b5(p)), p is \u00b5-underrated, i.e., has been already rejected by its match in \u00b5.\nLemma 4.1. Given two agents p and q, during the operation of an algorithm issuing proposals\nby both sides, starting with \u03baa = 0 \u2200a \u2208 A, there can be no state in which \u03bap > prp(q) and\n\u03baq > prq(p).\n\nProof. Assume p and q \ufb01nd themselves in such a position. Then one of the two, say p, must have\nexceeded its preference for the other, q, while q was already in such a position. Then q must have\nrejected a proposal from p while \u03baq > prq[p], i.e., p (cid:31)q (cid:96)q[\u03baq]; that cannot happen: q should have\naccepted the proposal from p, since p (cid:31)q (cid:96)q[\u03baq].\nTheorem 4.1 (Stability). When a procedure by SDA terminates, the outcome is a stable matching.\n\nProof. Suppose the resulting matching contains a blocking pair (x, y), i.e., \u03bax > prx[y] and \u03bay >\npry[x]; that is a violation of Lemma 4.1. Hence the theorem follows.\nLemma 4.2. By SDA, an \u00b5-overrated agent p may only form a couple with an \u00b5-underrated agent.\n\n3Note that at least one partner is always idle, as one must have proposed for the couple to be created.\n\n5\n\n\fProof. Let p be an \u00b5-overrated agent that forms a couple with q. If q is \u00b5-overrated, then (p, q) would\nbe a blocking pair in \u00b5, hence \u00b5 would not be stable. If q is pivotal, then p = \u00b5(q), hence q = \u00b5(p),\nthus p cannot be overrated. Hence the lemma follows.\nTheorem 4.2 (Universality). Starting from the state {\u03bai = 0,\u2200ai \u2208 A}, any stable matching \u00b5 is\nreachable by SDA proposals.\n\nProof. Let \u00b5 be any stable matching. Initially, all agents are \u00b5-overrated. Let each agent a propose\nto all preferences up to \u03baa = pra(\u00b5(a)); by Lemma 4.2, these proposals cannot be accepted, as they\nwould form pairs among \u00b5-overrated agents; then all agents are \u00b5-pivotal, hence produce \u00b5.\n\nExploiting SDA The following theorem shows how we can achieve termination by SDA.\nTheorem 4.3. Assume an algorithm operates on a set of agents A = {ai} in a two-sided market\nunder SDA, starting from a state in which all agents on one side, A, are idle. Then continuous\nproposals issued from the other side, B, lead, in O(n2) steps, to a stable matching \u00b5.\n\nProof. By Theorem 4.1, to show that the outcome is a stable matching, it suf\ufb01ces to show the\nprocedure terminates with all agents rendered idle. Each agent a on side A remains idle during\nproposals from side B, since it has either reached \u03baa = n as a single, or is matched to an idle\npartner b, and remains idle in case it accepts a proposal from another agent b(cid:48) on side B. As only\nagents on side B propose, eventually they all are rendered idle too. The process requires proposals at\nmost equal to the the length of the preference lists of side B, hence O(n2) proposals.\n\nWe call the total two-round process COMPROMISE. The critical point is that, by SDA, once all agents\non one side are idle, none of them can lose its partner, who is also idle. Contrariwise, by DA, an idle\nagent may be abandoned by a non-idle partner, hence termination does not come forth in two rounds.\nWe now develop an algorithm that terminates ef\ufb01ciently and caters to fairness too. We propose\nan initial phase in which the two sides both propose in turns, followed by a COMPROMISE phase.\nIf COMPROMISE is applied on the initial state with side A proposing \ufb01rst, it produces the B-side-\noptimal stable marriage. This may sound counterintuitive, given that the DA algorithm obtains a\nproposer-optimal matching [16]. Yet, in DA, the other side never proposes. When both sides propose\nin turns, the advantage is with the one that receives proposals \ufb01rst. Thus, it is fair to assign the role of\nproposers in each round to the side deemed to be better off, as measured by their \u03ba index values.\n\nAlgorithm 1 PowerBalance\nInput: A = M \u222a W (men and women), limit, cost\nOutput: stable matching \u00b5\n\nn = |M| = |W|; \u00b5 = \u2205; Rounds = 0\nfor all x \u2208 A do \u03bax = 0\nwhile (|\u00b5| < n) do\n\nRounds++; P = STRONGSIDE(M, W)\nfor all p \u2208 P do PROPOSE(p,\u00b5)\nif (Rounds > limit) then\n\n\u00b51 = COMPROMISE(M, \u00b5); \u00b52 = COMPROMISE(W, \u00b5)\nif (cost(\u00b51) \u2264 cost(\u00b52)) then \u00b5 = \u00b51 else \u00b5 = \u00b52\n\nw\u2208W \u03baw) then return M else return W\n\nreturn \u00b5\nfunction STRONGSIDE(M, W)\n\nm\u2208M \u03bam \u2264(cid:80)\n\nif ((cid:80)\n\nfunction COMPROMISE(C, \u00b5)\n\nif (C == M) then F = W else F = M\nwhile (\u2203x \u2208 C : \u00b5(x) = \u2205 \u2227 \u03bax < n) do\nwhile (\u2203x \u2208 F : \u00b5(x) = \u2205 \u2227 \u03bax < n) do\n\nfor all x \u2208 C do PROPOSE(x,\u00b5)\nfor all x \u2208 F do PROPOSE(x,\u00b5)\n\nprocedure PROPOSE(p, \u00b5)\n\nreturn \u00b5\nif (\u00b5(p) = \u2205 \u2227 \u03bap < n) then\n\nq = (cid:96)p[\u03bap]\nif accept(q, p) then\nif \u00b5(q) (cid:54)= \u2205 then\nr = \u00b5(q); \u00b5 = \u00b5 \\ {(cid:104)q, r(cid:105)}\n\u00b5 = \u00b5 \u222a {(cid:104)p, q(cid:105)}; \u03baq = prq(p)\n\nelse \u03bap = \u03bap + 1\n\n6\n\n(cid:46) Enforce termination after limit rounds\n\n(cid:46) side C, matching \u00b5\n(cid:46) Render side C idle\n(cid:46) Side F completes the matching\n\n(cid:46) proposer p, matching \u00b5\n\n(cid:46) p wants to propose to q\n\n(cid:46) break up q if married\n\n(cid:46) match p and q\n(cid:46) q rejects p\n\n\fAlgorithm 1, POWERBALANCE, applies this principle: it goes through a series of SDA proposal\niterations, in each of which the strongest side proposes; if the number of such matchmaking rounds\nexceeds a limit without termination, then POWERBALANCE enforces termination:\nit tries the\nCOMPROMISE procedure on both sides and chooses the solution that best \ufb01ts its goal, yielding a\nstable matching; as cost measure we use either the sex equality cost or the balance cost, introduced in\nSection 2. Moreover, we can control how fast we reach such a matching by tuning the limit of O(n)\nproposal rounds that it performs before enforcing the O(n2) termination procedure. We contend\nthat a few rounds can bring the two sides at a position of good balance, from which we can enforce\ntermination, with O(n2) overall runtime.\n\n5 Deferred Local Search\n\nThe algorithms discussed in Section 2 can be classi\ufb01ed into two types: (i) those that progressively\ntransform an unstable condition to a stable one; and (ii) those that move from one stable matching to a\nmore favorable one by local search. The former are more ef\ufb01cient, while the latter may achieve higher\nquality in terms of an equity measure, at the price of high worst-case complexity. We propose Deferred\nLocal Search (DLS), which \ufb01rst quickly converges to a fair stable matching by POWERBALANCE,\nand then improves upon this outcome with a few steps of local search in the lattice of all stable\nmatchings. This way, it achieves both ef\ufb01ciency and high quality in terms of equity measures.\n\nEnhancing Local Search The local search procedure in BILS [45] uses breakmarriage operations\n[34], each requiring O(n2) time, thus spends O(n3) per step to evaluate neighboring solutions\nproduced by breakmarriage on each of n agents. We reduce this cost by exploring the lattice via\n\ufb01ne-grained rotations [22] rather than bulk breakmarriage operations. We compute all rotations in\nO(n2) [19], and then, in each step, eliminate those exposed (i.e., amenable to elimination) in O(n2).\nWe also designed an enhanced, rotation-based version of BILS, which we term iBILS.\n\nApplying Local Search Our \ufb01rst Deferred Local Search (DLS) proposal, HYBRID, moves ahead\nfrom the output of POWERBALANCE, so as to reach a good neighboring solution in the lattice of\nstable matchings via rotation operations. Even in its re\ufb01ned form, BILS starts out from an extreme\nposition in the lattice and proceeds through several O(n2) local search steps, amounting to a O(n4)\nworst-case complexity. By contrast, HYBRID starts out from a middle position in the lattice, and\nperforms a controlled number of local search steps, with a O(n2) worst-case complexity. Our second\nDLS proposal, HYBRIDMULTISEARCH (HMS), enforces the termination of POWERBALANCE at\ndifferent rounds to yield several evenly placed solutions as starting points for local search. Instead\nof deciding on one of two sides when enforcing termination by COMPROMISE, we use both options\nas starting points. HMS takes O(rn + kmn2) time, where r is the number of POWERBALANCE\nproposal rounds, k the number of local searches, and m the maximum number of local search steps.\n\n6 Experimental Study\n\nWe conduct experiments measuring sex-equality cost, balance cost, and runtime. We use synthetic\ndatasets that draw preferences from three distributions: Uniform(U), with preferences created fully at\nrandom; Discrete(D), where for a Hot Set H \u2286 A, if ai \u2208 H then ai (cid:31)bk aj,\u2200aj \u2208 (A \u2212 H),\u2200bk \u2208\nA ; and Gaussian(G), in which ai (cid:31)bk aj iff i + X \u2265 j + Y,\u2200bk \u2208 A for X, Y = N (0, 0.4n). We\nalso generate asymmetric data set, in which one side follows the Uniform model, while the other\nside follows the Discrete; we set the Hot Set of Discrete distributions to include 40% of the agents.\nLast, we apply our solution on real data, reported at the end of this section. The algorithms are\nimplemented in Java4 and tested on an Intel Xeon 2.67GHz CPU with 28GB RAM.\n\nPowerBalance Parameter Tuning POWERBALANCE employs a limit parameter, which deter-\nmines the maximum number of matchmaking proposal rounds it performs before enforcing termi-\nnation. We experimentally determine a suf\ufb01cient value for limit as a function of dataset size. We\ngenerated 100 instances for every size n; for each instance, we tested a large number of limit values\nto \ufb01nd out the smallest value that suf\ufb01ces to get the best obtained results on sex-equality (SEq). We\nobserved that a suf\ufb01cient limit value grows in a fashion similar to n log2(n). In effect, we set the\n\n4Code and data are available at https://github.com/ntzia/stable-marriage\n\n7\n\n\fPOWERBALANCE limit to \u0398(n log2 n), yielding a complexity of O(n2 log2 n). We set the k and m\nparameters of HMS to \u0398(log n), so as to maintain the same asymptotic complexity bound.\n\nBILS Probability Parameter Viet et al. [45] suggest that their bidirectional local search execute\nrandom moves with probability p = 0.05. We shed light on the impact of p, measuring the sex-\nequality cost of the solution returned by both BILS and iBILS for three different sizes, 2000 instances\nper size, and a range of p values across distribution types. With iBILS, we observe an improvement\nin sex equality on Discrete data, peaking at around p = 0.125. Contrariwise, BILS did not bene\ufb01t\nby randomization, obtaining best results with p = 0. This difference is due to that fact that iBILS\nexplores the lattice by rotations, which are smaller steps than the breakmarriages used in BILS.\n\nPerformance Evaluation We compare the proposed algorithms against: APPROX, the lattice-\nbased approximation algorithm [25]; POLYMIN, which \ufb01nds the solutions minimizing the regret and\negalitarian cost and reports the best result; DACC, the proposal-based method of [11]; BILS, the\nlocal-search-based method [46, 45]; and iBILS, our own enhancement of BILS. We normalize cost\nresults, dividing by the corresponding best cost the DA algorithm can obtain. Algorithms using local\nsearch guide their search using a SEq or Bal cost function; POWERBALANCE selects the best of\ntwo outcomes with regard to cost when enforcing termination. DACC [11] does not specify an order\nof proposals; thus, we employ the proposal strategy of PowerBalance, letting all members of the\nadvantaged side act as proposers in each round. Given our analysis, we set the probability parameter\nin BILS to 0, in iBILS to 0.125, the limit parameter of POWERBALANCE to (cid:100)n log2\n2 n/10(cid:101), and the\nparameters in HMS to k = (cid:100)2 log n(cid:101) and m = (cid:100)log n(cid:101).\n\n(a) Bal-U\n\n(b) Bal-D\n\n(c) Bal-G\n\n(d) Bal-UD\n\n(e) SEq-U\n\n(f) SEq-D\n\n(g) SEq-G\n\n(h) SEq-UD\n\nFigure 1: Quality comparison against heuristics\n\n(a) Time-U\n\n(b) Time-D\n\n(c) Time-G\n\n(d) Time-UD\n\nFigure 2: Time comparison against heuristics\n\nComparison against other heuristics We compare our proposals against state-of-the-art heuristics,\non data sizes up to 4,000, with 50 instances per size and distribution and depict cost results with\nbox and whisker plots, with a black dot indicating the mean. On runtime, we plot mean values.\nFigures 1 and 2 show our results. DACC and POLYMIN perform poorly for both cost metrics.\nPOWERBALANCE is among the fastest, yet falls short cost-wise compared against the local search\nmethods. BILS performs the worst in runtime, while it is also weak in terms of balance and sex\n\n8\n\nPolyMinDACCPowerBalanceBiLSiBiLSHybridHybridMultiSearch20004000n0.140.170.20Bal Ratio over DA20004000n0.820.850.88Bal Ratio over DA20004000n0.700.80Bal Ratio over DA20004000n1.001.041.08Bal Ratio over DA20004000n106105104103102SEq Ratio over DA20004000n105104103102101100SEq Ratio over DA20004000n105104103102101SEq Ratio over DA20004000n1.101.300.90SEq Ratio over DAPolyMinDACCPowerBalanceBiLSiBiLSHybridHybridMultiSearch250500100020004000n102101100101102Time (sec)250500100020004000n102101100101102Time (sec)250500100020004000n102101100101102Time (sec)250500100020004000n102101100101102Time (sec)\fequality cost on Discrete data (Figures 1b, 1f). iBILS and HYBRID behave similarly, with HYBRID\nhaving a slight scalability advantage (Figure 2a). HMS achieves top quality across the board and\noutperforms others signi\ufb01cantly on Discrete data (Figure 1f). Most algorithms detect the same\none-side-biased solution on UniformDiscrete data (Figures 1d, 1h); due to the innate asymmetry\namong the two sides, a solution that favors one side over the other achieves good sex equality and\nbalance. Overall, POWERBALANCE is the most scalable, while HMS provides the highest quality.\n\n(a) U(n=4000)\n\n(b) D(n=4000)\n\n(c) G(n=4000)\n\n(d) UD(n=4000)\n\nFigure 3: Performance comparison against APPROX\n\nComparison against APPROX We now test our best methods, iBILS and HMS, against AP-\nPROX [24], whose \u03b5 parameter provides a sex-equality approximation guarantee with respect to the\nbest of the two DA outputs. We generate 50 data sets of size 4000 for each distribution, and explore\nthe range of \u03b5 to \ufb01nd values that yield competitive results. Figure 3 presents our results. The axes\non the left denote cost ratio (for APPROX, upper-bounded by \u03b5), while those on the right denote\nruntime. On Uniform and Gauss, iBILS and HMS signi\ufb01cantly outperform APPROX, while the cost\nratios they achieve put an overwhelming strain on the latter\u2019s runtime (Figures 3a, 3c). On Discrete\ndata, APPROX surpasses the ratios of iBILS at the cost of a runtime overhead, but does not reach\nthe ratios of HMS within reasonable runtime (Figure 3b). All algorithms \ufb01nd the same solution on\nUniformDiscrete, while APPROX needs an unnecessarily high runtime with ill-chosen \u03b5 (Figure 3d).\n\n(a) Bal\n\n(b) SEq\n(c) Time\nFigure 4: Real Data Experiment\n\n(d) Visualization (n = 100)\n\nApplication on real data. To investigate performance on real data, we extract distributions from\nthe data of an online dating service [10]. The data consists of 17,359,346 anonymous ratings, on the\n1 \u2212 10 scale, of 168,791 pro\ufb01les made by 135,359 LibimSeTi users, along with gender information.\nWe remove users of unknown gender and those who have not rated the opposite gender, and construct\na 2D distribution of the frequency of each pair of ratings (i, j). Drawing from this distribution, we\ngenerate data of n = 100. We resolve ties using 80% randomness and 20% popularity (P), i.e., the\nglobal ranking of agents by all ratings. We run 50 instances per size, and plot quality and runtime\nresults in Figures 4a, 4b, and 4c. We also visualize, Figure 4d, the process for POWERBALANCE\nwith the instance yielding the median sex equality cost; in each iteration, we measure the number of\nsingle agents and the sum of \u03ba index values for the two sides, i.e., \u2118(cid:48)\nw as de\ufb01ned in\nSection 2, which dictate which side proposes in the next round. The left-side axis marks the scale of\nsingles, while the right-side axis marks the scale for \u2118(cid:48)\nw; the vertical black dashed line shows\nthe round in which POWERBALANCE enters its termination procedure, COMPROMISE.\n\nm and women \u2118(cid:48)\n\nm and \u2118(cid:48)\n\n7 Conclusions\nWe revisited the NP-hard problem of \ufb01nding a stable matching optimizing an equity measure. We\nextended the Deferred Acceptance algorithm to a two-sided form, Monotonic Deferred Acceptance,\nproposed a simpler variant of its proposal acceptance criterion, Strongly Deferred Acceptance (SDA),\nand amended that with a few selective steps of ef\ufb01cient local search, Deferred Local Search (DLS).\nThese are the \ufb01rst, to our knowledge, procedures that reach stable matchings of good equity in\nquadratic time. Our experimental results demonstrate that DLS delivers both ef\ufb01ciency and high\nequity. In the future, we intend to study the problem under manipulation incentives, as in [36].\n\n9\n\nApproxAchievedRatioiBiLSAchievedRatioHMSAchievedRatioApproxTimeiBiLSTimeHMSTime0.0080.00850.0090.00950.010.00050.00100.00500.0100SEq Ratio over DA1101001000Time (sec)0.00010.00020.00030.00040.00050.000020.000100.00050SEq Ratio over DA1101001000Time (sec)0.030.040.050.060.0010.0100.100SEq Ratio over DA1101001000Time (sec)0.00250.0030.00350.0040.91.01.1SEq Ratio over DA1101001000Time (sec)PolyMinDACCPowerBalanceBiLSiBiLSHybridHybridMultiSearch50100n1.001.101.200.800.90Bal Ratio over DA50100n101100SEq Ratio over DA2030405060708090100n103102Time (sec)0100200300400500Rounds101000200400600800Singles0102\fReferences\n[1] Atila Abdulkadiro\u02d8glu and Tayfun S\u00f6nmez. School choice: A mechanism design approach. American\n\neconomic review, 93(3):729\u2013747, 2003.\n\n[2] Atila Abdulkadiro\u02d8glu, Parag A. Pathak, and Alvin Roth. The new york city high school match. 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