Shapley-Scarf Housing Markets: Respecting Improvement, Integer Programming, and Kidney Exchange
SShapley-Scarf Housing Markets: Respecting Improvement,Integer Programming, and Kidney Exchange ∗ P´eter Bir´o † Flip Klijn ‡ Xenia Klimentova § Ana Viana ¶ February 2, 2021
Abstract
In a housing market of Shapley and Scarf [41], each agent is endowed with one indivisibleobject and has preferences over all objects. An allocation of the objects is in the (strong)core if there exists no (weakly) blocking coalition. In this paper we show that in the caseof strict preferences the unique strong core allocation (or competitive allocation) “respectsimprovement”: if an agent’s object becomes more attractive for some other agents, thenthe agent’s allotment in the unique strong core allocation weakly improves. We obtaina general result in case of ties in the preferences and provide new integer programmingformulations for computing (strong) core and competitive allocations. Finally, we con-duct computer simulations to compare the game-theoretical solutions with maximum sizeand maximum weight exchanges for markets that resemble the pools of kidney exchangeprogrammes.
Keywords: housing market, respecting improvement, core, competitive allocations, in-teger programming, kidney exchange programmes.
Shapley and Scarf [41] introduced so-called “housing markets” to model trading in commoditiesthat are inherently indivisible. Specifically, in a housing market each agent is endowed with anobject (e.g., a house or a kidney donor) and has ordinal preferences over all objects, includingher own. The aim is to find plausible or desirable allocations where each agent is assigned oneobject. A standard approach in the literature is to discard allocations that can be blocked by acoalition of agents. Specifically, a coalition of agents blocks an allocation if they can trade their ∗ † Centre for Economic and Regional Studies, and Corvinus University of Budapest, 1097 Budapest, T´othK´alm´an utca 17, Hungary. P. Bir´o is supported by the Hungarian Scientific Research Fund – OTKA (no.K129086). ‡ Institute for Economic Analysis (CSIC) and Barcelona GSE, Campus UAB, 08193 Bellaterra (Barcelona),Spain; e-mail: [email protected] . F. Klijn gratefully acknowledges financial support from AGAUR–Generalitat de Catalunya (2017-SGR-1359) and the Spanish Ministry of Science and Innovation through grantECO2017-88130-P AEI/FEDER, UE and the Severo Ochoa Programme for Centres of Excellence in R&D(CEX2019-000915-S). § INESC TEC, Porto, Portugal. ¶ INESC TEC and ISEP – School of Engineering, Polytechnic of Porto, Porto, Portugal. a r X i v : . [ ec on . T H ] J a n ndowments so that each of the agents in the coalition obtains a strictly preferred allotment.Similarly, a coalition of agents weakly blocks an allocation if they can trade their endowmentsso that each of the agents in the coalition obtains a weakly preferred allotment and at leastone of them obtains a strictly preferred allotment. Thus, an allocation is in the (strong) coreif it is not (weakly) blocked. A distinct but also well-studied solution concept is obtainedfrom competitive equilibria, each of which consists of a vector of prices for the objects and a(competitive) allocation such that each agent’s allotment is one of her most preferred objectsamong those that she can afford. Interestingly, the three solution concepts are entwined: thestrong core is contained in the set of competitive allocations, and each competitive allocationpertains to the core.In a separate line of research, Balinski and S¨onmez [6] studied the classical two-sided collegeadmissions model of Gale and Shapley [20] and proved that the student-optimal stable matchingmechanism (SOSM) respects improvement of student’s quality. This means that under SOSM,an improvement of a student’s rank at a college will, ceteris paribus, lead to a weakly preferredmatch for the student. The natural transposition of this property to (one-sided) housing mar-kets requires that an agent obtains a weakly preferred allotment whenever her object becomesmore desirable for other agents. We study the following question: Do the most prominentsolution concepts for Shapley and Scarf’s [41] housing market “respect improvement”? Weobtain several positive answers to this question, which we describe in more detail in the nextsubsection.The respecting improvement property is important in many applications where centralisedclearinghouses use mechanisms to implement barter exchanges. A leading example are kidneyexchange programmes (KEPs), where end-stage renal patients exchange their willing but im-munologically incompatible donors (Roth et al. [37]). In the context of KEPs, the respectingimprovement property means that whenever a patient brings a “better” donor (e.g., youngeror with universal blood type 0 instead of A, B, or AB) or registers an additional donor, theKEP should assign her the same or a better exchange donor. However, in current KEPs, thetypical objective is to maximise the number of transplants and their overall qualities (see, e.g.,[11]) which can lead to violations of the respecting improvement property. As an illustration,consider the maximisation of the number of transplants in Figure 1, where each node repre-sents a patient-donor pair. Directed edges represent compatibility between the donor in onepair and the patient in another, and patients may have different levels of preference over theirset of compatible donors. Initially there are only continuous edges, where a thick (thin) edge12 3 4Figure 1: The maximisation of the number of transplants does not respect improvement.points to the most (least) preferred donor. For example, patient 3 has two compatible donors:donors 1 and 4, and donor 1 is preferred to donor 4. Obviously, the unique way to maximisethe number of (compatible) transplants is obtained by picking the three-cycle (1,2,3). Supposethat patient 4 receives antigen desensitisation treatment so that donor 3 becomes compatiblefor her, or patient 3 succeeds in bringing a second donor to the KEP and this donor turns outto be compatible for patient 4. Then, the discontinuous edge is included and patient-donor pair2 “improves.” But now the unique way to maximise the number of (compatible) transplantsis obtained by picking the 2 two-cycles (1,2) and (3,4), which means that patient 3 receives akidney that is strictly worse than the kidney she would have received initially.Similarly, the allocations induced by the standard objectives of KEPs need not be in the(strong) core. We refer to Example 1 for an illustration of this for the case of the maximisationof the number of transplants. As a consequence, blocking coalitions may exist. This is anundesirable feature because patient groups could make a potentially justified claim that thematching procedure is not in their best interest. A particular instance could occur in theorganisation of international kidney exchanges if a group of patient-donor pairs, all citizens ofthe same country, learn that an internal (i.e., national) matching would all of them give a morepreferred kidney.We conduct simulations to determine to what extent the typical KEP objectives lead to theviolation of the respecting improvement property and the existence of blocking coalitions. Thesimulations also include the three standard game-theoretical solution concepts (as they alsodo not “completely” satisfy the respecting improvement property), for which we first developnovel integer programming formulations to speed up the computations.Next, we describe our contributions in more detail and review the related literature. Section 3 contains our theoretical results on the respecting improvement property. First, weshow that for strict preferences the unique strong core allocation (which coincides with theunique competitive allocation) respects improvement (Theorem 1). In the case of preferenceswith ties, since the strong core can be empty, we focus on the set of competitive allocations.Since typically multiple competitive allocations exist, we have to make setwise comparisons.We obtain a natural extension of our first result: focusing on the agent’s allotments obtainedat competitive allocations, we establish that her most preferred allotment in the new marketis weakly preferred to her most preferred allotment in the initial market; and similarly, herleast preferred allotment in the new market is weakly preferred to her least preferred allotmentin the initial market (Proposition 1). Finally, we also prove that when preferences have tiesthe strong core respects improvement conditional on the strong core being non-empty. Moreprecisely, under the assumption that strong core allocations exist in both the initial and newmarkets, we show that the agent under consideration weakly prefers each allotment in the newstrong core to each allotment in the initial strong core (Theorem 2).Then we tackle an important assumption in the housing market of Shapley and Scarf, namelythat allocations can contain exchange cycles of any length, i.e., cycles are unbounded. As aconsequence, some allocations obtained from the theory of housing markets might be difficultto implement in the case of KEPs. The reason is that all transplants in a cycle are usuallycarried out simultaneously to avoid reneging. So, if the number of surgical teams and operationrooms is small, some of the transplants have to be conducted necessarily in a non-simultaneousway. In many countries, this “risky” solution is not allowed [8]. The definition of core, set ofcompetitive allocations, and strong core can be adjusted to the requirement that the lengthof exchange cycles does not exceed an exogenously given maximum. However, in this case thecore (and hence also the set of competitive allocations and the strong core) can be empty. Conditional on the existence of a core, competitive, or strong core allocation, we show that The corresponding decision problem is NP-hard [10, 23] even for tripartite graphs (also known as the cyclic3D stable matching problem [34]). price of fairness : the decrease in the number of transplants of maximumweight, core, competitive and strong core allocations, when compared with the maximum sizesolution. The analysis proceeds with an indirect assessment of how far other solutions wouldbe from the strong core. Such indicator provides some insight in/into the number of patientsin a pool that could get a strictly better match. The section concludes with a computationalanalysis of the frequency of violations of the respecting improvement property. It was observedthat even though there exist cases of violation of the property for (Wako-, strong) core, theirquantity is dramatically lower that those for the maximum size and maximum size solutions.
Housing markets
The non-emptiness of the core was proved in [41] by showing the bal-ancedness of the corresponding NTU-game, and also in a constructive way, by showing thatDavid Gale’s famous Top Trading Cycles algorithm (TTC) always yields competitive alloca-tions. Roth and Postlewaite [36] later showed that for strict preferences the TTC results in theunique strong core allocation, which coincides with the unique competitive allocation in thiscase. However, if preferences are not strict (i.e., ties are present), the strong core can be emptyor contain more than one allocation, but the TTC still produces all competitive allocations.Wako [43] showed that the strong core is always a subset of the set of competitive allocations.Quint and Wako [35] provided an efficient algorithm for finding a strong core allocation when-ever there exists one. Their work was further generalised and simplified by Cechl´arov´a andFleiner [14] who used graph models. Wako [45] showed that the set of competitive allocationscoincides with a core defined by antisymmetric weak domination. This equivalence is key for ourextension of the definition of competitive allocations to the case of bounded exchange cycles.
Respecting improvement
For Gale and Shapley’s [20] college admissions model, Balinskiand S¨onmez [6] proved that the student-optimal stable matching mechanism (SOSM) respectsimprovement of student’s quality. Kominers [28] generalised this result to more general settings.Balinski and So¨nmez [6] also showed that SOSM is the unique stable mechanism that respects4mprovement of student quality. Abdulkadiroglu and So¨nmez [1] proposed and discussed theuse of TTC in a model of school choice, which is closely related to the college admissions model.Klijn [24] proved that TTC respects improvement of student quality.Focusing on the other side of the market, Hatfield et al. [22] studied the existence ofmechanisms that respect improvement of a college’s quality. The fact that colleges can matchwith multiple students leads to a strong impossibility result: Hatfield et al. [22] proved thatthere is no stable nor Pareto-efficient mechanism that respects improvement of a college’squality. In particular, the (Pareto-efficient) TTC mechanism does not respect improvement ofa college’s quality.In the context of KEPs with pairwise exchanges, the incentives for bringing an additionaldonor to the exchange pool was first studied by Roth et al. [38]. In the model of housingmarkets their donor-monotonicity property boils down to the respecting improvement property.Roth et al. [38] showed that so-called priority mechanisms are donor-monotonic if each agent’spreferences are dichotomous, i.e., she is indifferent between all acceptable donors. However, ifagents have non-dichotomous preferences, then any mechanism that maximises the number ofpairwise exchanges (so, in particular any priority mechanism) does not respect improvement.This can be easily seen by means of Example 5 in Section 3.
IP formulations for matching
Quint and Wako [35] already gave IP formulations for find-ing core and strong core allocations, but the number of constraints in their paper is highlyexponential, as their formulations contain a no-blocking condition for each set of agents andany possible exchanges among these agents. Other studies provided IP formulations for othermatching problems. In particular, for Gale and Shapley’s [20] college admissions model, Ba¨ıouand Balinski [5] already described the stable admissions polytope, which can be used as abasic IP formulation. Further recent papers in this line of research focused on college admis-sions with special features [3], stable project allocation under distributional constraints [4], thehospital–resident problem with couples [9], and ties [30, 17].
Kidney exchange programmes
Starting from the seminal work by Saidman et al. [39],initial research on KEPs focused on integer programming (IP) models for selecting pairs fortransplantation in such a way that maximum (social) welfare, generally measured by the numberof patients transplanted, is achieved. Authors in [15, 18, 31] proposed new, compact formula-tions that, besides extending the models in [39] to accommodate for non-directed donors andpatients with multiple donors, also aimed to efficiently solve problems of larger size. Laterstudies [26, 19, 12, 32] modelled the possibility of pair dropout or cancellation of transplants(if e.g. new incompatibilities are revealed). While [26] and [19] aimed to find a solution thatmaximises expected welfare, [12] proposed robust optimisation models that search for a solutionthat, in the event of a cancellation, can be replaced by an alternative (recourse) solution thatin terms of selected patients is as “close” as possible to the initial solution. McElfresh et al. [32]also addressed the robustness of solutions, but they did not consider the possibility of recourse.In a different line of research, [13, 27, 33] studied KEPs where agents (e.g. hospitals, re-gional and national programmes) can collaborate. Allowing agents to control their internalexchanges, Carvalho et al. [13] studied strategic interaction using non-cooperative game theory.Specifically, for the two-agent case, they designed a game such that some Nash equilibriummaximises the overall social welfare. Considering multiple matching periods, Klimentova et al.[27] assumed agents to be non-strategic. Taking into account that at each period there can bemultiple optimal solutions, each of which can benefit different agents, the authors proposed an5nteger programming model to achieve an overall fair allocation. Finally, Mincu et al. [33] pro-posed integer programming formulations for the case where optimisation goals and constraintscan be distinct for different agents.A recent line of research acknowledges the importance of considering patients’ preferences(associated with e.g. graft quality) over matches. Bir´o and Cechl´arov´a [7] considered a modelfor unbounded length kidney exchanges, where patients most care about the graft they receive,but as a secondary factor (whenever there is a tie) they prefer to be involved in an exchangecycle that is as short as possible. The authors showed that although core allocations can still befound by the TTC algorithm, finding a core allocation with maximum number of transplants isa computationally hard problem (inapproximable, unless P = N P ). Recently, Klimentova etal. [25] provided integer programming formulations when each patient has preferences over theorgans that she can receive. The authors focused on allocations that among all (strong) coreallocations have maximum cardinality. Moving away from the (strong) core, they also analysedthe trade-off between maximum cardinality and the number of blocking cycles. Finally, thereader is referred to [11] for a recent review of KEPs.
We consider housing markets as introduced by Shapley and Scarf [41]. Let N = { , . . . , n } , n ≥
2, be the set of agents . Each agent i ∈ N is endowed with one object denoted by e i = i .Thus, N also denotes the set of objects . Each agent i ∈ N has complete and transitive (weak)preferences R i over objects. We denote the strict part of R i by P i , i.e., for all j, k ∈ N , jP i k if and only if jR i k and not kR i j . Similarly, we denote the indifference part of R i by I i , i.e.,for all j, k ∈ N , jI i k if and only if jR i k and kR i j . Let R ≡ ( R i ) i ∈ N . A (housing) market is apair ( N, R ). Object j ∈ N is acceptable to agent i ∈ N if jR i i . Agent i ’s preferences are called strict if they do not exhibit ties between acceptable objects, i.e., for all acceptable j, k ∈ N with j (cid:54) = k we have jP i k or kP i j . A housing market has strict preferences if each agent hasstrict preferences. A housing market where agents do not necessarily have strict preferences isreferred to as a housing market with weak preferences.Given a housing market M = ( N, R ) and a set S ⊆ N , the submarket M S is the housingmarket where S is the set of agents/objects and where the preferences ( R i ) i ∈ S are restricted tothe objects in S .The acceptability graph of a housing market M = ( N, R ) is the directed graph G M = ( N, E ),or G for short, where the set of nodes is N and where ( i, j ) is a directed edge in E if j is anacceptable object for i , i.e., jR i i . In particular, all self-loops ( i, i ) are in the graph (but forconvenience they are omitted in all figures). Let ˜ N ⊆ N and ˜ E ⊆ E ∩ ( ˜ N × ˜ N ). For each i ∈ ˜ N , the set of agent i ’s most preferred edges in graph ˜ G ≡ ( ˜ N , ˜ E ) or simply ˜ E is the set˜ E T,i ≡ { ( i, j ) : ( i, j ) ∈ ˜ E and for each ( i, k ) ∈ ˜ E, jR i k } . The most preferred edges in graph ˜ G is the set ∪ i ∈ ˜ N ˜ E T,i .Let M = ( N, R ) be a housing market. An allocation is a redistribution of the objects suchthat each agent receives exactly one object. Formally, an allocation is a vector x = ( x i ) i ∈ N ∈ N N such that:(i) for each i ∈ N , x i ∈ N denotes agent i ’s allotment , i.e., the object that she receives, and(ii) no object is assigned to more than one agent, i.e., ∪ i ∈ N { x i } = N. We will focus on individually rational allocations, i.e., allocations where each agent receives anacceptable object. Then, an allocation x can equivalently be described by its corresponding6 ycle cover g x of the acceptability graph G . Formally, g x = ( N, E x ) is the subgraph of G where( i, j ) ∈ E x if and only if x i = j . Thus, the graph g x consists of disconnected trading cycles or exchange cycles that cover G . We will often write an (individually rational) allocation incycle-notation, i.e., as a set of exchange cycles (where we sometimes omit self-cycles). We referto Example 1 for an illustration.An allocation x Pareto-dominates an allocation z if for each i ∈ N , x i R i z i , and for some j ∈ N , x j P j z j . An allocation is Pareto-efficient if it is not Pareto-dominated by any allocation.Two allocations x, z are welfare-equivalent if for each i ∈ N , x i I i z i .Next, we recall the definition of solution concepts that have been studied in the literature.A non-empty coalition S ⊆ N blocks an allocation x if there is an allocation z such that (1) { z i : i ∈ S } = S and (2) for each i ∈ S , z i P i x i . An allocation x is in the core of the market ifthere is no coalition that blocks x .A non-empty coalition S ⊆ N weakly blocks an allocation x if there is an allocation z suchthat (1) { z i : i ∈ S } = S , (2) for each i ∈ S , z i R i x i , and (3) for some j ∈ S , z j P j x j . Anallocation x is in the strong core of the market if there is no coalition that weakly blocks x .A price-vector is a vector p = ( p i ) i ∈ N ∈ R N where p i denotes the price of object i . Acompetitive equilibrium is a pair ( x, p ) where x is an allocation and p is a price-vector suchthat:(i) for each agent i ∈ N , object x i is affordable, i.e., p x i ≤ p i and(ii) for each agent i ∈ N , each object she prefers to x i is not affordable, i.e., jP i x i implies p j > p i .An allocation is a competitive allocation if it is part of some competitive equilibrium. Sincethere are n objects, we can assume, without loss of generality, that prices are integers in theset { , , . . . , n } . Remark 1.
If ( x, p ) is such that • for each i ∈ N , p x i ≤ p i , or • for each i ∈ N , p i ≤ p x i ,then for each i ∈ N , p x i = p i . This follows immediately by looking at each exchange cycleseparately (see, e.g., the proof of Lemma 1 in [14]). Hence, at each competitive equilibrium( x, p ), for each i ∈ N , p x i = p i .Wako [45] proved that the set of competitive allocations can be defined equivalently as adifferent type of core. Formally, a non-empty coalition S ⊆ N antisymmetrically weakly blocks an allocation x if there is an allocation z such that (1) { z i : i ∈ S } = S , (2) for each i ∈ S , z i R i x i , (3) for some j ∈ S , z j P j x j , and (4) for each i ∈ S , if z i I i x i then z i = x i . Requirements(1–3) say that coalition S weakly blocks x . The additional requirement (4) is that if an agent in S is indifferent between her allotments at x and z then she must get the very same object, i.e., z i = x i . An allocation x is in the core defined by antisymmetric weak domination if there is nocoalition that antisymmetrically weakly blocks x . Wako [45] proved that the set of competitiveallocations coincides with the core defined by antisymmetric weak domination. Henceforth, wewill often refer to the set of competitive allocations as the Wako-core . In the literature the core is sometimes called the weak core or “regular” core. In the literature the strong core is sometimes called the strict core. emma 1. The strong core, the set of competitive allocations (i.e., the Wako-core), and the coreconsist of individually rational allocations. Moreover, the cores are equivalently characterisedby the absence of blocking cycles in the acceptability graph G = ( N, E ) . In other words, in thedefinition of each of the three cores, it is sufficient to require no-blocking by coalitions S , say S = { i , . . . , i k } , such that for each l = 1 , . . . , k (mod k ), z i l = i l +1 and ( i l , i l +1 ) ∈ E .Proof. Individual rationality is immediate. To prove the statement for the strong core, let x be an individually rational allocation. Suppose there is a non-empty coalition T that weaklyblocks x through some allocation w . Let j ∈ T be such that w j P j x j . Let S ⊆ T be the agentsthat constitute the exchange cycle, say ( i , . . . , i k ), in w that involves agent j , i.e., without lossof generality, j = i . Since w is individually rational, S = { i , . . . , i k } weakly blocks x throughthe allocation z defined by z i ≡ (cid:26) w i if i ∈ S ; x i if i (cid:54)∈ S and for each l = 1 , . . . , k (mod k ), z i l = i l +1 and ( i l , i l +1 ) ∈ E . This proves the statement forthe strong core. The statements for the core and the Wako-core follow similarly.An individually rational allocation x is a maximum size allocation if for each individuallyrational allocation z , |{ i ∈ N : x i (cid:54) = i }| ≥ |{ i ∈ N : z i (cid:54) = i }| . Below we provide an example toillustrate the three cores and maximum size allocation. Example 1.
Let N = { , . . . , } and let preferences be given by Table 1, where each agent’s ownobject and all her unacceptable objects are not displayed. For instance, agent 1 is indifferentbetween objects 2 and 3, and strictly prefers both objects to object 5.1 2 3 4 5 62,3 1 2 3 2 15 3 4 2 6Table 1: Preferences 1234 56Figure 2: Acceptability graph x a = { (1 , , } x b = { (1 , , (3 , } x c = { (1 , , , (3 , } x d = { (1 , , , } x e = { (1 , , , (2 , , } Table 2: AllocationsFigure 2 displays the induced acceptability graph. Here, a thick edge denotes the most pre-ferred object(s) and a thin edge denotes the second most preferred object (if any).Consider the allocations defined in Table 2. For instance, x d (in cycle-notation, but withoutself-cycles) is the allocation x d = ( x d , x d , x d , x d , x d , x d ) = (3 , , , , , x a is the unique strong core allocation, x a and x b are the competitive allocations, while x a , x b , x c , and x d form the core. Hence, the strong core is a singleton and a proper subset of the setof competitive allocations, while the latter set is also a proper subset of the core. Finally, x e isthe unique maximum size allocation and does not pertain to the core. (cid:5) Shapley and Scarf [41] (see also page 135, Roth and Postlewaite [36]) showed that the set ofcompetitive allocations is non-empty and coincides with the set of allocations that are obtainedthrough David Gale’s Top Trading Cycles algorithm, which is discussed in the next subsection. Throughout the paper, self-loops are omitted from the acceptability graphs in the examples. If preferences are not strict, then the Top Trading Cycles algorithm is applied to the preference profilesthat can be obtained by breaking ties in all possible ways. Furthermore, it is easy to see that the set ofcompetitive allocations is always a subset of the core (Shapley and Scarf [41]).If preferences are strict, the unique competitive allocation is Pareto-efficient (because it isin the strong core) and Pareto-dominates any other allocation (Lemma 1, Roth and Postlewaite[36]); in particular, any other core allocation is Pareto-inefficient. If preferences are not strict,it is possible that each competitive allocation is Pareto-dominated by some allocation that isnot competitive. Finally, competitive allocations need not be welfare-equivalent: in fact, different agentscan strictly prefer distinct competitive allocations (see, e.g., Footnote 7). However, Wako [44]showed that all strong core allocations are welfare-equivalent. The latter result also immediatelyfollows from Quint and Wako’s [35] algorithm, which is discussed in the next subsection.
Definitions for bounded length exchanges
Motivated by kidney exchange programmes, here we consider housing markets where the lengthof allowed exchange cycles in allocations is limited. Assuming that blocking coalitions aresubject to the same limitation, the core and strong core can be adjusted straightforwardly (seealso [10]). In view of Wako’s [45] result we similarly adjust the set of competitive allocationsby using the (equivalent) Wako-core.For a housing market M = ( N, R ), let k be an integer that indicates the maximal allowedlength of exchange cycles. An allocation is a k -allocation if each exchange cycle has lengthat most k . Formally, an allocation x is a k -allocation if there exists a partition of N = S ∪ S ∪ · · · ∪ S q such that for each p ∈ { , . . . , q } , | S p | ≤ k and { x i : i ∈ S p } = S p . Thedefinition of the three cores can be adjusted accordingly as well. Specifically, the k -core consistsof the k -allocations for which there is no blocking coalition of size at most k ; the strong k -core consists of the k -allocations for which there is no weakly blocking coalition of size at most k ;the Wako- k -core consists of the k -allocations that are not antisymmetrically weakly dominatedthrough a coalition of size at most k . Due to the “nestedness” of the three blocking notions,it follows that the strong k -core is a subset of the Wako k -core, and that the Wako k -core is asubset of the k -core. It is also easy to verify that, similarly to the unbounded case, for strictpreferences the strong k -core coincides with the Wako- k -core.To keep notation as simple as possible, whenever the context is clear, we will omit “ k ” from k -allocation, k -core, etc. and instead refer to k -housing markets to invoke the above restrictionon exchange cycles, blocking coalitions, allocations, and cores.The absence of blocking coalitions is also called stability in the literature, that is widely usedespecially for bounded length exchanges. In the case of pairwise exchanges (i.e., for k = 2), the Wako [44] showed that the strong core coincides with the set of competitive allocations if and only ifany two competitive allocations are welfare-equivalent. Hence, whenever the set of competitive allocations is asingleton it coincides with the strong core. Example 1 in Sotomayor [42], which is attributed to Jun Wako, is illustrative: N = { , , } with 2 P P I P
2, 2 P P
3. The set of competitive allocations consists of x = { (1 , , (3) } and x (cid:48) = { (1) , (2 , } , whichare Pareto-dominated by core allocations { (1 , , } and { (1 , , } , respectively. Moreover, x P x (cid:48) and x (cid:48) P x .The strong core is empty. stable roommates problem , and stable marriage problem if the graph is bipartite, as introduced in [20]. For strict preferences the core, Wako-core, andstrong core are all equivalent and they correspond to the set of stable matchings . For weakpreferences the core and Wako-core are the same, and correspond to weakly stable matchings ,whilst the strong core corresponds to strongly stable matchings . See more about these conceptsin [16]. In this section we describe the TTC algorithm and its extension by Quint and Wako [35] forfinding strong core allocations. Our concise and standardised descriptions provide an easysummary of the current state of the art. Moreover, the graphs defined in the algorithms arecrucial tools in Section 3 where we prove that the strong core “respects improvement.” Weconsider housing markets with strict preferences and weak preferences separately. In the firstcase the strong core is always a singleton (which consists of the unique competitive allocation),while in the second case it can be empty.
Strict preferences
Let M = ( N, R ) be a housing market with strict preferences. We will construct a subgraph G CP of G by using the Top Trading Cycles (TTC) algorithm of David Gale [41]. The node setof G CP is N and its directed edges E CP = E C ∪ E P are partitioned into two sets E C and E P ,where E C will denote the edges in the TTC cycles and E P will denote a particular subset ofedges pointing to more preferred objects. TTC algorithm, construction of G CP Set E C ≡ ∅ , E P ≡ ∅ , and M ≡ M . Let G = ( N , E ) ≡ ( N, E ) denote the acceptability graphof M . We iteratively construct “shrinking” submarkets M t ( t = 2 , , . . . ) whose acceptabilitygraph will be denoted by G t = ( N t , E t ). Set t ≡ E Tt be the set of most preferred edges in G t .2. Let c t be a (top trading) cycle in ( N t , E Tt ). Let C t and E t denote the node set and edgeset of c t , respectively.3. Add the edges of c t to E C , i.e., E C ≡ E C ∪ E t .4. Let E Tt ( (cid:126)C t ) denote the subset of edges of E Tt pointing to C t from outside C t . Formally, E Tt ( (cid:126)C t ) ≡ { ( i, j ) ∈ E Tt : i ∈ N t \ C t and j ∈ C t } . Add E Tt ( (cid:126)C t ) to E P , i.e., E P ≡ E P ∪ E Tt ( (cid:126)C t ).5. If N t = C t , stop. Otherwise, let N t +1 ≡ N t \ C t , denote the submarket M N t +1 by M t +1 ,and go to step 1.When the algorithm terminates the set of cycles in E C is the unique competitive allocation andhence the unique strong core allocation.We classify the relation between any two agents through graph G CP as follows. Let i, j ∈ N with i (cid:54) = j . Then, exactly one of the following situations holds: • i and j are independent : there is no directed path from i to j or from j to i ;10 i and j are cycle-members : there is a path from i to j that entirely consists of edges in E C , i.e., i and j are in the same top trading cycle; • i is a predecessor of j (and j is a successor of i ): there is a path from i to j in G CP usingat least one edge from E P ; or • j is a predecessor of i (and i is a successor of j ): there is a path from j to i in G CP usingat least one edge from E P .In case i is a predecessor of j , we define the best path from i to j to be the path from i to j in G CP where at each node k (cid:54) = j on the path, the path follows agent k ’s (unique) mostpreferred edge in { ( k, l ) ∈ E CP : there is a path from l to j using edges in E CP } . Let p b ( i, j ) denote the unique best path from i to j in G CP . For each node k (cid:54) = j on p b ( i, j ),if there are multiple paths from k to j , then p b ( i, j ) follows the edge that points to the objectthat is part of the earliest top trading cycle. Weak preferences
Let M = ( N, R ) be a housing with weak preferences. We will now describe the efficientalgorithm of Quint and Wako [35] for finding a strong core allocation whenever there exists one.We use the simplified interpretation of Cechl´arov´a and Fleiner [14] and construct a subgraph G SP of G with node set N and edge set E SP ≡ E S ∪ E P , which will be useful for our lateranalysis.We first recall two definitions. A strongly connected component of a directed graph is asubgraph where there is a directed path from each node to every other node. An absorbing set is a strongly connected component with no outgoing edge. Note that each directed graph hasat least one absorbing set. Quint-Wako algorithm, construction of G SP Set E S ≡ ∅ , E P ≡ ∅ , and M = M . Let G = ( N , E ) ≡ ( N, E ) denote the acceptability graphof M . We iteratively construct “shrinking” submarkets M t ( t = 2 , , . . . ) whose acceptabilitygraph will be denoted by G t = ( N t , E t ). Set t ≡ E Tt be the set of most preferred edges in G t .2. Let S t be an absorbing set in ( N t , E Tt ). Let N t ( S t ) and E Tt ( S t ) denote the node set andedge set of S t .3. Add the edges of S t to E S , i.e., E S ≡ E S ∪ E Tt ( S t ).4. Let E Tt ( (cid:126)S t ) denote the subset of edges of E Tt pointing to N t ( S t ) from outside N t ( S t ).Formally, E Tt ( (cid:126)S t ) ≡ { ( i, j ) ∈ E Tt : i ∈ N t \ N t ( S t ) and j ∈ N t ( S t ) } . Add E Tt ( (cid:126)S t ) to E P ,i.e., E P ≡ E P ∪ E Tt ( (cid:126)S t ). Obviously, in this case there is also a path from j to i that consists of edges in E C . Note that in this case j was removed from the market before i . Hence, there is no path from i to j usingonly edges in E C and j is not a predecessor of i .
11. If N t = N t ( S t ), stop. Otherwise, let N t +1 ≡ N t \ N t ( S t ), denote the submarket M N t +1 by M t +1 , and go to step 1.Quint and Wako [35] proved that there is a strong core allocation for M if and only if foreach absorbing set S t defined in the above algorithm there exists a cycle cover, i.e., a set ofcycles covering all the nodes of S t . Finding a cycle cover, if one exists, can be done with theclassical Hungarian method [29] for finding a perfect matching for the corresponding bipartitegraph where the objects are on one side, the agents are on the other side, and there is anundirected arc between an object-agent pair if the object is among the agent’s most preferredobjects (which might include her own object). We refer to [2], [35], and [14] for further detailson this reduction. Remark 2.
If for each absorbing set S t defined in the above algorithm there exists a cyclecover, then the set of cycle covers (one cycle cover for each absorbing set) constitutes a strongcore allocation. Conversely, as shown in the proof of Theorem 5.5 in Quint and Wako [35],each strong core allocation can be written as a set of cycle covers (one for each absorbing set S t ). Hence, if the strong core is non-empty, all its allocations can be obtained by selecting allpossible cycle covers in the algorithm. (cid:5) Remark 3.
In the Quint-Wako algorithm, each agent obtains the same welfare at any two cyclecovers in which she is involved (because the agent is indifferent between any two of her outgoingedges in an absorbing set). Together with Remark 2, this immediately proves Theorem 2(2) inWako [44], which states that all strong core allocations are welfare-equivalent. (cid:5)
We classify the relation between any two agents through graph G SP as follows. Let i, j ∈ N with i (cid:54) = j . Then, exactly one of the following situations holds: • i and j are independent : there is no directed path from i to j or from j to i ; • i and j are absorbing set members : there is a path from i to j that entirely consists ofedges in E S , i.e., i and j are in the same absorbing set; • i is a predecessor of j (and j is a successor of i ): there is a path from i to j in G SP usingat least one edge from E P ; or • j is a predecessor of i (and i is a successor of j ): there is a path from j to i in G SP usingat least one edge from E P .In case i is a predecessor of j , a path from i to j in G SP is said to be a best path from i to j if at each node k (cid:54) = j on the path, the path follows one of agent k ’s most preferred edges in { ( k, l ) ∈ E SP : there is a path from l to j using edges in E SP } . Let P b ( i, j ) denote the set of best paths from i to j in G SP . Obviously, in this case there is also a path from j to i that consists of edges in E S . Note that in this case j was removed from the market before i . Hence, there is no path from i to j usingonly edges in E S and j is not a predecessor of i . Respecting Improvement
Let R, ˜ R be two preference profiles over objects N . Let i ∈ N . We say that ˜ R is an improvementfor i with respect to R if(1). ˜ R i = R i ;(2). for all j (cid:54) = i and all k with k R j j , i I j k = ⇒ i ˜ R j k and i P j k = ⇒ i ˜ P j k ; and(3). for all j (cid:54) = i and all k, l (cid:54) = i with k, l R j j , k R j l ⇐⇒ k ˜ R j l .In other words, (1) only agents different from i have possibly different preferences at ˜ R and R ,(2) for each agent j (cid:54) = i , object i can become more preferred than some acceptable objects, and(3) for each agent j (cid:54) = i and for each pair of acceptable objects different from i , preferencesremain unchanged.As a simple example with N = { , , , , } , let R be any preference profile such that4 P I I P
5. Let ˜ R be the preference profile where agents 1 , ,
3, and 4 have the samepreferences as at R and let ˜ R be defined by 1 I P I P
5. Then, ˜ R is an improvement foragent 1 with respect to R . For each profile of strict preferences R , let τ ( R ) denote the unique competitive allocation(or strong core allocation). We show that τ respects improvement on the domain of strictpreferences: Theorem 1.
For each i ∈ N and each pair of profiles of strict preferences R, ˜ R such that ˜ R isan improvement for i with respect to R , τ i ( ˜ R ) R i τ i ( R ) .Proof. Let x = τ ( R ) and ˜ x = τ ( ˜ R ). We can assume that there is a unique agent j (cid:54) = i with˜ R j (cid:54) = R j and prove that ˜ x i R i x i . (If there is more than one such agent, we repeatedly apply theone-agent result to obtain the result.) We can also assume that i ˜ R j x j . (Otherwise x j ˜ P j i andhence, from the TTC algorithm, ˜ x = x .)We distinguish among three cases, depending on the relation between agents i and j in thegraph G CP for the market ( N, R ), i.e., the graph that is obtained in the TTC algorithm for x . Case I: i and j are independent or j is a predecessor of i . Let F ( i ) be the set of followers of i in the graph G CP , where we use the convention i ∈ F ( i ). Then, j (cid:54)∈ F ( i ). In the TTCalgorithm for R , the agents in F ( i ) form, among themselves, trading cycles. Since for each agent k ∈ F ( i ), ˜ R k = R k , it follows that the trading cycles formed by F ( i ) in the TTC algorithm for R are also formed in the TTC algorithm for ˜ R . Hence, ˜ x i = x i . Case II: i and j are cycle-members. Let C be the cycle in the graph G CP that contains i and j . Let p ( i, j ) be the unique path from i to j in the graph G CP . Obviously, p ( i, j ) is part of C .Let N ( i, j ) be the nodes on p ( i, j ). (So, i, j ∈ N ( i, j ).) Let F ∗ ( N ( i, j )) be the followers outsideof N ( i, j ) that can be reached by some path in G CP that (1) starts from a node in N ( i, j ) and(2) does not contain edges in C . Then, the nodes in F ∗ ( N ( i, j )) constitute trading cycles at x . Moreover, the nodes in F ∗ ( N ( i, j )) are neither predecessors of j nor cycle-members with j .Hence, by the same arguments as in Case I, at ˜ x the nodes in F ∗ ( N ( i, j )) constitute the same We avoid the use of the usually equivalent nomenclature “successor” as the latter term has already aparticular (and different) meaning. i F ( i ) p ( i, j ) i. . .j F ∗ ( N ( i, j )) i l . . .j k p b ( i, j ) F ( k ) Case I Case II Case III
Figure 3: Graph G CP (simplified) in the proof of Theorem 1. Each ellipse represents a toptrading cycle.trading cycles as at x . Since i ˜ R j x j , the trading cycle of agent i at ˜ x is the cycle that consistsof the path p ( i, j ) and the edge ( j, i ). Since p ( i, j ) is part of C , it follows that ˜ x i = x i . Case III: i is a predecessor of j . Since for each k (cid:54) = j , ˜ R k = R k , i ˜ R j x j , and ˜ R j is obtained from R j by shifting i up, it follows that at some step in the TTC algorithm for ˜ R , agent j will startpointing to agent i and will keep doing so if and as long as agent i is present. Next, considerthe predecessor of j on the path p b ( i, j ) (in G CP for the market ( N, R )), say agent l . Let k ∈ N with kP l j . By definition of p b ( i, j ), k and j are not cycle-members nor is k a predecessor of j .From the same arguments as in Case I, the nodes in F ( k ) (the followers of k , where k ∈ F ( k ))form among themselves the same trading cycles at x and ˜ x . Hence, at some step in the TTCalgorithm for ˜ R , agent l will start pointing to agent j and will keep doing so if and as longas agent j is present. We can repeat the same arguments until we conclude that each node inthe cycle formed by p b ( i, j ) and the edge ( j, i ) will, at some step, start pointing to its followerand will keep doing so if and as long as the follower is present. Thus, the cycle is a tradingcycle at ˜ x . Let i (cid:48) be the follower of i in this cycle. Note that in the graph G CP , ( i, i (cid:48) ) ∈ E C or ( i, i (cid:48) ) ∈ E P . If ( i, i (cid:48) ) ∈ E C , then i (cid:48) = x i , in which case ˜ x i = i (cid:48) = x i . If ( i, i (cid:48) ) ∈ E P , then bydefinition of E P , ˜ x i = i (cid:48) P i x i . For each profile of preferences R , let T ( R ) denote the set of competitive allocations. Proposition 1.
For each i ∈ N and each pair of profiles of preferences R, ˜ R such that ˜ R is animprovement for i with respect to R , • there is ˜ x ∈ T ( ˜ R ) such that for each x ∈ T ( R ) , ˜ x i R i x i ; and • there is x ∈ T ( R ) such that for each ˜ x ∈ T ( ˜ R ) , ˜ x i R i x i . In other words, if agent i compares her best allotment at the allocations in T ( R ) with her bestallotment at the allocations in T ( ˜ R ), then she prefers the latter. Similarly, if agent i comparesher worst allotment at the allocations in T ( R ) with her worst allotment at the allocations in14 ( ˜ R ), then she prefers again the latter. Note that in general there is no competitive alloca-tion where each agent receives her most preferred allotment (among those that are obtainedat competitive allocations), i.e., agents do not unanimously agree on the “best” competitiveallocation (see, e.g., agents 3 and 4 and competitive allocations x a and x b in Example 1).Nonetheless, Proposition 1 shows that any individual agent that is systematically optimisticor pessimistic about the specific competitive allocation that is chosen subscribes to the thesisthat “the competitive mechanism” would respect any of her potential improvements. Proof.
Let R , R , . . . , R m be the profiles of strict preferences that are obtained from R bybreaking ties between acceptable objects in each possible way. Similarly, let ˜ R , ˜ R , . . . , ˜ R p be the profiles of strict preferences that are obtained from ˜ R by breaking ties between ac-ceptable objects in each possible way, where possibly p (cid:54) = m . Then, from Shapley andScarf [41] (see also page 306 in Wako [44]), T ( R ) = { τ ( R ) , τ ( R ) , . . . , τ ( R m ) } and T ( ˜ R ) = { τ ( ˜ R ) , τ ( ˜ R ) , . . . , τ ( ˜ R p ) } . It is not difficult to see that for each R k , there is some ˜ R l such that˜ R l is an improvement for i with respect to R k . Similarly, for each ˜ R l , there is some R k suchthat ˜ R l is an improvement for i with respect to R k .We can assume, without loss of generality, that for each k = 1 , . . . , m , τ i ( R ) R i τ i ( R k ). Let˜ R l be an improvement for i with respect to R . Then, from Theorem 1, τ i ( ˜ R l ) R i τ i ( R ). Since R i = R i , τ i ( ˜ R l ) R i τ i ( R ). This proves the first statement.We can also assume, without loss of generality, that for each l = 1 , . . . , p , τ i ( ˜ R l ) R i τ i ( ˜ R p ).Let R k be such that ˜ R p is an improvement for i with respect to R k . Then, from Theorem 1, τ i ( ˜ R p ) R ki τ i ( R k ). Since R ki = R i , τ i ( ˜ R p ) R i τ i ( R k ). This proves the second statement.The following example illustrates Proposition 1. Example 2.
Let N = { , , , , } and let preferences R and ˜ R be given by Tables 3 and 4,where only acceptable objects are displayed. Note that ˜ R is an improvement for agent 3 withrespect to R .1 2 3 4 54 1 4 1 22 3,5 2 4 51 2 3Table 3: R
22 1 2 4 51 5 32Table 4: ˜
R x a = { (1 , , (2 , } x b = { (1 , , (2 , } x c = { (1 , , (3 , } Table 5: CompetitiveallocationsBy applying the TTC algorithm to the strict preferences obtained by breaking all ties in allpossible ways, we compute the competitive allocations (Table 5). In the case of R , the twocompetitive allocations are x a and x b , and in the case of ˜ R , the two competitive allocations are x b and x c . Hence, both of agent 3’s best and worst competitive allotment strictly improve, andat both R and ˜ R her best allotment is different from her worst allotment. (cid:5) In Example 2, for the “improving agent” (agent 3), each competitive allotment in the newmarket is weakly preferred to each competitive allotment in the initial market. However, it iseasy to construct housing markets without this feature.Next, we turn to the strong core, which in the case of weak preferences is a (possiblyproper) subset of the set of competitive allocations. Formally, for a profile of preferences R ,15et SC ( R ) denote the (possibly empty) strong core of R . Since strong core allocations arewelfare-equivalent (Remark 3), we can show that the correspondence SC conditionally respectsimprovement : Theorem 2.
Let i ∈ N . Let R, ˜ R be a pair of profiles of preferences such that ˜ R is animprovement for i with respect to R . If SC ( R ) , SC ( ˜ R ) (cid:54) = ∅ , then for each ˜ x ∈ SC ( ˜ R ) and foreach x ∈ SC ( R ) , ˜ x i R i x i .Proof. Let x ∈ SC ( R ). From Remark 3 it follows that it is sufficient to show that there exists˜ x ∈ SC ( ˜ R ) with ˜ x i R i x i . We can assume that there is a unique agent j (cid:54) = i with ˜ R j (cid:54) = R j .(If there is more than one such agent, we repeatedly apply the one-agent result to obtain theresult.) We can also assume that i ˜ R j x j . (Otherwise x j ˜ P j i and hence, from the Quint-Wakoalgorithm, x ∈ SC ( R ) = SC ( ˜ R ).)We distinguish among three cases, depending on the relation between agents i and j in thegraph G SP for the market ( N, R ), i.e., the graph that is generated in the Quint-Wako algorithmto obtain x . Case I: i and j are independent or j is a predecessor of i . Let F ( i ) be the followers of i inthe graph G SP , where we use the convention i ∈ F ( i ). Then, j (cid:54)∈ F ( i ). In the Quint-Wakoalgorithm for R , the nodes in F ( i ) are exactly the nodes of a collection of absorbing sets. Sincefor each agent k ∈ F ( i ), ˜ R k = R k , it follows that in the Quint-Wako algorithm for ˜ R the nodesin F ( i ) are exactly the nodes of the same collection of absorbing sets. Since SC ( ˜ R ) (cid:54) = ∅ , itfollows from Remark 2 that there exists ˜ x ∈ SC ( ˜ R ) such that for each agent k ∈ F ( i ), ˜ x k = x k .In particular, ˜ x i = x i . Case II: i and j are absorbing set members. Let S t be the absorbing set that contains i and j in the Quint-Wako algorithm for R . Let ( i, l ) be an edge in S t , where possibly l = i . Let F ∗ ( N t ( S t )) be the followers outside of N t ( S t ) that can be reached by some path in G SP thatstarts from a node in N t ( S t ). Then, the nodes in F ∗ ( N t ( S t )) are exactly the nodes of a collectionof absorbing sets in the Quint-Wako algorithm for R . Moreover, the nodes in F ∗ ( N t ( S t )) areneither predecessors of j nor absorbing set members with j . Hence, by the same arguments asin Case I, in the Quint-Wako algorithm for ˜ R the nodes in F ∗ ( N t ( S t )) are again the nodes ofthe same collection of absorbing sets. Therefore, when the Quint-Wako algorithm is applied to˜ R , the absorbing set that contains i will again contain l . Since SC ( ˜ R ) (cid:54) = ∅ , at each ˜ x ∈ SC ( ˜ R ),agent i will receive an object ˜ x i such that ˜ x i I i l . Since also x i I i l , we obtain ˜ x i I i x i . Case III: i is a predecessor of j . Since for each k (cid:54) = j , ˜ R k = R k , i ˜ R j x j , and ˜ R j is obtainedfrom R j by shifting i up, it follows that at some step in the Quint-Wako algorithm for ˜ R , agent j will start pointing to agent i and will keep doing so if and as long as agent i is present. Next,consider the predecessor of j on a best path p b ( i, j ) ∈ P b ( i, j ) (in G SP for the market ( N, R )),say agent l . Let k ∈ N with kP l j . By definition of p b ( i, j ), k and j are not absorbing setmembers nor is k a predecessor of j . From the same arguments as in Case I, the nodes in F ( k )(the followers of k , where k ∈ F ( k )) form among themselves the same absorbing sets in theQuint-Wako algorithm for both R and ˜ R . Hence, during the Quint-Wako algorithm for ˜ R , atsome step agent l will start pointing to agent j and will keep doing so if and as long as agent j is present. We can repeat the same arguments until we conclude that each node in the cycleformed by p b ( i, j ) and the edge ( j, i ) will, at some step, start pointing to its follower and willkeep doing so if and as long as the follower is present. Hence, at some step of the algorithm thecycle formed by p b ( i, j ) and the edge ( j, i ) is part of an absorbing set. Let i b denote the followerof agent i in path p b ( i, j ). Since SC ( ˜ R ) (cid:54) = ∅ , at each ˜ x ∈ SC ( ˜ R ), agent i will receive an object16 x i such that ˜ x i I i i b . Note that in the graph G SP , ( i, i b ) ∈ E S or ( i, i b ) ∈ E P . If ( i, i b ) ∈ E S ,then i b I i x i , in which case ˜ x i I i x i . If ( i, i b ) ∈ E P , then by definition of E P , i b R i x i , in which case˜ x i R i x i . Corollary 1.
For each i ∈ N and each pair of profiles of strict preferences R, ˜ R such that SC ( R ) , SC ( ˜ R ) (cid:54) = ∅ and ˜ R is an improvement for i with respect to R , • there is ˜ x ∈ SC ( ˜ R ) such that for each x ∈ SC ( R ) , ˜ x i R i x i ; and • there is x ∈ SC ( R ) such that for each ˜ x ∈ SC ( ˜ R ) , ˜ x i R i x i . In this subsection we provide several examples to demonstrate the possible violations of therespecting improvement property (or variants/extensions of the property) in the setting ofbounded length exchanges.
Pairwise exchanges
As mentioned in Section 1, the maximisation of the number of pairwise exchanges does notrespect improvement. Example 5 below proves this formally. A consequence is that the prioritymechanisms studied by Roth et al. [38] need not be donor-monotonic if agents’ preferences canbe non-dichotomous.
Example 3.
Let N = { , . . . , } and let preferences R be given by Table 6 and the newpreferences ˜ R by Table 7, where the only improvement is that agent 1 becomes acceptable foragent 3.1 2 3 42 1 3 23 4 41 2Table 6: R
23 4 3 41 2Table 7: ˜ R R , there are two ways to maximise the number of pairwise exchanges, namelyby picking either of the two-cycles (1 ,
2) and (2 , ,
2) is selected. (In case (2 ,
4) is selected, similar arguments can be employed.) Supposethe discontinuous edge (in Figure 4) is included so that agent 1 “improves” and we obtain ˜ R .Then, the unique way to maximise the number of pairwise exchanges is obtained by picking the2 two-cycles (1,3) and (2,4), which means that agent 1 is strictly worse off than in the initialsituation. (cid:5) We illustrate that the respecting improvement property can be violated in a weak sense ,namely the best allotment remains the same, but a worse allotment is created for the improvingagent. 17 xample 4.
Let N = { , . . . , } and let preferences R be given by Table 8 and the newpreferences ˜ R by Table 9, where the only improvement is that agent 1 becomes acceptable foragent 3.1 2 3 42 4 4 33 1 3 21 2 4Table 8: R
33 1 4 21 2 3 4Table 9: ˜ R R , the unique (strong) core allocation is x a = { (1 , , (3 , } . Suppose the discon-tinuous edge (in Figure 5) is included so that agent 1 “improves” and we obtain ˜ R . Then,another (strong) core solution is created, x b = { (1 , , (2 , } , which is strictly worse for 1. (cid:5) The following example illustrates violation of best-Ri property for pairwise exchanges withties for core and Wako-core.
Example 5.
Let N = { , . . . , } and let preferences R be given by Table 10. In the newpreferences ˜ R , Table 11, agent 4 makes an improvement and becomes acceptable for agent 1.1 2 3 43 4 1,4 134Table 10: R R R there exists two (Wako-) core allocations x a = { (3 , } and x b = { (1 , , (2 , } , i.e., C ( R ) = WC ( R ) = { x a , x b } . The best allotment for agent 4 agent 3,i.e. allocation x a is to be chosen.For preferences ˜ R , newly formed cycle (1 ,
4) is blocking for allocation x a , while (1 ,
3) isblocking for allocation (1 , C ( ˜ R ) = WC ( ˜ R ) = { x b } , hence the improvement is notrespected. Three-way exchanges
The following example exhibits three housing markets where for each housing market the threecores coincide (and are non-empty). Subsequently, we will use the example to show that thethree cores do not respect improvement when the maximal allowed length of exchange cyclesis 3.For a housing market (
N, R ), let, with a slight abuse of notation, SC ( R ), WC ( R ), and C ( R )denote the (possibly empty) strong core, Wako-core, and core of R , respectively. Example 6.
Throughout the example we focus on the core. However, since all blockingarguments can be replaced by weak blocking arguments, all statements also hold for the strongcore, and hence also for the Wako-core. Let N = { , . . . , } be the set of agents. We considerthree different housing markets that only differ in preferences. First, consider the housingmarket ( N, R ), or simply R for short, with the following “cyclic” strict preferences (whereunacceptable objects are not displayed): 18references R i, i + 1) (mod 10) where agents i and i + 1 swap their objects. Thecore C ( R ) = { x a , x b } consists of the following two allocations: x a = { (1 , , (3 , , (5 , , (7 , , (9 , } and x b = { (10 , , (2 , , (4 , , (6 , , (8 , } . Next, we create an extended housing markets R b by inserting one three-cycle in R .Preferences R b are provided in Table 12, where the changes with respect to R are bold-facedand depicted in Figure 7.1 2 3 4 5 6 7 8 9 10
10 12 1 2 R b Figure 7: Acceptability graph for R b Apart from the earlier mentioned self-cycles and two-cycles, the only additional exchangecycle with only acceptable objects in R b is c b = (1 , , x b is in the core of R b because c b does not block x b : agent 4 obtains object 8 in c b , which is strictly less preferred toher assigned object at x b . In fact, x b is the unique core allocation of R b . To see this, note firstthat x a is not in the core of R b as c b blocks it. And second, the only new exchange cycle createdin R b , i.e., c b , cannot be part of a core allocation, because if it were, then to avoid blocking cycle(4 , ,
6) would have to be part of the allocation, in which case 7 wouldremain unmatched (i.e., be a self-cycle) and cycle (6 ,
7) would block the allocation. Therefore, x b is the unique core allocation of R b , i.e., C ( R b ) = { x b } . (cid:5) Using the above example we can easily prove the following result. For i ∈ N , preferences R i , and a set of allocations X , agent i ’s most preferred allotment in X is her most preferredallotment among those that she receives at the allocations in X . Proposition 2.
Suppose the maximum allowed length of exchange cycles is 3. Then, there are3-housing markets with strict preferences ( N, R ) and ( N, ˜ R ) with • X ( R ) ≡ SC ( R ) = WC ( R ) = C ( R ) (cid:54) = ∅ and • X ( ˜ R ) ≡ SC ( ˜ R ) = WC ( ˜ R ) = C ( ˜ R ) (cid:54) = ∅ So, the core coincides with the set of stable matchings of the corresponding “roommate problem” (Galeand Shapley, 1962). uch that for some i ∈ N , ˜ R is an improvement for i with respect to R but • X ( ˜ R ) ⊆ X ( R ) , • for the unique x ∈ X ( R ) \ X ( ˜ R ) and for each ˜ x ∈ X ( ˜ R ) \ X ( R ) , x i P i ˜ x i .Proof. Let ( N, ˜ R ) be the 3-housing market with N = { , . . . , } and ˜ R = R b from Exam-ple 6. Let ( N, R ) be the 3-housing market that is obtained from ( N, ˜ R ) by making object 1unacceptable for agent 8. Obviously, ˜ R is an improvement for agent 1 with respect to R . Asshown in Example 6, SC ( ˜ R ) = WC ( ˜ R ) = C ( ˜ R ) = { x b } (cid:54) = ∅ . One also easily verifies that SC ( R ) = WC ( R ) = C ( R ) = { x a , x b } (cid:54) = ∅ . Finally, agent 1’s most preferred allotment in SC ( R ) = WC ( R ) = C ( R ) is object 2, while agent 1’s unique (hence, most preferred) allotmentin SC ( ˜ R ) = WC ( ˜ R ) = C ( ˜ R ) is object 10. Since agent 1 strictly prefers object 2 to object 10,the result follows. In this section we propose novel integer programming (IP) formulations for the core, the setof competitive allocations (the
Wako-core ), and the strong core. First, we propose modelsfor the unbounded case, for the three solution concepts. Second, we propose alternative cycleformulations for the Wako-Quint formulations for core and strong core. Finally, we propose anew formulation for the competitive allocations.
Let (
N, R ) be a housing market and G ≡ G ( N, R ) = (
N, E ) its acceptability graph. Since allthree cores only contain individually rational allocations, we can restrict attention to the edgesof the acceptability graph. Specifically, with each edge ( i, j ) ∈ E we associate a variable y ij asfollows: y ij = (cid:26) i receives object j ;0 otherwise.Then, the base model reads as follows: (cid:88) j :( i,j ) ∈ E y ij = 1 ∀ i ∈ N (1) (cid:88) j :( j,i ) ∈ E y ji = 1 ∀ i ∈ N (2) y ij ∈ { , } ∀ ( i, j ) ∈ E (3)Constraints (1) guarantee that agent i receives exactly one (acceptable) object (possibly herown). Constraints (2) guarantee that object i is given to exactly one agent. Each vector( y ij ) ( i,j ) ∈ E that satisfies (1), (2), and (3) yields an allocation x defined by x i = j if and onlyif y ij = 1. Moreover, each allocation can be obtained in this way. So, there is a one-to-onecorrespondence between allocations and vectors that satisfy (1), (2), and (3).We introduce for each i ∈ N an additional integer variable p i that represents the price ofobject i . p i ∈ { , . . . , n } ∀ i ∈ N (4)20n what follows we give our IP formulations for the general case of weak preferences andexplain how they can be simplified for strict preferences. We tackle the core, the set of com-petitive allocations, and the strong core (in this order), by subsequently adding constraints.Given an allocation x , we say that x dominates an edge ( i, j ) in the acceptability graph G ifagent i weakly prefers her allotment x i to object j , i.e., x i R i j . IP for the core
It follows from Lemma 1 that an allocation x is in the core if and only if each cycle in G containsan edge that is dominated by x . Or equivalently, there exists no cycle in G that consists ofundominated edges. Note that the undominated edges form a cycle-free subgraph if and only ifthere is a topological order of the objects. The existence of this topological order is equivalentto the existence of prices of the objects such that for each undominated edge ( i, j ), p i < p j .Therefore, an allocation x is in the core if and only if there exist prices ( p i ) i ∈ N such that( i, j ) ∈ E is not dominated by x = ⇒ p i < p j . (*)Thus, core allocations are characterised by constraints (1)–(4) together with (5) below: p i + 1 ≤ p j + n · (cid:88) k : kR i j y ik ∀ ( i, j ) ∈ E (5) Proposition 3.
Let x be an allocation. Let y be the corresponding vector that satisfies (1) , (2) ,and (3) . Allocation x is in the core if and only if there are prices ( p i ) i ∈ N such that (4) and (5) hold.Proof. First observe that for each ( i, j ) ∈ E ,( i, j ) is dominated by x ⇐⇒ x i R i j ⇐⇒ there is k ∈ N with k R i j and y ik = 1 ⇐⇒ (cid:88) k : kR i j y ik = 1 . (**)Suppose x is in the core. Then, there exist prices ( p i ) i ∈ N that satisfy (4) and (*). We verifythat (5) holds. Let ( i, j ) ∈ E . If ( i, j ) is not dominated by x , then (5) follows immediatelyfrom (*). Suppose ( i, j ) is dominated by x . From (**), (cid:80) k : kR i j y ik = 1. Hence, p i + 1 ≤ n + 1 ≤ p j + n = p j + n · (cid:88) k : kR i j y ik . Suppose that there exist prices ( p i ) i ∈ N such that (4) and (5) hold. We verify that (*) holds.Let ( i, j ) ∈ E and suppose it is not dominated by x . From (**), (cid:80) k : kR i j y ik = 0. Hence, from(5), p i + 1 ≤ p j + n ·
0, i.e., p i < p j . IP for the set of competitive allocations, i.e., the Wako-core
The set of competitive allocations is characterised by constraints (1)–(5) together with (6)below: p i ≤ p j + n · (1 − y ij ) ∀ ( i, j ) ∈ E (6)21 roposition 4. Let x be an allocation. Let y be the corresponding vector that satisfies (1) , (2) ,and (3) . Allocation x is competitive if and only if there exist prices ( p i ) i ∈ N such that (4) , (5) ,and (6) hold. Moreover, if such prices exist, then together with x they constitute a competitiveequilibrium.Proof. Suppose x is competitive. Let ( p i ) i ∈ N be prices such that ( x, p ) is a competitive equilib-rium. Then, (4) and (*) hold. From the first part of the proof of Proposition 3 it follows that(5) holds. We now prove that (6) holds as well. Let ( i, j ) ∈ E . If y ij = 0, then immediately p i ≤ p j + n = p j + n · (1 − y ij ). If y ij = 1, then x i = j , and since ( x, p ) is a competitiveequilibrium it follows from Remark 1 that p i = p x i = p j .Suppose that there exist prices ( p i ) i ∈ N such that (4), (5), and (6) hold. We verify that ( x, p )is a competitive equilibrium. First, it follows from (6) that for each i ∈ N , taking j = x i yields p i ≤ p x i + n · (1 −
1) = p x i , i.e., p i ≤ p x i . Hence, from Remark 1, for each i ∈ N , p i = p x i .Second, let j ∈ N be an object such that j P i x i . Then, ( i, j ) ∈ E is not dominated by x .From the second part of the proof of Proposition 3 it follows that (*) holds. Hence, we obtain p i < p j . IP for the strong core
The strong core is characterised by constraints (1)–(6) together with (7) below: p i ≤ p j + n · (cid:32) (cid:88) k : kP i j y ik (cid:33) ∀ ( i, j ) ∈ E (7) Proposition 5.
Let x be an allocation. Let y be the corresponding vector that satisfies (1) , (2) , and (3) . Allocation x is in the strong core if and only if there exist prices ( p i ) i ∈ N such that (4) , (5) , (6) , and (7) hold. Moreover, if such prices exist, then together with x they constitutea competitive equilibrium.Proof. Suppose x is in the strong core. By Remark 2, x can be obtained in the Quint-Wakoalgorithm by choosing for each absorbing set in the algorithm a particular cycle cover. Hence,there exist price ( p i ) i ∈ N such that (i) constraints (4) are satisfied, (ii) all objects in the sameabsorbing set have the same price, and (iii) an absorbing set that is processed earlier by thealgorithm has a strictly higher associated price (of its objects). It is easy to verify that ( x, p )is a competitive allocation. Hence, from the first part of the proof of Proposition 4 it followsthat (5) and (6) hold. Finally, to see that (7) holds note that from the definition of the pricesit follows that (i) if jR i x i then p i ≤ p j and (ii) if x i P i j then p i ≤ n = n ( (cid:80) k : kP i j y ik ).Suppose that there exist prices ( p i ) i ∈ N such that (4), (5), (6), and (7) hold. It followsfrom Proposition 4 that ( x, p ) is a competitive equilibrium. We prove that x is a strong coreallocation. Suppose there is a coalition S that weakly blocks x through an allocation z . FromLemma 1 it follows that we can assume, without loss of generality, that S = { , . . . , r } and thatfor each i = 1 , . . . , r − z i = i + 1, z r = 1, and z P x . Since x is individually rational, r > x, p ) is a competitive equilibrium, p < p . Since 3 = z R x , we have (cid:80) k : kP y k = 0.Hence, from (7), p ≤ p + n · (cid:32) (cid:88) k : kP y k (cid:33) = p . p ≤ p . By repeatedly applying the same arguments we find p ≤ p ≤ · · · ≤ p r ≤ p .Since p < p , we obtain a contradiction. Therefore, there is no coalition that weakly blocks x .Hence, x is a strong core allocation. Remark 4.
We note that in the case of strict preferences, constraints (7) are satisfied by anycompetitive equilibrium ( x, p ). To see this note that if y ij = 1 then (6) implies (7), since1 − y ij = 0, and hence p i ≤ p j + n · (1 − y ij ) = p j ≤ p j + n · (cid:32) (cid:88) k : kP i j y ik (cid:33) . Otherwise, if y ij = 0 then (5) implies (7), since for strict preferences (cid:80) k : kP i j y ik = (cid:80) k : kP i j y ik + y ij = (cid:80) k : kR i j y ik , and hence p i < p i + 1 ≤ p j + n · (cid:32) (cid:88) k : kR i j y ik (cid:33) = p j + n · (cid:32) (cid:88) k : kP i j y ik (cid:33) . Therefore, in either case, constraints (7) are satisfied. This reflects the fact that for strictpreferences the strong core is a singleton that consists of the unique competitive allocation. (cid:5)
To compare our IP formulations with the IP formulations for the core and the strong core givenby Quint and Wako [35], we describe the latter IP formulations using our notation.First, for both the core and the strong core, Quint and Wako [35] used the “basic” constraints(1), (2), and (3). We refer to their formulas (9.2), (9.3), (9.4), as well as (8.2), (8.3), (8.4),together with an integrality condition.Next, to obtain the core Quint and Wako [35] imposed the following additional no-blockingcondition (see (9.1) in [35]): (cid:88) i ∈ S (cid:32) (cid:88) j : jR i π i y ij (cid:33) ≥ ∀ S ⊆ N, π ∈ Π S (8)Finally, to obtain the strong core Quint and Wako [35] imposed the following additionalno-blocking condition (see (8.1) in [35]): (cid:88) i ∈ S (cid:32) (cid:88) j : jP i π i y ij + 1 | S | (cid:88) j : jI i π i y ij (cid:33) ≥ ∀ S ⊆ N, π ∈ Π S , (9)where Π S is the set of allocations in the submarket M S (so that π is an allocation in M S ).Constraints (8) and (9) directly describe that no coalition S can block / weakly blockthrough an allocation π , respectively. Both sets of constraints are highly exponential (in thenumber of agents), since they are required not only for all subsets S of N , but also for allpossible redistributions within each S . Alternative cycle-formulations
23n view of Lemma 1, it is sufficient to impose constraints (8) and (9) for the cycles of theacceptability graph G . Based on this observation and results in [25], we will describe alterna-tive cycle-formulations for the core and the strong core. Furthermore, we will provide a newproposition and IP formulation for the Wako-core.Let M = ( N, R ) be a housing market. Let C denote the set of exchange cycles in G ( N, R ).For a cycle c ∈ C , let N ( c ) and A ( c ) denote the set of nodes and edges in c , respectively, andlet | c | denote the size/length of c . We write c i = j if agent i receives object j in the exchangecycle c , i.e., ( i, j ) ∈ A ( c ). Proposition 6 ([25]) . An allocation x is in the core if and only if for each cycle c ∈ C , forsome agent i ∈ N ( c ) , x i R i c i . The corresponding IP constraints, which reduce the constraints (8) to cycles, are as follows: (cid:88) ( i,j ) ∈ A ( c ) (cid:88) k : kR i j y ik ≥ ∀ c ∈ C (10)Next, we describe the alternative cycle-formulation for the strong core. First we focus onthe special case of strict preferences. Proposition 7 ([25]) . Suppose preferences are strict. Then, an allocation x is in the strongcore if and only if for each cycle c ∈ C , c is an exchange cycle in x or for some agent i ∈ N ( c ) , x i P i c i . Proposition 7 leads to the following constraints: (cid:88) ( i,j ) ∈ A ( c ) y ij + | c | · (cid:88) ( i,j ) ∈ A ( c ) (cid:88) k : kP i j y ik ≥ | c | ∀ c ∈ C (11)The alternative cycle-formulation for the strong core in the general case (where preferencescan have ties) is as follows. Proposition 8 ([25]) . An allocation x is in the strong core if and only if for each cycle c ∈ C ,(i) c is an exchange cycle in x , or(ii) for some agent i ∈ N ( c ) , x i P i c i , or(iii) for each agent i ∈ N ( c ) , c i I i x i . The corresponding IP constraints, which reduce the constraints (9) to cycles, are as follows: (cid:88) ( i,j ) ∈ A ( c ) (cid:88) k : kI i j y ik + | c | · (cid:88) ( i,j ) ∈ A ( c ) (cid:88) k : kP i j y ik ≥ | c | ∀ c ∈ C (12)Finally, similarly to the core and strong core, we provide a new alternative characterisationfor the Wako-core. Proposition 9.
An allocation x is in the Wako-core if and only if for each cycle c ∈ C ,(i) c is an exchange cycle in x , or(ii) for some agent i ∈ N ( c ) , x i P i c i , or(iii) for some agent i ∈ N ( c ) , c i I i x i and c i (cid:54) = x i . (cid:88) ( i,j ) ∈ A ( c ) y ij + | c | · (cid:88) ( i,j ) ∈ A ( c ) (cid:88) k : kR i j,k (cid:54) = j y ik ≥ | c | ∀ c ∈ C (13)To see the correctness of this new formulation, observe that the first term of (13) is equal to | c | if condition (i) of Proposition 9 is satisfied and less than | c | otherwise; and the second term hasvalue at least | c | if condition (ii) or (iii) of Proposition 9 is satisfied and 0 otherwise. Therefore,constraint (13) is satisfied if and only if at least one of the three conditions of Proposition 9holds. Note that the above cycle-formulations are not very practical due to the exponentially largenumber of cycles. In fact, this justified the novel IP formulations proposed in Section 4.1.However, the cycle-formulations are practical for the case of bounded length exchanges.One easily verifies that Lemma 1 can be extended to bounded length exchanges in a naturalway: the strong core, Wako-core, and core of a k -housing market can be defined equivalently bythe absence of corresponding blocking cycles of size at most k . In fact, Klimentova et al. [25]proposed associated IP formulations by adapting constraints (10) and (12) to bounded exchangecycles. One can similarly adapt constraints (13) to obtain an IP formulation for the Wako-coreof a k -housing market. In our simulations we used the most efficient cycle-edge formulationsby Klimentova et al. (see the detailed description in section 3.3 of [25]). In this section we perform a computational analysis of the models proposed in Section 4 andcompare them with the models for bounded length exchange cycles, proposed in [25]. Further-more, we estimate the frequency of violations of the respecting improvement property for allmodels by computational simulations. The models are run for both strict and weak preferencesand considering two objective functions: maximisation of the size of the exchange (correspond-ing to the maximisation of the number of transplants in the context of KEPs), denoted byMax t , and maximisation of total weight (where weights can mean the scores given to the corre-sponding transplants in a KEP, reflecting the qualities of the transplants), denoted by Max w .For bounded length exchanges, the maximum length considered are k = 2 and k = 3.In Section 5.1, we compare the size/weight of the maximum size/weight allocation to thecore, competitive and strong core allocations under the same objective. For the unboundedcase, we further analyse the price of fairness : the difference in percentage in the number oftransplants of the maximum total weight solution, and the core, competitive and strong coreallocations for both objectives, when compared to the maximum size solution.In Section 5.2 we calculate the average number of weakly blocking cycles for allocationsprovided by each formulation. By doing so, we give a rough indication of how far each solutionis from the strong core. We complement that analysis with the quantification of the average25umber of vertices of an instance that may obtain a strictly better allotment in at least oneweakly blocking cycle.Average CPU times required to solve an instance of a given size for each of the formulationsin Section 4 are presented in subsection 5.3.Finally, in subsection 5.4 we provide results on the frequency of violations of the respectingimprovement property for all of our models.All programs were implemented using Python programming language and tested usingGurobi as optimisation solver [21]. Tests were executed on a MacMini 8 running macOS version10.14.3 in a Intel Core i7 CPU with 6 cores at 3.2 GHz with 8GB of RAM.Test instances were generated with the generator proposed in [40, 27] and are available fromhttp: TBA. The number of pairs of an instance ranges from 20 to 150; 50 instances of each sizewere generated. The weights associated to the arcs of the graph were generated randomly withinthe interval (0 , | V | wereconsidered equally preferable. Figure 8 presents average results for the maximum size and maximum weight objectives forweak preferences under different settings: no stability requirements (Max), core, competitiveand strong core allocations. We refrain ourselves from presenting the results for the case ofstrict preferences as all curves are similar, except that for strict preferences the competitiveand strong core allocations are the same.As expected, both the number of transplants and total weight decrease by increasing thenumber of constraints from Max to Core, then to Competitive, and then to Strong Core allo-cations. The strong core curve is non-monotonic, which is explained by the absence of feasiblesolutions for several instances. Next to the curve we present the number of instances out of thetotal 50 where a feasible solution existed.Figure 9 makes a similar analysis for the bounded case, when k = 2 and k = 3. Max k = ∞ refers to the unbounded exchange problem, whilst Max k =2 and Max k =3 correspond to thebounded problem for k = 2 and k = 3, respectively. The same reasoning is used for thenotation associated to the Wako-core (W.-Core) and strong core (S.Core). For easiness ofcomparison between the bounded and the unbounded cases, we again plot the two curves fromFigure 8 associated with maximum utility (Max) which, in both cases, represent an upperbound for our solutions. Naturally, the curves associated with k = 2 are dominated by thoseassociated with k = 3. We can observe that the maximum number of transplants for k = 3and for unbounded k are very similar (see Figure 9 (left)). Notice also that even though somecurves overlap and seem identical, there are minor differences among them, except for the caseof core and Wako-core allocations for k = 2, that coincide. Again, we only present results forweak preferences, as this is the more general case. For strict preferences, for k = 3 the curvesare similar, for k = 2, core, competitive and strong core allocations coincide, and the latter twoare also the same for unbounded exchanges.From a practical point of view an interesting question is to study the impact of stabilityrequirements on the number of transplants achievable. Although KEPs have many other key As mentioned before, for the unbounded case competitive allocations are equivalent to Wako-core, and forbounded case we only have Wako-core. t r a n s p l a n t s
33 24 23 21 13 15 12 6 5 7 6 5 2 2
Max
20 30 40 50 60 70 80 90 100 110 120 130 140 150 |V| T o t a l w e i g h t o f t r a n s p l a n t s
33 24 23 21 13 15 12 6 5 7 6 5 2 2
Max total weight
Max Core Competitive Strong core
Figure 8: Number of transplants (left) and total weight of transplants (right) for unboundedlength and weak preferences. The numbers in the chart reflect the number of instances wherea feasible solution existed.
20 30 40 50 60 70 80 90 100 110 120 130 140 150|V| 020406080100 t r a n s p l a n t s Max
20 30 40 50 60 70 80 90 100 110 120 130 140 150 |V| T o t a l w e i g h t o f t r a n s p l a n t s Max total weight
Max k = Max k =2 Core k =2 W.-Core k =2 S.Core k =2 Max k =3 Core k =3 W.-Core k =3 S.Core k =3 Figure 9: Comparison of the number of transplants (left) and the total weight of transplants(right) for bounded length exchanges ( k = 2 ,
3) and weak preferences. A solid line is used forthe unbounded case, dotted lines are used for k = 2 and dashed lines for k = 3.27erformance indicators, this is unarguably the most relevant one, as this criterion is optimisedas a first objective in all the European KEPs [11]. Figure 10 presents the price of fairness ,that is difference in percentage in the number of transplants for Max w allocation, and forCore, Competitive and Strong Core allocations under both objectives, when compared to themaximum number of transplants achievable (Max t ). Subscripts t and w identify the objectivefunctions used for each allocation. As shown, the price of fairness for competitive and strongcore allocations is extremely high, when compared to the core. It decreases with problem sizefor both objective functions and for all allocation models, being slightly higher for the core forthe total weight objective (see curve Core w ). For the maximisation of the number of transplants(curve Core t ), for instances with more than 50 nodes the reduction is of less than 3%, decreasingto 1% for the largest instance. Such result is of major practical relevance as it indicates thatwith increasing size of the programs one can consider pairs’ preferences in the matching withno significant reduction in the number of pairs transplanted.
20 30 40 50 60 70 80 90 100 110 120 130 140 150|V|0510152025 P r i c e o f f a i r n e ss ( % ) Strict preferences
20 30 40 50 60 70 80 90 100 110 120 130 140 150|V|
Weak preferences
Core t Compet. t S.Core t Max w Core w Compet. w S.Core w Figure 10: Price of fairness with respect to maximum number of transplants for core, compet-itive and strong core allocations, with maximum number of transplants and maximum totalweight, and solution with maximum total weight; for strict (left) and weak (right) preferences.
Figure 11 (left) presents the average number of weakly blocking cycles of size 2 in Max, Core,and Competitive (Wako-core) allocations. We denote the maximum length of the blockingcycles considered by l . For the bounded case, following the same reasoning as in [25], the figurealso reports the minimum average number of weakly blocking cycles for the cases where thestrong core does not exist, i.e., for the maximum number of transplants/total weight solutionwith minimum number of weakly blocking cycles. Interestingly, when the objective functionis the number of transplants, the “unstability” of the solutions barely depends on the size ofexchanges allowed. The same does not hold for the core, where the number of blocking cycles28 w e a k l y b l o c k i n g c y c l e s Max
20 30 40 50 60 70 80 90 100 110 120 130 140 150|V|
Max total weight
Max k = Core k = Compet. k = Max k =3 Core k =3 W.-Core k =3 S.Core k =3 Max k =2 Core k =2 W.-Core k =2 S.Core k =2 Figure 11: Number of weakly blocking cycles of size l = 2 for solutions with maximum numberof transplants (left) and maximum total weight of transplants (right), for unbounded exchangesand exchanges of size up to k = 2 and k = 3 for weak preferences. A solid line is used for theunbounded case, dotted lines are used for k = 2 and dashed lines for k = 3.is considerably smaller for k = 2. For this and all the remaining cases, the average number ofweakly blocking cycles is very low, in most cases below 1. It is worth to note that the averagenumber of blocking cycles tends to be smaller when the objective is to maximise the totalweight (Figure 11 (right)). A plausible justification for this is that the weights reflect patientspreferences and therefore a solution obtained by considering that objective will be closer to astable solution.Figure 12 presents the same analysis, now considering weakly blocking cycles of size up to3. Naturally, the solutions for k = 2 are excluded from this analysis, as they are fully reflectedin Figure 11. The conclusions drawn for l = 2 remain valid for this case. For the unboundedcase, the number of blocking cycles is larger, since one must consider also the cases when l > l > w e a k l y b l o c k i n g c y c l e s Max
20 30 40 50 60 70 80 90 100 110 120 130 140 150|V|
Max total weight
Max k = Core k = Compet. k = Max k =3 Core k =3 W.-Core k =3 S.Core k =3 Figure 12: Number of weakly blocking cycles of size l = 3 for solutions with maximum numberof transplants (left) and maximum total weight of transplants (right), for unbounded exchangesand exchanges of size up to k = 3 for weak preferences.
20 30 40 50 60 70 80 90 100 110 120 130 140 150|V|10 w e a k l y b l o c k i n g c y c l e s Max
20 30 40 50 60 70 80 90 100 110 120 130 140 150|V|
Max total weight
Max l =4 Core l =4 Compet. l =4 Max l =5 Core l =5 Compet. l =5 Figure 13: Number of weakly blocking cycles of size l = 4 and l = 5 for solutions with maximumnumber of transplants (left) and maximum total weight of transplants (right), for unboundedexchanges and weak preferences. 30 v e r t i c e s Max
20 30 40 50 60 70 80 90 100 110 120 130 140 150 |V|
Max total weight
Max Core Compet.
Figure 14: Average number of agents for an instance for those there exists at least one weaklyblocking cycles, where this agent receives a strictly better allotment for weak preferences. Thegrey line is a reference line showing the number of vertices in an instance.
In Table 13 we present the average CPU time for solving an instance of a given size with oneof the tree newly proposed IP models for unbounded case.The instances with the weak preferences are more complicated, for core and, in particular,for the competitive allocation model. However, it was faster to find the strong core for weak,rather than for strict preferences. Moreover, surprisingly, finding the strong core is the mosttime consuming task for strict preferences, while it is least time consuming for weak preferences.Finally, we can notice that models for finding core and strong core allocations are performingwithin the same ranges of magnitude with respect to the CPU time if compared with thecorresponding models for the bounded case, analysed in [25].
In this section we will make a computational analysis on how often the respecting improvement(RI) property is violated for different models, for both unbounded and bounded cases. To doso, for each model and for instances with 20 and 30 vertices we run the following procedure,presented in Algorithm 1. For the unbounded case we considered the Max and Core modelsunder both objectives.Let r ik , r ∈ { , . . . , | V |} be the rank of good k for agent i , that will reflect preferences of i ,31 ax | V | Core Compet. S.Core Core Compet. S.Core Core Compet. S.Core Core Compet. S.Core
Strict preferences Weak preferences
20 0.00 0.03 0.01 0.00 0.02 0.01 0.00 0.04 0.01 0.00 0.03 0.0130 0.03 0.13 0.04 0.02 0.11 0.03 0.02 0.28 0.04 0.02 0.17 0.0340 0.08 0.48 0.12 0.06 0.25 0.11 0.09 0.63 0.10 0.06 0.44 0.0850 0.24 1.74 0.38 0.16 0.58 0.34 0.20 2.15 0.25 0.17 1.06 0.2160 0.47 2.39 0.87 0.28 0.91 0.79 0.52 6.03 0.44 0.26 2.87 0.3970 1.06 3.91 1.94 0.66 2.29 1.50 0.84 16.99 1.09 0.53 7.35 0.7780 1.62 6.54 3.26 0.82 3.39 2.32 1.41 32.21 1.63 0.76 17.47 1.0190 3.14 36.34 5.31 3.27 5.38 3.59 3.29 167.15 2.36 1.82 80.88 1.49100 3.53 16.19 19.26 2.43 6.15 9.81 4.51 188.35 8.87 3.08 95.39 4.62110 8.73 21.42 28.26 4.97 9.01 13.79 6.68 331.64 16.40 5.92 159.12 7.24120 17.84 72.87 57.36 6.81 15.36 24.32 20.14 392.88 19.60 6.79 218.58 10.87130 14.34 46.92 84.49 14.24 22.68 34.11 14.78 586.27 21.75 12.32 438.23 10.42140 29.50 61.99 110.82 21.51 34.33 46.67 41.59 708.92 40.97 16.43 539.56 14.89150 41.99 161.10 214.32 30.66 52.61 70.77 57.13 786.43 61.79 27.82 682.99 23.91
Table 13: Average CPU time (in seconds) for solving an instance of a given size with theproposed formulation.i.e. if r ik ≤ ( <, =) r ij , then kR i ( P i , I i ) j . Algorithm 1:
Procedure for Checking RI property
Result: N number of violations of RI property N ← for i ∈ V , j ∈ V , i (cid:54) = j do Let R be the current preferences of agents;Find allocation with the best allotment for i with respect to R , denote the solutionby ¯ y ;For ¯ y il = 1 denote ¯ r ← r il ; while ∃ kP j i do Let k be the first strictly preferred agent for j that precedes i in R ; if Strict preferences then
Swap i with k in the list of preferences of j ; endif Weak preferences then
Let i become equally preferred for j as k (i.e. r ji = r jk ); end Denote modified preferences by ˜ R ;Find allocation with the best allotment for i with respect to ˜ R , denote solutionby ˜ y ;For ˜ y it = 1, denote ˜ r = r it ; if ¯ r < ˜ r then The respecting improvement property is violated: N ← N + 1; end ¯ r ← ˜ r ; R ← ˜ R ; endend For each pair of agents i and j , agent i is consecutively making improvements, moving up in32 Max t Max w Max t Max w Max t Max w Core t , w W.-,S.Core t , w |V| = 20|V| = 30 k = k = 2 k = 3 Figure 15: Number of violations of the respecting improvement property for all instances intotal of a given size, | V | = 20 ,
30, for strict preferences.the preference list of agent j until its top. In each step (see while loop in the algorithm) for thecase of strict preferences, i is swapped with k , who is the first strictly preferred agent by j to i .For the case of ties, agent i first becomes equally preferred for j as k . After the improvements,the best allocations for the original ( R ) and improved ( ˜ R ) preferences are compared for i . It isconsidered that there is a violation of the RI property if i obtains a strictly worse allotment inallocation for ˜ R .Figures 15 and 16 present box plots for the number of violations of the RI property forinstances of a given size for strict and weak preferences, respectively, for those models wherethe RI property is violated at least once. Models whose results are the same, independentlyof the objective considered, are plotted together. That is the case, for example, of Core t and Core w , for k = 3 and strict preferences (see figure 15), or of Wako-core and Core, weakpreferences and k = 2 and k = 3 (see figure 16).For (Wako-, Strong) Core models there were few cases of violations of the RI property, asreflected in the figure. To give an indication, the total number of violations for all instances with30 vertices for the weak preferences and k = 3 was 4549 for Max t , 3145 for Max w , 10 for Core t,w ,20 for W.-Core t,w , and 2 for S.Core t,w . For maximum size and maximum weight solutions (Max t and Max w , respectively), both for the unbounded and the bounded cases, one can observe asignificant number of violations. Those numbers increase with instance size. Interestingly, forthe unbounded case the number of violations for Max t was lower than that for Max w . Thiscan be explained by the fact that the former problem has a larger number of alternative bestallocations, while for the weighted objective problem the solution is usually unique. On thecontrary, for the bounded case maximum weighted solutions violated the property less times,compared to maximum size solutions. 33 Max t Max w Max t Max w (W.-)Core t , w Max t Max w Core t , w W.-Core t , w S.Core t , w |V| = 20|V| = 30 k = k = 2 k = 3 Figure 16: Number of violations of the respecting improvement property for all instances intotal of a given size, | V | = 20 ,
30, for weak preferences.
This paper advances current state of the art in several lines of research. We first prove that incase of strict preferences the unique competitive allocation respects improvement; an extensionof that result is provided for the case of ties.We also advance the work in the housing market of Shapley and Scarf presented in [35] byproviding Integer Programming models that do not require exponential number of constraintsfor the weak core, strong core, and the set of competitive allocations. These models assumethat there is no limit on the maximum size of an exchange cycle. However, since there areproblems where such assumption may be difficult to hold (e.g. Kidney Exchange Programmes)we further propose alternative IP models for bounded cycles. This contribution is inspired bythe definition of competitive equilibrium allocations provided in [45].We proceed with computational experiments that provide insights on the trade-off betweenstability requirements and maximum number of transplants. Results show that with increasingsize of the instances, such trade-off decreases: for instances with more than 50 nodes coreallocations impact on the reduction of transplants is less than 3%, decreasing to 1% for thelargest instance.Furthermore, results show that when the objective is to maximise the number of transplants,the “unstability” of the solutions, measured by the number of weakly blocking cycles barelydepends on the length of the exchanges. Additionally, the maximisation of total weight insteadof the number of transplants, leads to solutions where patients’ preferences matter more.As the main open question we left open whether the respecting improvement propertywith regard to the best allotment holds for a) the core for unbounded exchanges b) for stablematchings in the roommates problem with strict preferences. It would also be interestingto study whether the respecting improvement property can be used to characterise the TTCmechanism for the classical housing markets with strict preferences.34 cknowledgements
We thank Antonio Nicol´o for his contribution to an earlier version of this paper, and TayfunS¨onmez, and Utku ¨Unver for valuable comments.