Housing markets is a classic model in economics where agents are initially endowed with one unit of an indivisible good, called a house, and agents may trade their houses according to their preferences without using monetary transfers. In such markets, trading results in a reallocation of houses in a way that each agent ends up with exactly one house. Motivation for studying housing markets comes from applications such as kidney exchange [roth-sonmez-unver-2004, biro-kidney-exchange-survey, biroetal2021] and on-campus housing [adbulkadiroglu-sonmez].
In their seminal work Shapley and Scarf [shapley-scarf-1974] examined housing markets where agents’ preferences are weak orders. They proved that such markets always admit a core allocation, that is, an allocation where no coalition of agents can strictly improve their situation by trading only among themselves. They also described the Top Trading Cycles (TTC) algorithm, proposed by David Gale, and proved that the set of allocations that can be obtained through the TTC algorithm coincides with the set of competitive allocations; hence the TTC always produces an allocation in the core. When preferences are strict, the TTC produces the unique allocation in the strict core, that is, an allocation where no coalition of agents can weakly improve their situation by trading among themselves [roth-postlewaite].
Although the core of housing markets has been the subject of considerable research, there are still many challenges which have not been addressed. Consider the following question: given an agent and a house , does there exist an allocation in the core where obtains ? Or one where does not obtain ? Can we determine whether may receive a house better than her own in some core allocation? Similar questions have been extensively studied in the context of the Stable Marriage and the Stable Roommates problems [Knuth1976, GusfieldIrving-book, Dias-2003, Fleiner-Irving-Manlove, Cseh-Manlove-SR-2016], but have not yet been considered in relation to housing markets.
Even less is known about the core of housing markets in cases where the market is not static. Although some researchers have addressed certain dynamic models, most of these either focus on the possibility of repeated allocation [roth-postlewaite, kamijo-kawasaki, kawasaki-2015], or consider a situation where agents may enter and leave the market at different times [Unver-2010, bloch-cantala, kurino-2014]. Recently, Biró et al. [BKKV-arxiv] have investigated how a change in the preferences of agents affects the housing market. Namely, they considered how an improvement of the house belonging to agent affects the situation of . Following their lead, we aim to answer the following question: if the value of the house belonging to agent increases, how does this affect the core of the market from the viewpoint of ? Is such a change bound to be beneficial for , as one would expect? This question is of crucial importance in the context of kidney exchange: if procuring a new donor with better properties (e.g., a younger or healthier donor) does not necessarily benefit the patient, then this could undermine the incentive for the patient to find a donor with good characteristics, damaging the overall welfare.
1.1 Our contribution
We consider the computational complexity of deciding whether the core of a housing market contains an allocation where a given agent obtains a certain house. In Theorem 1 we prove that this problem is -complete, as is the problem of finding a core allocation where does not receive a certain house. Even worse, it is already -complete to decide whether a core allocation can assign any house to other than her own. Various generalizations of these questions can be answered efficiently in both the Stable Matching and Stable Roommates settings [Knuth1976, GusfieldIrving-book, Dias-2003, Fleiner-Irving-Manlove, Cseh-Manlove-SR-2016], so we find these intractability results surprising.
Instead of asking for a core allocation where a given agent can trade her house, one can also look at the optimization problem which asks for an allocation in the core with the maximum number of agents involved in trading. This problem is known to be -complete [Cechlarova-Repisky-2011]. We show in Theorem 2 that for any , approximating this problem with ratio for a set of agents is -hard. We complement this strong inapproximability result in Proposition 1 by pointing out that a trivial approach yields an approximation algorithm with ratio .
Turning our attention to the question of how an increase in the value of a house affects its owner, we show the following result in Theorem 1. If the core of a housing market contains an allocation where receives a house , and the market changes in a way such that some agents perceive an increased value for the house owned by (and nothing else changes in the market), then the resulting housing market admits an allocation in its core where receives either or a house that she prefers to . We prove this in a constructive way, by presenting an algorithm that finds such an allocation. This settles an open question by Biró et al. [BKKV-arxiv] who ask whether the core respects improvement in the sense that the best allocation achievable for an agent in a core allocation can only (weakly) improve for as a result of an increase in the value of ’s house.
It is clear that an increase in the value of ’s house may not always yield a strict improvement for (as a trivial example, some core allocation may assign her top choice even before the change), but one may wonder if we can efficiently determine when a strict improvement for becomes possible. This problem turns out to be closely related to the question whether can obtain a given house in a core allocation; in fact, we were motivated to study the latter problem by our interest in determining the possibilities for a strict improvement. Although one can formulate several variants of the problem depending on what exactly one considers to be a strict improvement, by Theorem 2 each of them leads to computational intractability (-hardness or -hardness).
Finally, we also answer a question raised by Biró et al. [BKKV-arxiv] regarding the property of respecting improvements in the context of the Stable Roommates problem. An instance of Stable Roommates contains a set of agents, each having preferences over the other agents; the usual task is to find a matching between the agents that is stable, i.e., no two agents prefer each other to their partners in the matching. It is known that a stable matching need not always exist, but if it does, then Irving’s algorithm [Irving-SR] finds one efficiently. In Theorem 1 we show that if some stable matching assigns agent to agent in a Stable Roommates instance, and the valuation of increases (that is, if she moves upward in other agents’ preferences, with anything else remaining constant), then the resulting instance admits a stable matching where is matched either to or to an agent she prefers to . This result is a direct analog of the one stated in Theorem 1 for the core of housing markets; however, the algorithm we propose in order to prove it uses different techniques.
We remark that we use a model with partially ordered preferences (a generalization of weak orders), and describe a linear-time implementation of the TTC algorithm in such a model.
1.2 Related work
Most works relating to the core of housing markets aim for finding core allocations with some additional property that benefits global welfare, most prominently Pareto optimality [Jaramillo-Manjunath, Alcalde-Unzu-Mollis, Aziz-deKeijzer, Plaxton-2013, Saban-Sethuraman-2013]. Another line of research comes from kidney exchange where the length of trading cycles is of great importance and often plays a role in agents’ preferences [cechlarova-romero-medina, cechlarova-hajdukova-2003, cechlarova-fleiner-manlove-kidney, Biro-Cechlarova-2007, cechlarove-lacko-2012] or is bounded by some constant [abraham-blum-sandholm, biro-manlove-rizzi-kidney, biro-mcdermid-2010-3cycles, huang-3wayKE, Cechlarova-Repisky-2011]. None of these papers deal with problems where a core allocation is required to fulfill some constraint regarding a given agent or set of agents—that they be trading, or that they obtain (or not obtain) a certain house. Nevertheless, some of them focus on finding a core allocation where the number of agents involved in trading is as large as possible. Cechlárová and Repiský [Cechlarova-Repisky-2011] proved that this problem is -hard in the classical housing market model, while Biró and Cechlárová [Biro-Cechlarova-2007] considered a special model where agents care first about the house they receive and after that about the length of their trading cycle (shorter being better); they prove that for any , it is -hard to approximate the number of agents trading in a core allocation with a ratio (where is the set of agents).
The property of respecting improvement has first been studied in a paper by Balinski and Sönmez [balinski-sonmez] on college admission, who proved that the student-optimal stable matching algorithm respects the improvement of students, so a better test score for a student always results in an outcome weakly preferred by the student (assuming other students’ scores remain the same). Hatfield et al. [hatfield-kojima-narita] contrasted their findings by showing that no stable mechanism respects the improvement of school quality. Sönmez and Switzer [sonmez-switzer] applied the model of matching with contracts to the problem of cadet assignment in the United States Military Academy, and have proved that the cadet-optimal stable mechanism respects improvement of cadets. Recently, Klaus and Klijn [klaus-klijn-RI] have obtained results of a similar flavor in a school-choice model with minimal-access rights.
Roth et al. [roth-sonmez-unver-2005] deal with the property of respecting improvement in connection to kidney exchange: they show that in a setting with dichotomous preferences and pairwise exchanges priority mechanisms are donor monotone, meaning that a patient can only benefit from bringing an additional donor on board. Biró et al. [BKKV-arxiv] focus on the classical Shapley-Scarf model and investigate how different solution concepts behave when the value of an agent’s increases. They prove that both the strict core and the set of competitive allocations satisfy the property of respecting improvements, however, this is no longer true when the lengths of trading cycles are bounded by some constant.
Preferences as partial orders. In the majority of the existing literature, preferences of agents are usually considered to be either strict or, if the model allows for indifference, weak linear orders. Weak orders can be described as lists containing ties, a set of alternatives considered equally good for the agent. Partial orders are a generalization of weak orders that allow for two alternatives to be incomparable for an agent. Incomparability may not be transitive, as opposed to indifference in weak orders. Formally, an (irreflexive)111Throughout the paper we will use the term partial ordering in the sense of an irreflexive (or strict) partial ordering. partial ordering on a set of alternatives is an irreflexive, antisymmetric and transitive relation.
Partially ordered preferences arise by many natural reasons; we give two examples motivated by kidney exchanges. For example, agents may be indifferent between goods that differ only slightly in quality. Indeed, recipients might be indifferent between two organs if their expected graft survival times differ by less than one year. However, small differences may add up to a significant contrast: an agent may be indifferent between and , and also between and , but strictly prefer to
. Partial preferences also emerge in multiple-criteria decision making. The two most important factors for estimating the quality of a kidney transplant are the HLA-matching between donor and recipient, and the age of the donor.222In fact, these are the two factors for which acceptability thresholds can be set by the patients in the UK program [biro-kidney-exchange-survey]. An organ is considered better than another if it is better with respect to both of these factors, leading to partial orders.
Housing markets. Let be a housing market with agent set and with the preferences of each agent represented by a partial ordering of the agents. For agents , , and , we interpret as agent preferring the house owned by agent to the house of agent . We will write as equivalent to , and we write if and . We say that agent finds the house of acceptable, if , and we denote by the set of agents whose house is acceptable for . We define the acceptability graph of the housing market as the directed graph with ; we let . Note that for each . The submarket of on a set of agents is the housing market where is the partial order restricted to ; the acceptability graph of is the subgraph of induced by , denoted by . For a set of agents, let be the submarket obtained by deleting from ; for we may write simply .
For a set of arcs in and an agent we let denote the set of agents such that ; whenever is a singleton we will abuse notation by writing . We also define and as the number of in-going and out-going arcs of in , respectively. For a set of agents, we let denote the set of arcs in that run between agents of .
We define an allocation in as a subset of arcs in such that for each , that is, forms a collection of cycles in containing each agent exactly once. Then denotes the agent whose house obtains according to allocation . If , then is trading in . For allocations and , we say that prefers to if .
For an allocation in , an arc is -augmenting, if . We define the envy graph of as the subgraph of containing all -augmenting arcs. A blocking cycle for in is a cycle in , that is, a cycle where each agent on prefers to . An allocation is contained in the core of , if there does not exist a blocking cycle for it, i.e., if is acyclic. A weakly blocking cycle for is a cycle in where for each agent on and for at least one agent on . The strict core of contains allocations that do not admit weakly blocking cycles.
Organization. Section 3 contains an adaptation of the TTC algorithm for partially ordered preferences, followed by our results on finding core allocations with various arc restrictions and on maximizing the number of agents involved in trading. In Section 4 we present our results on the property of respecting improvements in relation to the core of housing markets, including our main technical result, Theorem 1. In Section 5 we study the respecting improvement property in the context of the Stable Roommates problem. We conclude with some questions for future research in Section 6.
3 The core of housing markets: some computational problems
We investigate a few computational problems related to the core of housing markets. In Section 3.1 we describe our adaptation of TTC to partially ordered preferences. In Section 3.2 we turn our attention to the problem of finding an allocation in the core of a housing market that satisfies certain arc restrictions, requiring that a given arc be contained or, just the opposite, not be contained in the desired allocation. In Section 3.3 we look at the most prominent optimization problem in connection to the core: given a housing market, find an allocation in its core where the number of agents that are trading is as large as possible.
3.1 Top Trading Cycles for preferences with incomparability
Strict preferences. If agents’ preferences are represented by strict orders, then the TTC algorithm [shapley-scarf-1974] produces the unique allocation in the strict core. TTC creates a directed graph where each agent points to her top choice, that is, to the agent owning the house most preferred by . In the graph each agent has out-degree exactly 1, since preferences are assumed to be strict. Hence, contains at least one cycle, and moreover, the cycles in do not intersect. TTC selects all cycles in the graph as part of the desired allocation, deletes from the market all agents trading along these cycles, and repeats the whole process until there are no agents left.
Preferences as partial orders. When preferences are represented by partial orders, one can modify the TTC algorithm by letting each agent in point to her undominated choices: is undominated for , if there is no agent such that . Notice that an agent’s out-degree is then at least 1. Thus, contains at least one cycle, but in case it contains more than one cycle, these may overlap.
A simple approach is to select a set of mutually vertex-disjoint cycles in each round, removing the agents trading along them from the market and proceeding with the remainder in the same manner. It is not hard to see that this approach yields an algorithm that produces an allocation in the core: by the definition of undominated choices, any arc of a blocking cycle leaving an agent necessarily points to an agent that was already removed from the market at the time when a cycle containing got selected. Clearly, no cycle may consist of such “backward” arcs only, proving that the computed allocation is indeed in the core.
Implementation in linear time. Abraham et al. [ACMM-2004] describe an implementation of the TTC algorithm for strict preferences that runs in time. We extend their ideas to the case when preferences are partial orders as follows.
For each agent we assume that ’s preferences are given using a Hasse diagram which is a directed acyclic graph that can be thought of as a compact representation of . The vertex set of is , and it contains an arc if and only if and there is no agent with . Then the description of our housing market has length which we denote by . If preferences are weak or strict orders, then .
Throughout our variant of TTC, we will maintain a list containing the undominated choices of among those that still remain in the market, as well as a subgraph of spanned by all arcs with . Furthermore, for each agent in the market, we will keep a list of all occurrences of as someone’s undominated choice. Using we can find the undominated choices of in time, so initialization takes time in total.
Whenever an agent is deleted from the market, we find all agents such that , and we update by replacing with its in-neighbors in . Notice that the total time required for such deletions (and the necessary replacements) to maintain is . Hence, we can efficiently find the undominated choices of each agent at any point during the algorithm, and thus traverse the graph consisting of arcs with .
To find a cycle in , we simply keep building a path using arcs of , until we find a cycle (perhaps a loop). After recording this cycle and deleting its agents from the market (updating the lists as described above), we simply proceed with the last agent on our path. Using the data structures described above the total running time of our variant of TTC is .
3.2 Allocations in the core with arc restrictions
We now focus on the problem of finding an allocation in the core that fulfills certain arc constraints. The simplest such constraints arise when we require a given arc to be included in, or conversely, be avoided by the desired allocation.
We define the Arc in Core problem as follows: given a housing market and an arc in , decide whether there exists an allocation in the core of that contains , or in other words, where agent obtains the house of agent . Analogously, the Forbidden Arc in Core problem asks to decide if there exists an allocation in the core of not containing .
By giving a reduction from Acyclic Partition [BokalEtAl-2002], we show in Theorem 1 that both of these problems are computationally intractable, even if agents have a strict ordering over the houses. In fact, we cannot even hope to decide for a given agent in a housing market whether there exists an allocation in the core of where is trading; we call this problem Agent Trading in Core.
Theorem 1 (333Proofs marked by an asterisk can be found in Appendix 0.a. )
Each of the following problems is -complete, even if agents’ preferences are strict orders:
Arc in Core,
Forbidden Arc in Core, and
Agent Trading in Core.
3.3 Maximizing the number of agents trading in a core allocation
Perhaps the most natural optimization problem related to the core of housing markets is the following: given a housing market , find an allocation in the core of whose size, defined as the number of trading agents, is maximal among all allocations in the core of ; we call this the Max Core problem. Max Core is -hard by a result of Cechlárová and Repiský [Cechlarova-Repisky-2011]. In Theorem 2 below we show that even approximating Max Core is -hard. Our result is tight in the following sense: we prove that for any , approximating Max Core with a ratio of is -hard, where is the number of agents in the market. By contrast, a very simple approach yields an approximation with ratio .
We remark that Biró and Cechlárová [Biro-Cechlarova-2007] proved a similar inapproximability result, but since they considered a special model where agents not only care about the house they receive but also about the length of their exchange cycle, their result cannot be translated to our model, and so does not imply Theorem 2. Instead, our reduction relies on the ideas we use to prove Theorem 1.
Theorem 2 ()
For any constant , the Max Core problem is -hard to approximate within a ratio of where is the set of agents, even if agents’ preferences are strict orders.
We contrast Theorem 2 with the observation that an algorithm that outputs any allocation in the core yields an approximation for Max Core with ratio .
Proposition 1 ()
Max Core can be approximated with a ratio of in polynomial time, where is the number of agents in the input.
4 The effect of improvements in housing markets
Let be a housing market containing agents and . We consider a situation where the preferences of are modified by “increasing the value” of for without altering the preferences of over the remaining agents. If the preferences of are given by a strict or weak order, then this translates to shifting the position of in the preference list of towards the top. Formally, a housing market is called a -improvement of , if for any , and is such that (i) iff for any , and (ii) if , then for any . We will also say that a housing market is a -improvement of , if it can be obtained by a sequence of -improvements for a series of agents for some .
To examine how -improvements affect the situation of in the market, one may consider several solution concepts such as the core, the strict core, and so on. We regard a solution concept as a function that assigns a set of allocations to each housing market. Based on the preferences of , we can compare allocations in . Let denote the set containing the best houses can obtain in :
Similarly, let be the set containing the worst houses can obtain in .
Following the notation used by Biró et al. [BKKV-arxiv], we say that respects improvement for the best available house or simply satisfies the RI-best property, if for any housing markets and such that is a -improvement of for some agent , for every and . Similarly, respects improvement for the worst available house or simply satisfies the RI-worst property, if for any housing markets and such that is a -improvement of for some agent , for every and .
Notice that the above definition does not take into account the possibility that a solution concept may become empty as a result of a -improvement. To exclude such a possibility, we may require the condition that an improvement does not destroy all solutions. We say that strongly satisfies the RI-best (or RI-worst) property, if besides satisfying the RI-best (or, respectively, RI-worst) property, it also guarantees that whenever , then also holds where is a -improvement of for some agent .
We prove that the core of housing markets satisfies the RI-best property. In fact, Theorem 1 (proved in Section 4.2) states a slightly stronger statement. By contrast, Proposition 2 shows that the core of housing markets violates the RI-worst property.
Given an allocation in the core of the housing market and a -improvement of , there exists an allocation in the core of such that either or prefers to . Moreover, given , and , it is possible to find such an allocation in polynomial time.
The core of housing markets strongly satisfies the RI-best property.
The core of housing markets violates the RI-worst property.
Let be the set of agents. The preferences indicated in Figure 1 define a housing market and a -improvement of .
We claim that in every allocation in the core of , agent obtains the house of . To see this, let be an allocation where . If agent is not trading in , then and form a blocking cycle; therefore, . Now, if , then and form a blocking cycle for ; otherwise, and form a blocking cycle for . Hence, obtains her top choice in all core allocations of .
However, it is easy to verify that the core of contains an allocation where obtains only her second choice (’s house), as shown in Figure 1.
We describe our algorithm for Theorem 1 in Section 4.1, and prove its correctness in Section 4.2. In Section 4.3 we look at the problem of deciding whether a -improvement leads to a situation strictly better for .
4.1 Description of algorithm HM-Improve
Before describing our algorithm for Theorem 1, we need some notation.
Pre-allocations and their envy graphs. Given a housing market and two distinct agents and in , we say that a set of arcs in is a pre-allocation from to in , if
for each , and
for each .
Note that is a collection of vertex-disjoint cycles and a unique path in , with leading from to . We call the source of and its sink.
Given a pre-allocation from to in , an arc is -augmenting, if and . We define the envy graph of as where is the set of -augmenting arcs in . A blocking cycle for is a cycle in ; notice that such a cycle cannot contain the sink , since no -augmenting arc leaves . We say that the pre-allocation is stable, if no blocking cycle exists for , that is, if its envy graph is acyclic.
We are now ready to propose an algorithm called HM-Improve that given an allocation in the core of outputs an allocation as required by Theorem 1. Observe that we can assume w.l.o.g. that is a -improvement of for some agent , as we can apply such a single-agent version of Theorem 1 repeatedly to obtain the theorem for -improvements involving multiple agents.
Algorithm HM-Improve. First, HM-Improve checks whether belongs to the core of , and if so, outputs . Hence, we may assume that admits a blocking cycle in . Observe such a cycle must contain the arc , as otherwise it would block in as well. This implies that .
HM-Improve proceeds by modifying the housing market: it adds a new agent to , with taking the place of in the preferences of ; the only house that agent prefers to her own will be the house of . Let be the housing market obtained. Then the acceptability graph of can be obtained from the acceptability graph of by subdividing the arc with a new vertex corresponding to agent . Let , and let be the set of arcs in .
Initialization. Let in . Observe that is a pre-allocation from the source to the sink in . Additionally, we define a set of irrelevant agents, initially empty. We may think of irrelevant agents as temporarily deleted from the market.
Iteration. Next, algorithm HM-Improve iteratively modifies the pre-allocation and the set of irrelevant agents. It will maintain the property that is a pre-allocation in ; we denote its envy graph by , having vertex set . While the source of changes during the iteration, the sink remains fixed.
At each iteration, HM-Improve performs the following steps:
Let be the source of . If , then the iteration stops.
Otherwise, if there exists a -augmenting arc in entering (note that ), then let . The algorithm modifies by deleting the arc and adding the arc to . Note that thus becomes a pre-allocation from to in .
Otherwise, no arc in enters ; let . The algorithm adds to the set of irrelevant agents, and modifies by deleting the arc . Again, becomes a pre-allocation from to in .
Output. Let be the pre-allocation at the end of the above iteration, its source, and the set of irrelevant agents. HM-Improve applies the variant of the TTC algorithm described in Section 3.1 to the submarket of when restricted to the set of irrelevant agents. Let denote the obtained allocation in the core of . Then HM-Improve outputs an allocation defined as
4.2 Correctness of algorithm HM-Improve
We begin proving the correctness of algorithm HM-Improve with the following.
At each iteration, pre-allocation is stable in .
The proof is by induction on the number of iterations performed. For , observe that initially for each agent , and by we know that prefers to . Note also that neither nor the arcs and are contained in the envy graph . Thus, a cycle in would be present in the envy graph of in as well. Since is in the core of , it follows that is stable in . Note that initially .
For , assume that the algorithm has performed iterations so far. Let and be as defined at the beginning of the -th iteration, with being the source of , and let and be the pre-allocation and the set of irrelevant agents obtained after the modifications in this iteration. Assume that is stable in , so is acyclic. In case HM-Improve does not stop in Step 1 but modifies and possibly , we distinguish between two cases:
the algorithm modifies in Step 2, by using a -augmenting arc ; then . Note that prefers to , and for any other agent we know . Hence, this modification amounts to deleting all arcs from the envy graph where .
the algorithm modifies in Step 3, by adding the source to the set of irrelevant agents, i.e., . Then for each agent , so the envy graph is obtained from by deleting .
Since deleting some arcs or a vertex from an acyclic graph results in an acyclic graph, the stability of is clear.
We proceed with the observation that an agent’s situation in may only improve, unless it becomes irrelevant: this is a consequence of the fact that the algorithm only deletes arcs and agents from the envy graph .
Let and be two pre-allocations computed by algorithm HM-Improve, with computed at an earlier step than , and let be an agent that is not irrelevant at the end of the iteration when is computed. Then either or prefers to .
We need an additional lemma that will be useful for arguing why irrelevant agents may not become the cause of instability in the housing market.
At the end of algorithm HM-Improve, there does not exist an arc such that , and .
Suppose for contradiction that is such an arc, and let and be as defined at the end of the last iteration. Let us suppose that HM-Improve adds to during the -th iteration, and let be the pre-allocation at the beginning of the -th iteration. By Proposition 3, either or . The assumption yields by the transitivity of . Thus, is a -augmenting arc entering , contradicting our assumption that the algorithm put into in Step 3 of the -th iteration.
The following lemma, the last one necessary to prove Theorem 1, shows that HM-Improve runs in linear time; the proof relies on the fact that in each iteration but the last either an agent or an arc is deleted from the envy graph, thus limiting the number of iterations by .
Lemma 3 ()
Algorithm HM-Improve runs in time.
Proof (of Theorem 1)
By Lemma 3 it suffices to show that algorithm HM-Improve is correct. Let and be the pre-allocation and the set of irrelevant agents, respectively, at the end of algorithm HM-Improve, and let be the source of . To begin, we prove it formally that is an allocation for .
First assume . This means that is the union of disjoint cycles covering each agent in exactly once; note that no arc of enters or leaves . Hence, is an allocation not only in , but also in the submarket of on agent set , i.e., . Second, assume that ; in this case , because can be entered only through . So the arc set is an allocation in . Consequently, is indeed an allocation in in both cases.
Now, let us prove that the allocation is in the core of by showing that the envy graph of is acyclic. First, the subgraph is exactly the envy graph of in and hence is acyclic.
Let and let be an -augmenting arc in . Then is -augmenting as well, i.e., .
Proof (of Claim)
If , then is an arc in , and thus the claim follows immediately from except for the case and ; in this latter case implies that prefers to in as well, that is, is -augmenting.
We finish the proof of the claim by showing that is not -augmenting if . Let be the source of . If , then this is clear by . If , then let us now consider the penultimate iteration in which the source of is moved to either in Step 2 or in Step 3. Recall that the only arc entering is . If became the source of in Step 2, then we know . By the construction of , this means that prefers to in , so is not -augmenting, a contradiction. Finally, if became the source of in Step 3, then we get , which contradicts our assumption .
As a consequence of our claim, we obtain that is a subgraph of and therefore it is acyclic by Lemma 1. Hence, any cycle in must contain agents both in and in (recall that is acyclic as well). However, contains no arcs from to , since such arcs cannot be -augmenting by Lemma 2. Thus is acyclic and is in the core of .
4.3 Strict improvement
Looking at Theorem 1 and Corollary 1, one may wonder whether it is possible to detect efficiently when a -improvement leads to a situation that is strictly better for . For a solution concept and housing markets and such that is a -improvement of for some agent , one may ask the following questions:
Possible Strict Improvement for Best House or PSIB:
is it true that for some and ?
Necessary Strict Improvement for Best House or NSIB:
is it true that for every and ?
Possible Strict Improvement for Worst House or PSIW:
is it true that for some and ?
Necessary Strict Improvement for Worst House or NSIW:
is it true that for every and ?
Focusing on the core of housing markets, it turns out that all of the above four problems are computationally intractable, even in the case of strict preferences.
Theorem 2 ()
With respect to the core of housing markets, PSIB and NSIB are -hard, while PSIW and NSIW are -hard, even if agents’ preferences are strict orders.
5 The effect of improvements in Stable Roommates
In the Stable Roommates problem we are given a set of agents, and a preference relation over for each agent ; the task is to find a stable matching between the agents. A matching is stable if it admits no blocking pair, that is, a pair of agents such that each prefers the other over her partner in the matching. Notice that an input instance for Stable Roommates is in fact a housing market. Viewed from this perspective, a stable matching in a housing market can be thought of as an allocation that (i) contains only cycles of length at most 2, and (ii) does not admit a blocking cycle of length at most 2.
For an instance of Stable Roommates, we assume mutual acceptability, that is, for any two agents and , we assume that holds if and only if holds. Consequently, it will be more convenient to define the acceptability graph of an instance of Stable Roommates as an undirected simple graph where agents and are connected by an edge if and only if they are acceptable to each other and . A matching in is then a set of edges in such that no two of them share an endpoint.
Biró et al. [BKKV-arxiv] have shown the following statements.
Proposition 4 ([BKKV-arxiv])
Stable matchings in the Stable Roommates model
violate the RI-worst property (even if agents’ preferences are strict), and
violate the RI-best property, if agents’ preferences may include ties.
Complementing Proposition 4, we show that a -improvement can lead to an instance where no stable matching exists at all. This may happen even in the case when preferences are strict orders; hence, stable matchings do not strongly satisfy the RI-best property. For an illustration of Propositions 4 and 5 by simple examples see Appendices 0.A.3 and 0.A.4, respectively.
Proposition 5 ()
Stable matchings in the Stable Roommates model do not strongly satisfy the RI-best property, even if agents’ preferences are strict.
Contrasting Propositions 4 and 5, it is somewhat surprising that if agents’ preferences are strict, then the RI-best property holds for the Stable Roommates setting. Thus, the situation of cannot deteriorate as a consequence of a -improvement unless instability arises. The proof of Theorem 1 is provided at the end of this section.
Let be a housing market where agents’ preferences are strict orders. Given a stable matching in and a -improvement of for two agents , either admits no stable matchings at all, or there exists a stable matching in such that . Moreover, given , and it is possible to find such a matching in polynomial time.
Stable matchings in the Stable Roommates model satisfy the RI-best property.
Structural ingredients. To prove Theorem 1 we are going to rely on the concept of proposal-rejection alternating sequences introduced by Tan and Hsueh [Tan-Hsueh-1995], originally used as a tool for finding a stable partition in an incremental fashion by adding agents one-by-one to a Stable Roommates instance. We somewhat tailor their definition to fit our current purposes.
Let be an agent in a housing market , and let be a stable matching in . A sequence of agents is a proposal-rejection alternating sequence starting from , if there exists a sequence of matchings such that for each
is the agent most preferred by among those who prefer to their partner in or are unmatched in ,
is a matching in .
We say that the sequence starts from , and that the matchings are induced by . We say that stops at , if there does not exist an agent fulfilling condition (i) in the above definition for , that is, if no agent prefers to her current partner in and no unmatched agent in finds acceptable. We will also allow a proposal-rejection alternating sequence to take the form , in case conditions (i), (ii), and (iii) hold for each , and is an unmatched agent in satisfying condition (i) for . In this case we define the last matching induced by the sequence as , and we say that the sequence stops at agent .
We summarize the most important properties of proposal-rejection alternating sequences in Lemma 4 as observed and used by Tan and Hsueh.444The first claim of the lemma is only implicit in the paper by Tan and Hsueh [Tan-Hsueh-1995], we prove it for the sake of completeness in Appendix 0.A.4.
Lemma 4 ([Tan-Hsueh-1995] )
Let be a proposal-rejection alternating sequence starting from a stable matching and inducing the matchings in a housing market . Then the following hold.
is a stable matching in for each .
If for some and , then does not admit a stable matching; in such a case we say that sequence has a return.
If the sequence stops at or , then is a stable matching in .
For any agent prefers to .
For any agent prefers to .
Description of algorithm SR-Improve. Let be the stable matching given for the housing market , and let be a -improvement of for two agents and in (recall that unless ). We now propose algorithm SR-Improve that computes a stable matching in with , whenever admits some stable matching.
First, SR-Improve checks whether is stable in , and if so, returns the matching . Otherwise, must be a blocking pair for in .
Second, the algorithm checks whether admits a stable matching and if so, computes any stable matching in using Irving’s algorithm [Irving-SR]; if no stable matching exists for , algorithm SR-Improve stops. Now, if , then SR-Improve returns , otherwise proceeds as follows.
Let be the housing market obtained from by deleting all agents from the preference list of (and vice versa, deleting from the preference list of these agents). Notice that in particular this includes the deletion of as well as of from the preference list of (recall that ).
Let us define and . Notice that is a stable matching in : clearly, any possible blocking pair must contain , but any blocking pair that is blocking in would also block by . Observe also that is unmatched in .
Finally, SR-Improve builds a proposal-rejection alternating sequence of agents in starting from , and inducing matchings until one of the following cases occurs:
: in this case SR-Improve outputs ;
stops: in this case SR-Improve outputs .
Correctness of algorithm SR-Improve. The proof that algorithm SR-Improve is correct relies on the following two facts.
Lemma 5 ()
The sequence cannot have a return. Furthermore, if stops, then it stops at with .
Lemma 6 ()
If SR-Improve outputs a matching , then is stable in and .
Proof (of Theorem 1)
From the description of SR-Improve and Lemma 6 it is immediate that any output the algorithm produces is correct. It remains to show that it does not fail to produce an output. By Lemma 5 we know that the sequence built by the algorithm cannot have a return and can only stop at , implying that SR-Improve will eventually produce an output. Considering the fifth statement of Lemma 4, we also know that the length of is at most . Thus, the algorithm finishes in time.
6 Further research
Even though the property of respecting improvement is important in exchange markets, many solution concepts have not been studied from this aspect. For instance, in the Stable Roommates setting with weakly or partially ordered preferences, do strongly stable matchings satisfy the RI-best property? What about stable half-matchings (or equivalently, stable partitions) in instances of Stable Roommates without a stable matching? Although Appendix 0.A.5 contains an example about stable half-matchings where improvement of an agents’ house damages her situation, perhaps a more careful investigation may shed light on some interesting monotonicity properties.
Appendix 0.A Appendix
We present all proofs missing from Sections 3 and 4 in Sections 0.A.1 and 0.A.2, respectively. We provide examples for Proposition 4 in Section 0.A.3. Section 0.A.4 contains all proofs missing from Section 5. We close the appendix by some notes on the respecting improvement property in relation to stable half-matchings in a Stable Roommates instance in Section 0.A.5.
0.a.1 Missing proofs from Section 3
It is easy to see that all of these problems are in , since given an allocation for , we can check in linear time whether it admits a blocking cycle: taking the envy graph of , we only have to check that it is acyclic, i.e., contains no directed cycles (this can be decided using, e.g., some variant of the depth-first search algorithm).
To prove the -hardness of Arc in Core, we present a polynomial-time reduction from the Acyclic Partition problem: given a directed graph , decide whether it is possible to partition the vertices of into two acyclic sets and . Here, a set of vertices is acyclic, if is acyclic. This problem was proved to be -complete by Bokal et al. [BokalEtAl-2002].
Given our input , we construct a housing market as follows (see Fig. 2 for an illustration). We denote the vertices of by , and we define the set of agents in as
The preferences of the agents’ are as shown below; for each agent we only list those agents whose house finds acceptable. Here, for any set of agents we let denote an arbitrary fixed ordering of .
We finish the construction by defining our instance of Arc in Core as the pair . We claim that there exists an allocation in the core of containing if and only if the vertices of can be partitioned into two acyclic sets.
“”: Let us suppose that there exists an allocation that does not admit any blocking cycles and contains .
We first show that contains every arc for . To see this, observe that the only possible cycle in that contains is the cycle of length 2, because the arc is the only arc going into . Hence, if for some the arc is not in , then the cycle is a blocking cycle. As a consequence, exactly one of the arcs and must be contained in for any , and similarly, exactly one of the arcs and