The purpose of this article is to propose a new solution to the roommate problem with strict references. Therefore, to solve the roommate problem, we propose matchings that lie at the intersection of the maximum irreversible matchings and the maximum stable matchings, which are called Q-stable matchings. The purpose of this article is to propose a new solution to the roommate problem with strict preferences.
Second, the inability to find a stable match and the more complex structure of the roommate problem may have deterred researchers from analyzing it. The goal of this paper is to propose a new solution to the strict preference roommate problem.3 It is essential to require a solution that provides stable matching when dealing with solvable problems and some matching otherwise. Therefore, we believe that the largest irreversible set of pairs should form part of the matching chosen to solve any roommate problem.
Accordingly, we select the set of couplings that lie at the intersection of the two solutions and call them Q-stable couplings. Therefore, we proceed to investigate whether our proposal is one of the elements of an absorbing set. Thus, a stable partition is a partition of the set of agents, such that each set in a stable partition is either a ring, a pair of mutually acceptable individuals, or a singleton, and the partition satisfies the (usual) stability condition between any two individuals. sets and also within each set.8 The following statements are proven by.
Maximum irreversibility
The problem (N,) has no stable matchings if and only if there exists a stable partition with an odd ring. ii) All stable partitions have exactly the same odd rings and singles. iii). All even rings in a stable partition can be partitioned into pairs of mutually acceptable agents without disturbing the stability. The notion of a stable partition plays an important role in this work and is not so easy to interpret.
Therefore, in the appendix we informally describe the algorithm presented by Tan and Hsueh [33] for computing stable patitions and illustrate it with a numerical example that we believe clarifies its meaning. In this section, we first introduce the notion of strong stability, which we believe is appropriate to consider when searching for a match that is as stable as possible. The set of largest irreversible matches is consistent with the core, and the larger the set of irreversible pairs, the more selective this criterion will be.
In the following, we present two central consistent solutions that have been proposed in the relevant literature.
Two previous solution concepts
This is because it is blocked by a group of agents who are better off with a different matching.10 However, Pareto optimality by itself is not a convincing criterion for selecting matchings in the roommate problem for two reasons. First, it requires that when two agents block a matching by forming a new pair, their possible partners must not be worse off in the new matching. However, in our situation it is sufficient if two agents are better off by forming a blocking pair, without considering the welfare of their abandoned partners in the newly formed matching.11 Second, it may select too many matchings: for solvable problems can The optimal Pareto solution is core inclusive, that is, it selects all stable and some unstable matches.
2] study matches with the minimum number of blocking pairs and call them quasi-stable matches. A matchingµ is almost stableif|bp(µ)| ≤ |bp(µ0)|for allµ06=µ, where|bp(µ)|denotes the number of matching blocking pairs µ. Maximum internal stability. A maximally stable matching if it excludes the minimum number of agents such that the non-excluded ones form a stable complete matching see Tan.
Then he defines the problem restricted to the set of non-deleted agents, preserving their original preferences. This new problem is solvable and the computation of a stable fit gives a maximally stable fit. Tan's solution is applied to a setting in which a match is defined as a set of pairs while isolated agents never form part of that match.
To adapt Tan's definition of a maximal consistent matching to our setup, where a matching is a set of disjoint and unique pairs formed by all agents of a given setN, we need to add the deleted agents as single to the maximally stable matching, such that all agents in the problem are part of that matching. All these matches are equally close to stable in the sense that they contain the same number of stable pairs. The reader may have noticed some similarities between P-stable matches and maximally stable matches as defined above.
It turns out that the set of P-stable matchings coincides with the set of maximal internally stable matchings computable by Tan's algorithm. The following result states some features of P-stable matchings obtained from the same stable partition P.
Incompatibilities between solutions
Proposition 1 The intersection of Pareto optimal matches and maximal internally stable matches can be empty. Proposition 3 The intersection of nearly stable matchings and maximally irreversible matchings can be empty. Given the incompatibility between near-stable fits and the other two solutions demonstrated in the previous section, a natural question is to ask whether the intersection of the other three solutions is not empty.
As mentioned in the introduction, our goal in this article is to provide policymakers with a procedure for calculating a Q-stable matching. Then a stable partition with a maximum set of irreversible pairs is derived, from which a Q∗-stable matching is finally obtained. If the problem is solvable, the output of the algorithm is a stable matching and therefore it is immediately the case that it is maximally internally stable and maximally irreversible.
First, we present some claims necessary to show that the algorithm produces a matching with a set of irreversible pairs of maximum size. But then the set of odd rings and singletons would not be the same in P and P0, which contradicts Remark 1(ii). If Q∗-stable matching is maximally irreversible according to claim 3 and maximally internally stable according to remark 3, then the following theorem can be stated.
Theorem 6 A Q∗-stable matching can be calculated in O(mn) time, where nis is the number of agents andm is the total length of the preference lists. The execution of each step takes linear time inm, which is the total length of the preference lists, since a stable partition can be found with Tan's algorithm [31] in O(m) time, and so does the cleanup process in phase 1 can be performed in linear time. Consider, for example, the problem of dividing agents into a fixed number of doubles, or the kidney exchange problem.17 In those cases, we can merge the individual agents from the Q∗-stable matching outcome of the algorithm.
We give an algorithm for computing a matching that lies at the intersection of the set of largest internal stable matchings and the set of largest irreversible stable matchings.
Q-stable matchings and the absorbing sets
In case 1, all matches with at least one pair of agents are Q-stable, but the match µ does not belong to the unique absorption set. Let P∗ be the largest irreversible stable partition (one of such partitions can be obtained in step 2 of the algorithm) and let I be the set of agents that irreversibly match in P∗. Thus V1 is the set of agents that block some matching in M|P∗ with a single odd ring agent P∗ and their subµ partners.18 Note that V16=∅otherwise S∗=I and we are done.
18 By the definition of stable partitioning, no inV agent prefers a single partition piece to its partition partner, and therefore this type of pairs cannot block any stable matching P. 19 Remark 4 can be extended to any matching set such that the agents in the odd rings are paired as in the consistent matching set P and the remaining agents are evenly paired. By Remark 7 and by the definition of the absorbing group there is a consistent correspondence in P to an absorbing group A such that µRTµ∗ but not µ∗RTµ.
In this paper, we have argued that Q-stable matchings are good proposals in a class of strict-preference roommate problems, and we have presented an efficient algorithm for finding one of these matchings. On the other hand, it is easy to check that absorbing sets are included in the set of maximal irreversible correspondences.20 Given the inclusion relation, one may ask why we did not consider the intersection between absorbing sets and the set of maximal internally stable correspondences instead. The reason is that it is not clear whether the results of the algorithm are the only reasonable suggestions.
First, by a similar reasoning as in claim 4, it can be deduced that from any P-stable matching that is not maximally irreversible, there is a sequence of blocking pairs for aP-stable matching that is maximally irreversible. Next, suppose that there is a matching µ in an absorbing set which is not maximally irreversible. Then µ∗RTµ0RTµwhere µ∗is aP-stable matching which is maximally irreversible and µ0is an aP-stable matching of the same absorbing set as µ(possibly µ∗=µ0).
Consider for example the hedonic games, Bogomolnaia and Jackson [6], which constitute an immediate generalization of the roommate problem. This negative result indicates that the roommate problem has a very special structure that makes it difficult to extend the results in this paper to more complex models. Let (Nk,()k) be the bounded problem where Nk ={a1, .., ak}and ()kis the list of agent preferences in Nk in which N\Nk agents are deleted.
All agents involved in the subsequent cycle form an odd ring and a stable partition Pk+1 is constructed for the (Nk+1,()k+1) problem.
