Optimal Stable Matching Research Articles

Application of Gale-Shapley algorithm in optimal matching for healthcare facilities to elderly population: the case of Hangzhou, China

ABSTRACT This paper serves as the first empirical research applying the Nobel Prize winning Gale-Shapley algorithm to study optimal stable matching of healthcare facilities to elderly population living in residential neighbourhoods in main urban districts of Hangzhou, China. Through bilateral considerations of supply side attributes of healthcare facilities and demand side preferences of elderly population, this research shows that the Gale-Shapley matching is superior to unilateral spatial or non-spatial utility matching and can optimize stable matching for elderly population in a way similar to the Pareto optimal. Further, this study provides a conceptual model classifying urban spaces for healthcare services at various levels.

Content Service Oriented Resource Allocation for Space–Air–Ground Integrated 6G Networks: A Three-Sided Cyclic Matching Approach

Since the existing terrestrial fifth generation (5G) network has limited coverage, it is difficult to meet the growing demand for seamless network connection. Meanwhile, current network resource allocation methods mainly research on how to improve system performance only from the perspective of resource utilization, but rarely take users’ specific needs for network content into consideration. This brings severe challenges to efficient network service and flexible resource allocation. Therefore, we construct the content service-oriented resource allocation model for space–air–ground integrated sixth generation networks (SAGIN 6G), and formulate the three-sided matching issue among the space–air–ground integrated network equipment (SAGINE), content sources, and users. In this model, users request to establish connection with SAGIN which forwards users’ request to content service provider (CSP). CSP manages the creation of content data, and finally returns the requested content to users through SAGIN. For the content service-oriented resource allocation in SAGIN, finding the optimal stable three-sided matching with the largest cardinality is an NP-complete problem. Therefore, to efficiently solve the above issue, we design some reasonable restrictions and convert it to a restricted three-sided matching problem with size and cycle preferences. We further develop the content-oriented resource allocation algorithm (COR2A) and the user-oriented resource allocation algorithm (UOR2A) in a distributed manner. Extensive simulations verify our approach outperforms traditional benchmark resource allocation schemes in terms of system throughput, CSP revenue, and user experience.

Improving efficiency in school choice under partial priorities

We provide a school choice model where the student priority orders for schools are allowed not to be total. We introduce a class of algorithms, each of which derives a student optimal stable matching once we have an initial stable matching, when the priorities are represented by partial orders. Since a method to derive a stable matching exists when the priorities are partial orders, we can use it to derive a student optimal stable matching. Moreover, we show that any student optimal stable matchings that Pareto dominate the starting stable one are obtained via an algorithm within this class. For the problem of improving efficiency by allowing some priorities to be violated, the algorithms can also be applied, with a weaker assumption on the violations than in the previous study. Finally, we suggest some specific rules of priorities that can be introduced by weakening the requirement of total ordering.

Stable fractional matchings

We study a generalization of the classical stable matching problem that allows for cardinal preferences (as opposed to ordinal) and fractional matchings (as opposed to integral). In this cardinal setting, stable fractional matchings can have much larger social welfare than stable integral ones. Our goal is to understand the computational complexity of finding an optimal (i.e., welfare-maximizing) stable fractional matching. We consider both exact and approximate stability notions, and provide simple approximation algorithms with weak welfare guarantees. Our main result is that, somewhat surprisingly, achieving better approximations is computationally hard. To the best of our knowledge, these are the first computational complexity results for stable fractional matchings in the cardinal model. En route to these results, we provide a number of structural observations that could be of independent interest.

Matching with myopic and farsighted players

We introduce the new notion of the pairwise myopic-farsighted stable set to study stable matchings under the assumption that players can be both myopic and farsighted. For the special case where all players are myopic, our concept predicts the set of matchings in the core. When all players are farsighted, we provide the characterization of pairwise myopic-farsighted stable sets: a set of matchings is a pairwise myopic-farsighted stable set if and only if it is a singleton consisting of a core element. This result confirms the result obtained by Mauleon et al. (2011) with a completely different effectivity function and provides a new special case where the farsighted stable set is absolutely maximal (Ray and Vohra, 2019) and coincides with the Strong Rational Expectations Farsighted Stable Set (Dutta and Vohra, 2017). When myopic and farsighted players interact, matchings outside the core can be stable and the most farsighted side can achieve its optimal stable matching.

Improving Efficiency in School Choice Under Partial Priorities

We provide a school choice model where the student priority orders are allowed not to be total. We introduce a class of algorithms each of which derive a student optimal stable matching once we have an initial stable matching. Since there is a method to derive a stable matching, we can derive a student optimal stable matching of this model. Moreover, any student optimal stable matchings that Pareto dominate the starting stable one are shown to be obtained via an algorithm within this class. For the problem of improving efficiency by allowing some priorities to be violated, the algorithms can also be applied, with a weaker assumption on the violations than in the previous study.

The Iterative Deferred Acceptance Mechanism

The last few years have seen an increasing use of sequential online mechanisms, instead of the traditional direct counterparts, in college admissions in countries such as Germany, Brazil, and China. We describe those, and identify shortcomings they have in terms of incentives and outcome properties. We introduce a new family of mechanisms for one-sided matching markets, which improve upon these shortcomings. Unlike most mechanisms available in the literature, which ask students for a full preference ranking over all colleges, they are instead sequentially asked to make choices or submit partial rankings from sets of colleges. These are used to produce, in each step, a tentative allocation. If at some point it is determined that a student cannot be accepted into a college anymore, then she is asked to make another choice among those which would tentatively accept her. Participants following the simple strategy of choosing the most preferred college in each step is a robust equilibrium that yields the Student Optimal Stable Matching.

A stable and Pareto efficient update of matching in school choice

We consider a situation where a student–school matching has already been temporarily decided, but unexpectedly the quotas of some schools are increased and/or some new schools enter. Furthermore, the school priority orders are allowed to be weak. We provide a polynomial-time algorithm to derive a student optimal stable matching under the profile of increased quotas that Pareto dominates the stable matching under that of initial quotas.

Hidden Coordination in Matching: The Role of Sequentiality

This paper explores sequential mechanisms for two-sided matching problems: The agents belonging to a side of the market (individuals or institutions) determine an eligibility restriction; and the agents on the other side (institutions or individuals, resp.) select their preferred mates, constrained by the above eligibility.We find out some asymmetries, as well as some coincidences, related to the cases where the first decision is made either by the individuals or the institutions. In particular, for the two mechanisms, stable allocations are decentralized. This can be interpreted as if the sequentiality in which agents decide takes the place of the coordination among the whole society. An 'extra' (hidden) coordination among individuals is exercised only when these agents determine the eligibility restriction. A consequence of such an 'over-coordination' is that the unique allocation supported by any equilibrium is the optimal stable matching from the individuals' point of view. A dual result, implementing the institutions' optimal stable allocation, is not obtained when eligibility is imposed by institutions.

Solving stable matching problems using answer set programming

AbstractSince the introduction of the stable marriage problem (SMP) by Gale and Shapley (1962), several variants and extensions have been investigated. While this variety is useful to widen the application potential, each variant requires a new algorithm for finding the stable matchings. To address this issue, we propose an encoding of the SMP using answer set programming (ASP), which can straightforwardly be adapted and extended to suit the needs of specific applications. The use of ASP also means that we can take advantage of highly efficient off-the-shelf solvers. To illustrate the flexibility of our approach, we show how our ASP encoding naturally allows us to select optimal stable matchings, i.e. matchings that are optimal according to some user-specified criterion. To the best of our knowledge, our encoding offers the first exact implementation to find sex-equal, minimum regret, egalitarian or maximum cardinality stable matchings for SMP instances in which individuals may designate unacceptable partners and ties between preferences are allowed.

Coalitions and cliques in the school choice problem

The school choice mechanism design problem focuses on assignment mechanisms matching students to public schools in a given school district. The well-known Gale Shapley Student Optimal Stable Matching Mechanism (SOSM) is the most efficient stable mechanism proposed so far as a solution to this problem. However its inefficiency is well-documented, and recently the Efficiency Adjusted Deferred Acceptance Mechanism (EADAM) was proposed as a remedy for this weakness. In this note we describe two related adjustments to SOSM with the intention to address the same inefficiency issue. In one we create possibly artificial coalitions among students where some students modify their preference profiles in order to improve the outcome for some other students. Our second approach involves trading cliques among students where those involved improve their assignments by waiving some of their priorities. The coalition method yields the EADAM outcome among other Pareto dominations of the SOSM outcome, while the clique method yields all possible Pareto optimal Pareto dominations of SOSM. The clique method furthermore incorporates a natural solution to the problem of breaking possible ties within preference and priority profiles. We discuss the practical implications and limitations of our approach in the final section of the article.

Optimal Matching Using SMA

Gale and Shapley proposed a GS algorithm to solve Marriage problem. This algorithm results a man optimal or women optimal stable matching. In this paper a new algorithm called SMA is applied to variants of Marriage problem and results are compared with GS algorithm.

A Novel Mutual-Preference-Based Admission Control Mechanism for Hybrid Femtocells System

Femtocells have been assuming an increasingly important role to improve the coverage and capacity of cellular networks over the next few years. Different from macrocellular networks, femtocells, which are run by an operator, can provide three access modes, i.e., open, closed, and hybrid access. As a compromise of both open and closed access mode, hybrid mode is considered the most efficient for both operators and customers. In hybrid access mode, a femtocell normally prioritizes different types of users, and users always prefer to choose the femtocell with the best signal quality for better service and saving power, giving rise to a unique mutual preference. This unique mutual preference relationship between femtocells and users makes the admission control much more challenging, and no current approach considers this mutual preference when hybrid femtocells are accessed by many users. In this paper, we model a mutual selection admission (MSA) problem for hybrid femtocells and users, aiming to find a stable matching, where both the preference of femtocells and that of users are satisfied under a backhaul capacity constraint. We show that the problem of seeking an optimal stable matching with maximum cardinality is NP-hard, and we propose an efficient heuristic algorithm for solving the MSA problem by seeking a matching with different levels of stability. With extensive simulations, we show that the MSA algorithm can always find a stable matching and have a great rate gain compared with hybrid access for a wide range of users and femtocells.

Equilibria under deferred acceptance: Dropping strategies, filled positions, and welfare

We study many-to-one matching markets where hospitals have responsive preferences over students. We study the game induced by the student-optimal stable matching mechanism. We assume that students play their weakly dominant strategy of truth-telling.Roth and Sotomayor (1990) showed that equilibrium outcomes can be unstable. We prove that any stable matching is obtained in some equilibrium. We also show that the exhaustive class of dropping strategies does not necessarily generate the full set of equilibrium outcomes. Finally, we find that the ‘rural hospital theorem’ cannot be extended to the set of equilibrium outcomes and that welfare levels are in general unrelated to the set of stable matchings. Two important consequences are that, contrary to one-to-one matching markets, (a) filled positions depend on the equilibrium that is reached and (b) welfare levels are not bounded by the optimal stable matchings (with respect to the true preferences).

Cyclic stable matching for three-sided networking services

Three-sided relationship is very common in the social and economic area, e.g., the supplier–firm–buyer relationship, kidney exchange problem. The three-sided relationship can also be found in many scenarios of computer networking systems involving three types of agents, which we regard as three-sided networks. For example, in sensor networks, data are retrieved from data sources (sensors) and forwarded to users through a group of servers. In such three-sided networks, users always prefer to receive the best data services from data sources, data sources would choose servers that are more efficient to deliver their data, and servers try to satisfy more users. Such preferences form a specific cyclic relationship and how to optimally allocate network resources to satisfy preferences of all parties becomes a great challenge. In this paper, inspired by the three-sided stable matching, we model the Three-sided Matching with Size and Cyclic preference (TMSC) problem for data sources, servers and users, aiming to find a stable matching for them, where all their preferences are satisfied. TMSC is different from the traditional three-sided matching models, as each server may normally serve more than one users. We show that the problem of seeking an optimal stable matching with maximum cardinality is NP-hard and propose efficient algorithms for the restricted model of TMSC problem to find a stable matching. The effectiveness of the proposed algorithms is validated through extensive simulations.

The Blocking Lemma for a many-to-one matching model

The Blocking Lemma identifies a particular blocking pair for each non-stable and individually rational matching that is preferred by some agents of one side of the market to their optimal stable matching. Its interest lies in the fact that it has been an instrumental result to prove key results on matching. For instance, the fact that in the college admissions problem the workers-optimal stable mechanism is group strategy-proof for the workers and the strong stability theorem in the marriage model follow directly from the Blocking Lemma. However, it is known that the Blocking Lemma and its consequences do not hold in the general many-to-one matching model in which firms have substitutable preference relations. We show that the Blocking Lemma holds for the many-to-one matching model in which firms’ preference relations are, in addition to substitutable, quota q-separable. We also show that the Blocking Lemma holds on a subset of substitutable preference profiles if and only if the workers-optimal stable mechanism is group strategy-proof for the workers on this subset of profiles.

The Cooperative Theory of Two Sided Matching Problems: A Re-examination of Some Results

We show that, given two matchings of which say the second is stable, if (a) no firm prefers the first matching to the second, and (b) no firm and the worker it is paired with under the second matching prefer each other to their respective assignments in the first matching, then no worker prefers the second matching to the first. This result is a strengthening of a result originally due to Knuth (1976). A theorem due to Roth and Sotomayor (1990), says that if the number of workers increases, then there is a non-empty subset of firms and the set of workers they are assigned to under the F - optimal stable matching, such that given any stable matching for the old two-sided matching problem and any stable matching for the new one, every firm in the set prefers the new matching to the old one and every worker in the set prefers the old matching to the new one. We provide a new proof of this result using mathematical induction. This result requires the use of a theorem due to Gale and Sotomayor (1985 a,b), which says that with more workers around, firms prefer the new optimal stable matchings to the corresponding ones of the old two-sided matching problem, while the opposite is true for workers. We provide an alternative proof of the Gale and Sotomayor theorem, based directly on the deferred acceptance procedure.

An algorithm to compute the full set of many-to-many stable matchings

The paper proposes an algorithm to compute the full set of many-to-many stable matchings when agents have substitutable preferences. The algorithm starts by calculating the two optimal stable matchings using the deferred-acceptance algorithm. Then, it computes each remaining stable matching as the firm-optimal stable matching corresponding to a new preference profile, which is obtained after modifying the preferences of a previously identified sequence of firms.

On the invariance of male optimal stable matching

The stable matching problem is that of matching two sets of agents in such a manner that no two unmatched agents prefer each other to their actual partners under the matching. In this paper, we present a set of sufficient conditions on the preference lists of any given stable matching instance, under which the optimality of the original male optimal stable matching is still preserved.

The necessary and sufficient condition for the worst-case male optimal stable matching

Gale and Shapley [l] introduced and solved the stable matching problem. That problem involves two disjoint sets of equal cardinality n, the men and the women. Each person ranks all members of the opposite sex in order of preference. A stable matching is defined as a complete matching between men and women with the property that no man and woman who are not partners both prefer each other to their actual partners under the matching. Several stable matching algorithms [l-3, S-101 were proposed to solve the problem by returning the male optimal stable solution as its answer. Any sequential one of those algorithms will be called a stable matching algorithm in this paper. The purpose of this paper is to consider the worst-case choice for the sequential stable matching problem. This problem has been investigated by some researchers since the early paper of Wilson [ll]. Itoga [5] presented some conclusions about the nature of the worst-case situation. Tseng and Lee [lo] gave a necessary condition for the worst-case execution of the stable matching problem which takes the maximum number of proposals for the McVitie-Wilson’s algorithm [8,9]. Kapur and Krishnamoorthy [6] presented a worst-case choice which takes the maximum number of stages for Gale-Shapley’s algorithm [l]. In this paper we give the necessary and sufficient condition for the worst-case execution, which leads the sequential stable matching algorithm to take the maximum number of proposals. We then point out that the probability that the worst-case execution occurs when a sequential stable matching algorithm is employed is extremely small.