Matching under Preferences

Matching theory studies how agents and/or objects from different sets can be matched with each other while taking agents\u2019 preferences into account. The theory originated in 1962 with a celebrated paper by David Gale and Lloyd Shapley (1962), in which they proposed the Stable Marriage Algorithm as a solution to the problem of two-sided matching. Since then, this theory has been successfully applied to many real-world problems such as matching students to universities, doctors to hospitals, kidney transplant patients to donors, and tenants to houses. This chapter will focus on algorithmic as well as strategic issues of matching theory. Many large-scale centralized allocation processes can be modelled by matching problems where agents have preferences over one another. For example, in China, over 10 million students apply for admission to higher education annually through a centralized process. The inputs to the matching scheme include the students\u2019 preferences over universities, and vice versa, and the capacities of each university. The task is to construct a matching that is in some sense optimal with respect to these inputs. Economists have long understood the problems with decentralized matching markets, which can suffer from such undesirable properties as unravelling, congestion and exploding offers (see Roth and Xing, 1994, for details). For centralized markets, constructing allocations by hand for large problem instances is clearly infeasible. Thus centralized mechanisms are required for automating the allocation process. Given the large number of agents typically involved, the computational efficiency of a mechanism's underlying algorithm is of paramount importance. Thus we seek polynomial-time algorithms for the underlying matching problems. Equally important are considerations of strategy: an agent (or a coalition of agents) may manipulate their input to the matching scheme (e.g., by misrepresenting their true preferences or underreporting their capacity) in order to try to improve their outcome. A desirable property of a mechanism is strategyproofness, which ensures that it is in the best interests of an agent to behave truthfully

Envy-free Matchings with Lower Quotas

While every instance of the Hospitals/Residents problem admits a stable matching, the problem with lower quotas (HR-LQ) has instances with no stable matching. For such an instance, we expect the existence of an envy-free matching, which is a relaxation of a stable matching preserving a kind of fairness property. In this paper, we investigate the existence of an envy-free matching in several settings, in which hospitals have lower quotas. We first provide an algorithm that decides whether a given HR-LQ instance has an envy-free matching or not. Then, we consider envy-freeness in the Classified Stable Matching model due to Huang (2010), i.e., each hospital has lower and upper quotas on subsets of doctors. We show that, for this model, deciding the existence of an envy-free matching is NP-hard in general, but solvable in polynomial time if quotas are paramodular

Local search for stable marriage problems

The stable marriage (SM) problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools, or more generally to any two-sided market. In the classical formulation, n men and n women express their preferences (via a strict total order) over the members of the other sex. Solving a SM problem means finding a stable marriage where stability is an envy-free notion: no man and woman who are not married to each other would both prefer each other to their partners or to being single. We consider both the classical stable marriage problem and one of its useful variations (denoted SMTI) where the men and women express their preferences in the form of an incomplete preference list with ties over a subset of the members of the other sex. Matchings are permitted only with people who appear in these lists, an we try to find a stable matching that marries as many people as possible. Whilst the SM problem is polynomial to solve, the SMTI problem is NP-hard. We propose to tackle both problems via a local search approach, which exploits properties of the problems to reduce the size of the neighborhood and to make local moves efficiently. We evaluate empirically our algorithm for SM problems by measuring its runtime behaviour and its ability to sample the lattice of all possible stable marriages. We evaluate our algorithm for SMTI problems in terms of both its runtime behaviour and its ability to find a maximum cardinality stable marriage.For SM problems, the number of steps of our algorithm grows only as O(nlog(n)), and that it samples very well the set of all stable marriages. It is thus a fair and efficient approach to generate stable marriages.Furthermore, our approach for SMTI problems is able to solve large problems, quickly returning stable matchings of large and often optimal size despite the NP-hardness of this problem.Comment: 12 pages, Proc. COMSOC 2010 (Third International Workshop on Computational Social Choice

Local search for stable marriage problems with ties and incomplete lists

The stable marriage problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools, or more generally to any two-sided market. We consider a useful variation of the stable marriage problem, where the men and women express their preferences using a preference list with ties over a subset of the members of the other sex. Matchings are permitted only with people who appear in these preference lists. In this setting, we study the problem of finding a stable matching that marries as many people as possible. Stability is an envy-free notion: no man and woman who are not married to each other would both prefer each other to their partners or to being single. This problem is NP-hard. We tackle this problem using local search, exploiting properties of the problem to reduce the size of the neighborhood and to make local moves efficiently. Experimental results show that this approach is able to solve large problems, quickly returning stable matchings of large and often optimal size.Comment: 12 pages, Proc. PRICAI 2010 (11th Pacific Rim International Conference on Artificial Intelligence), Byoung-Tak Zhang and Mehmet A. Orgun eds., Springer LNA

Dynamic capacities and priorities in stable matching

Cette thèse aborde les facettes dynamiques des principes fondamentaux du problème de l'appariement stable plusieurs-à-un. Nous menons notre étude dans le contexte du choix de l'école et de l'appariement entre les hôpitaux et les résidents. Dans la première étude, en utilisant le modèle résident-hôpital, nous étudions la complexité de calcul de l'optimisation des variations de capacité des hôpitaux afin de maximiser les résultats pour les résidents, tout en respectant les contraintes de stabilité et de budget. Nos résultats révèlent que le problème de décision est NP-complet et que le problème d'optimisation est inapproximable, même dans le cas de préférences strictes et d'allocations de capacités disjointes. Ces résultats posent des défis importants aux décideurs qui cherchent des solutions efficaces aux problèmes urgents du monde réel. Dans la seconde étude, en utilisant le modèle du choix de l'école, nous explorons l'optimisation conjointe de l'augmentation des capacités scolaires et de la réalisation d'appariements stables optimaux pour les étudiants au sein d'un marché élargi. Nous concevons une formulation innovante de programmation mathématique qui modélise la stabilité et l'expansion des capacités, et nous développons une méthode efficace de plan de coupe pour la résoudre. Des données réelles issues du système chilien de choix d'école valident l'impact potentiel de la planification de la capacité dans des conditions de stabilité. Dans la troisième étude, nous nous penchons sur la stabilité de l'appariement dans le cadre de priorités dynamiques, en nous concentrant principalement sur le choix de l'école. Nous introduisons un modèle qui tient compte des priorités des frères et sœurs, ce qui nécessite de nouveaux concepts de stabilité. Notre recherche identifie des scénarios où des appariements stables existent, accompagnés de mécanismes en temps polynomial pour leur découverte. Cependant, dans certains cas, nous prouvons également que la recherche d'un appariement stable de cardinalité maximale est NP-difficile sous des priorités dynamiques, ce qui met en lumière les défis liés à ces problèmes d'appariement. Collectivement, cette recherche contribue à une meilleure compréhension des capacités et des priorités dynamiques dans les scénarios d'appariement stable et ouvre de nouvelles questions et de nouvelles voies pour relever les défis d'allocation complexes dans le monde réel.This research addresses the dynamic facets in the fundamentals of the many-to-one stable matching problem. We conduct our study in the context of school choice and hospital-resident matching. In the first study, using the resident-hospital model, we investigate the computational complexity of optimizing hospital capacity variations to maximize resident outcomes, while respecting stability and budget constraints. Our findings reveal the NP-completeness of the decision problem and the inapproximability of the optimization problem, even under strict preferences and disjoint capacity allocations. These results pose significant challenges for policymakers seeking efficient solutions to pressing real-world issues. In the second study, using the school choice model, we explore the joint optimization of increasing school capacities and achieving student-optimal stable matchings within an expanded market. We devise an innovative mathematical programming formulation that models stability and capacity expansion, and we develop an effective cutting-plane method to solve it. Real-world data from the Chilean school choice system validates the potential impact of capacity planning under stability conditions. In the third study, we delve into stable matching under dynamic priorities, primarily focusing on school choice. We introduce a model that accounts for sibling priorities, necessitating novel stability concepts. Our research identifies scenarios where stable matchings exist, accompanied by polynomial-time mechanisms for their discovery. However, in some cases, we also prove the NP-hardness of finding a maximum cardinality stable matching under dynamic priorities, shedding light on challenges related to these matching problems. Collectively, this research contributes to a deeper understanding of dynamic capacities and priorities within stable matching scenarios and opens new questions and new avenues for tackling complex allocation challenges in real-world settings

Constrained School Choice

Recently, several school districts in the US have adopted or consider adopting the Student-Optimal Stable mechanism or the Top Trading Cycles mechanism to assign children to public schools. There is evidence that for school districts that employ (variants of) the so-called Boston mechanism the transition would lead to efficiency gains. The first two mechanisms are strategy-proof, but in practice student assignment procedures typically impede a student to submit a preference list that contains all his acceptable schools. We study the preference revelation game where students can only declare up to a fixed number of schools to be acceptable. We focus on the stability and efficiency of the Nash equilibrium outcomes. Our main results identify rather stringent necessary and sufficient conditions on the priorities to guarantee stability or efficiency of either of the two mechanisms. This stands in sharp contrast with the Boston mechanism which has been abandoned in many US school districts but nevertheless yields stable Nash equilibrium outcomes.school choice, matching, stability, Gale-Shapley deferred acceptance algorithm, top trading cycles, Boston mechanism, acyclic priority structure, truncation