Preliminary Definitions, Results and Motivation Stable Matching Problems: The Stable Marriage Problem: An Update SM and HR with Indifference The Stable Roommates Problem Further Stable Matching Problems Other Optimal Matching Problems: Pareto Optimal Matchings Popular Matchings Profile-Based Optimal Matchings.

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Algorithmics of Matching Under Preferences

The deferred acceptance algorithm proposed by Gale and Shapley (1962) has had a profound influence on market design, both directly, by being adapted into practical matching mechanisms, and, indirectly, by raising new theoretical questions. Deferred acceptance algorithms are at the basis of a number of labor market clearinghouses around the world, and have recently been implemented in school choice systems in Boston and New York City. In addition, the study of markets that have failed in ways that can be fixed with centralized mechanisms has led to a deeper understanding of some of the tasks a marketplace needs to accomplish to perform well. In particular, marketplaces work well when they provide thickness to the market, help it deal with the congestion that thickness can bring, and make it safe for participants to act effectively on their preferences. Centralized clearinghouses organized around the deferred acceptance algorithm can have these properties, and this has sometimes allowed failed markets to be reorganized.

Deferred Acceptance Algorithms: History, Theory, Practice, and Open Questions

A survey of the literature on matching market design regarding house allocation, kidney exchange, and school choice problems.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1860278_code286615.pdf?abstractid=1311517

Matching, Allocation, and Exchange of Discrete Resources

Many-to-one matching markets exist in numerous different forms, such as college admissions, matching medical interns to hospitals for residencies, assigning housing to college students, and the classic firms and workers market. In all these markets, externalities such as complementarities and peer effects severely complicate the preference ordering of each agent. Further, research has shown that externalities lead to serious problems for market stability and for developing efficient algorithms to find stable matchings. In this paper we make the observation that peer effects are often the result of underlying social connections, and we explore a formulation of the many-to-one matching market where peer effects are derived from an underlying social network. The key feature of our model is that it captures peer effects and complementarities using utility functions, rather than traditional preference ordering. With this model and considering a weaker notion of stability, namely twosided exchange stability, we prove that stable matchings always exist and characterize the set of stable matchings in terms of social welfare. To characterize the efficiency of matching markets with externalities, we provide general bounds on how far the welfare of the worst-case stable matching can be from the welfare of the optimal matching, and find that the structure of the social network (e.g. how well clustered the network is) plays a large role.

/pdf/peer-effects-and-stability-in-matching-markets-4ipghh50su.pdf

Peer effects and stability in matching markets

Fair assignment of indivisible objects under ordinal preferences

We study Pareto optimal matchings in the context of house allocation problems. We present an $O(\sqrt{n}m)$ algorithm, based on Gale's Top Trading Cycles Method, for finding a maximum cardinality Pareto optimal matching, where n is the number of agents and m is the total length of the preference lists. By contrast, we show that the problem of finding a minimum cardinality Pareto optimal matching is NP-hard, though approximable within a factor of 2. We then show that there exist Pareto optimal matchings of all sizes between a minimum and maximum cardinality Pareto optimal matching. Finally, we introduce the concept of a signature, which allows us to give a characterization, checkable in linear time, of instances that admit a unique Pareto optimal matching.

/pdf/pareto-optimality-in-house-allocation-problems-41b6lydgsm.pdf

Pareto optimality in house allocation problems

The exchange-stable marriage problem

The model of a housing market, introduced by Shapley and Scarf in 1974 [14], captures a fundamental situation in an economy where each agent owns exactly one unit of some indivisible good: a house. We focus on an extension of this model where duplicate houses may exist. As opposed to the classical setting, the existence of an economical equilibrium is no longer ensured in this case. Here, we study the deficiency of housing markets with duplicate houses, a notion measuring how close a market can get to an economic equilibrium. We investigate the complexity of computing the deficiency of a market, both in the classical sense and also in the context of parameterized complexity.

Computing the Deficiency of Housing Markets with Duplicate Houses

Stable matchings of teachers to schools

Several countries successfully use centralized matching schemes for school or higher education assignment, or for entry-level labour markets. In this paper we explore the computational aspects of a possible similar scheme for assigning teachers to schools. Our model is motivated by a particular characteristic of the education system in many countries where each teacher specializes in two subjects. We seek stable matchings, which ensure that no teacher and school have the incentive to deviate from their assignments. Indeed we propose two stability definitions depending on the precise format of schools' preferences. If the schools' ranking of applicants is independent of their subjects of specialism, we show that the problem of deciding whether a stable matching exists is NP-complete, even if there are only three subjects, unless there are master lists of applicants or of schools. By contrast, if the schools may order applicants differently in each of their specialization subjects, the problem of deciding whether a stable matching exists is NP-complete even in the presence of subject-specific master lists plus a master list of schools. Finally, we prove a strong inapproximability result for the problem of finding a matching with the minimum number of blocking pairs with respect to both stability definitions.

Katarína Cechlárová

Papers

Pareto optimality in house allocation problems

The exchange-stable marriage problem

Computing the Deficiency of Housing Markets with Duplicate Houses

Stable matchings of teachers to schools

Stable matchings of teachers to schools