Existence Of Stable Matchings Research Articles

Not all is lost: Sorting and self-stabilizing sets

It is well known that in the absence of transfers, stable matchings are positive assortative when agents' preferences are strictly monotonic in each other's types and types are commonly known. Instead, monotonicity is consistent with the existence of stable matchings that exhibit negative sorting when types on one side of the market are private information. This paper sheds light on the scope and meaning of this consistency by showing that within familiar monotonic markets we cannot conclude that a stable matching is negative, and not positive, assortative.

Matching with Externalities

AbstractWe incorporate externalities into the stable matching theory of two-sided markets. Extending the classical substitutes condition to markets with externalities, we establish that stable matchings exist when agent choices satisfy substitutability. We show that substitutability is a necessary condition for the existence of a stable matching in a maximal-domain sense and provide a characterization of substitutable choice functions. In addition, we extend the standard insights of matching theory, like the existence of side-optimal stable matchings and the deferred acceptance algorithm, to settings with externalities even though the standard fixed-point techniques do not apply.

Minimal instances with no weakly stable matching for three-sided problem with cyclic incomplete preferences

Given n men, n women, and n dogs, each man has an incomplete preference list of women, each woman has an incomplete preference list of dogs, and each dog has an incomplete preference list of men. We understand a family as a triple consisting of one man, one woman, and one dog such that the dog belongs to the preference list of the woman, who, in turn, belongs to the preference list of the man, while the latter belongs to the preference list of the dog. We understand a matching as a collection of nonintersecting families (some agents, possibly, remain single). A matching is said to be nonstable, if one can find a man, a woman, and a dog who do not live together currently but each of them would become “happier” if they do. Otherwise, the matching is said to be stable (a weakly stable matching). We give an example of this problem for [Formula: see text] where no stable matching exists. Moreover, we prove the absence of such an example for [Formula: see text]. Such an example was known earlier only for [Formula: see text] [P. Biró and E. McDermid, Three-sided stable matchings with cyclic preferences, Algorithmica 58 (2010) 5–18]. The constructed examples also allow one to halve the size of the recently constructed analogous example for complete preference lists [C.-K. Lam and C.G. Plaxton, On the existence of three-dimensional stable matchings with cyclic preferences, in Algorithmic Game Theory, Lecture Notes in Computer Science, Vol. 11801 (Springer, 2019), pp. 329–342].

Stability in Matching with Couples Having Non-Responsive Preferences

The paper studies matching markets where institutions are matched with possibly more than one individual. The matching market contains some couples who view the pair of jobs as complements. We specify that the couples have a weak'' preference to be matched together. We first assume that the institutions have common preference over all the individuals. We then characterize under which weak preferences of couples a stable matching exists. We then impose further conditions on the common preference of institutions over the individuals and prove existence of stable matching for unrestricted couple preferences. Finally, we establish a result on stability by relaxing the condition on common preference of institutions over individuals and assuming different preferences for different institutions.

Matching with ownership

We consider a hybrid model at the intersection of the standard two-sided matching market as proposed by Gale and Shapley (1962) and a housing market as proposed by Shapley and Scarf (1974). Two sets of agents have to be matched in pairs to a common set of objects. Agents of one type have preferences that depend on not only the object they are matched to but also the agent of the other type matched to this object. The crucial difference lies in the fact that the common side is interpreted as an object and has no intrinsic preferences over the agents matched to it. We introduce a natural definition of the ownership of the objects that determines which agent owns the object he is matched to. Ownership restricts the objections of agents who are not owners and defines a notion of stability. We consider two natural ownership structures and show that stable matchings exist in both structures. The first ownership structure, i.e., one-side ownership, always gives ownership to agents of the same side. Even if this structure shares similarities with the classical two-sided matching framework, we show the following important difference: stable matchings and Pareto-efficient matchings can be disjoints, implying that the core can be empty. We also propose two subdomains of preferences, i.e., lexicographic and couple preferences, where core matchings exist in one-sided ownership structures. The second notion is joint ownership, where any reallocation of objects must be jointly agreed upon by the two agents initially assigned to them. As discussed in Morrill (2010), this notion is equivalent to Pareto-efficient matchings, and we discuss possible algorithms that can be used to check Pareto efficiency. Finally, we propose a general definition of ownership structures and show that one-sided ownerships are not the only ones that can guarantee the existence of stable matchings. To further investigate the link with the housing market literature, we also introduce an initial allocation to objects and define a core notion with respect to this initial allocation. We also show that in contrast to the standard setting, this housing market core can be empty. However, we show that in this housing market framework, there always exists a Pareto-efficient matching that is not blocked by any coalition of size two. In both settings, pairwise stability is the only minimal requirement that one can ensure.

Matching with Externalities

We incorporate externalities into the stable matching theory of two-sided markets. Extending the classical substitutes condition to allow for externalities, we establish that stable matchings exist when agent choices satisfy substitutability. Furthermore, we show that substitutability is a necessary condition for the existence of a stable matching in a maximal-domain sense and provide a characterization of substitutable choice functions. In addition, we establish novel comparative statics on externalities and show that the standard insights of matching theory, like the existence of side-optimal stable matchings and the deferred acceptance algorithm, remain valid despite the presence of externalities even though the standard fixed-point techniques do not apply.

Matching with floor constraints

Floor constraints are a prominent feature of many matching markets, such as medical residency, teacher assignment, and military cadet matching. We develop a theory of matching markets under floor constraints. We introduce a stability notion, which we call floor respecting stability, for markets in which (hard) floor constraints must be respected. A matching is floor respecting stable if there is no coalition of doctors and hospitals that can propose an alternative matching that is feasible and an improvement for its members. Our stability notion imposes the additional condition that a coalition cannot reassign a doctor outside the coalition to another hospital (although she can be fired). This condition is necessary to guarantee the existence of stable matchings. We provide a mechanism that is strategy‐proof for doctors and implements a floor respecting stable matching.

Matching with Subjective and Objective Evaluations

There are matching markets in which objective evaluations such as exam scores, skill qualifications, and priorities, are available in addition to subjective evaluations over agents. To examine these situations, we extend a college admission model by allowing that colleges have two different types of ordinal rankings over students, i.e., common priority order and individual preferences. A matching is called double stable if it is both priority stable and preference stable. While the existence of a double stable matching is not always guaranteed, we provide its characterization through the existing well-known mechanisms in the literature; a double stable matching exists if and only if the resulting outcome of the serial dictatorship mechanism coincides with that of the student-proposing deferred acceptance mechanism.

Matching with Budget Constraints

We study a matching model with salaries in which firms face budget constraints. Due to budget constraints, the existence of a stable matching is not guaranteed. We show that if workers are homogeneous, from the firms' point of view, then a weak stable matching always exists; furthermore, when a strong stable matching does not exist, there is a nearby budget vector for firms such that a strong stable matching exists for the problem with perturbed budgets. On the flip side, if firms are homogeneous from the workers' point of view, a stable matching may not exist; however, one can reach a stable matching by changing the budget of firms where the total budget remains the same and each firm's budget change is bounded by the value of at most one worker to that firm.

The roommate problem with externalities

This paper extends the roommate problem to include externalities, allowing preferences for a partner to depend on the situation of others. Stability concepts for matchings and partitions of the set of agents are proposed and characterized, conditional on all agents having prudent expectations about other agents’ reactions to deviations. We prove that any roommate problem with externalities has a stable partition and that a stable matching exists if there is a stable partition without odd rings. These results allow us to find restrictions on the space of preferences ensuring the existence of a stable matching. We also show that some classical properties are lost in the presence of externalities: the existence of paths to stability from any unstable matching, the coincidence of the core with the set of stable matchings, and the invariance of the set of agents who are alone in a stable matching.

Matching with Floor Constraints

Floor constraints are a prominent feature of many matching markets, such as medical residency, teacher assignment and military cadet matching.We develop a theory of matching markets under floor constraints. We introduce a stability notion, which we call floor respecting stability, for markets in which (hard) floor constraints must be respected. A matching is floor respecting stable if there is no coalition of doctors and hospitals that can propose an alternative matching that is feasible and an improvement for its members. Our stability notion imposes the additional condition that a coalition cannot reassign a doctor outside the coalition to another hospital (although she can be fired). This condition is necessary to guarantee the existence of stable matchings. We provide a mechanism that is strategy-proof for doctors and implements a floor respecting stable matching.

Stable Matching with Double Infinity of Workers and Firms

Abstract In this paper we analyze the existence of stable matchings in a two-sided large market in which workers are assigned to firms. The market has a continuum of workers while the set of firms is countably infinite. We show that, under certain reasonable assumptions on the preference correspondences, stable matchings not only exist but are also Pareto optimal.

Stable Matching in Large Economies

We study stability of two‐sided many‐to‐one matching in which firms' preferences for workers may exhibit complementarities. Although such preferences are known to jeopardize stability in a finite market, we show that a stable matching exists in a large market with a continuum of workers, provided that each firm's choice is convex and changes continuously as the set of available workers changes. We also study the existence and structure of stable matchings under preferences exhibiting substitutability and indifferences in a large market. Building on these results, we show that an approximately stable matching exists in large finite economies. We extend our framework to ensure a stable matching with desirable incentive and fairness properties in the presence of indifferences in firms' preferences.

Stability in Matching With Couples Having Non-Responsive Preferences

We study many-to-one matching problems between institutions and individuals where an institution can possibly be matched to more than one individual. The matching market contains some couples, who view pairs of jobs as complements. Institutions' preferences over sets of individuals are assumed to satisfy responsiveness. However, couples' preferences over pairs of institutions are allowed to violate responsiveness. In this setting, first, we assume that institutions have a common preference over the individuals, and (i) we provide a complete characterization of all preferences of couples such that a stable matching exists under the additional assumption that couples violate responsiveness in order to be matched at the same institution, and (ii) we provide a necessary and sufficient condition on the common preference of institutions so that a stable matching exists when couples can violate responsiveness in an arbitrary manner. Finally, we relax the common preference assumption on institutions' preferences and provide a sufficient condition on the same for the existence of a stable matching.

Roommate Problems with Externalities

This paper extends the roommate problem to include externalities, allowing preferences for a partner to depend on the situation of the others. Assuming that everyone has prudent expectations about other agents' reactions to deviations, stability concepts for matchings and partitions of the set of agents are proposed and characterized. We prove that any roommate problem with externalities has a stable partition and that a stable matching exists if there is a stable partition without odd rings. These results allow us to find restrictions on the space of preferences ensuring the existence of a stable matching. We also show that some classical properties are lost in the presence of externalities: the existence of paths to stability from any unstable matching, the coincidence of the core with the set of stable matchings, and the invariance of the set of agents who are alone in a stable matching.

Matching with contracts, substitutes and two-unit demand

In the context of many-to-one matching with contracts, we show that for any choice function that satisfies the irrelevance of rejected contracts condition (Aygün and Sönmez, 2013) and selects at most two contracts from any given set of contracts (two-unit demand), bilateral substitutability and weak substitutability are equivalent. As a corollary, we obtain a new maximal domain for the existence of stable matchings in the unit-capacity couples model. Finally, we show with an example that the equivalence between bilateral and weak substitutability crucially depends on the two-unit demand condition.

On sufficient conditions for the existence of stable matchings with contracts

We introduce two new sufficient conditions for the existence of stable outcomes in many-to-one matching with contracts. The conditions subsume the observable substitutability of Hatfield et al. (2015) and the substitutable completability of Hatfield and Kominers (2016) as special cases. We also prove that unilaterally substitutability and irrelevance of rejected contracts imply substitutable completability.

Stable Marriage with General Preferences

We propose a generalization of the classical stable marriage problem. In our model, the preferences on one side of the partition are given in terms of arbitrary binary relations, which need not be transitive nor acyclic. This generalization is practically well-motivated, and as we show, encompasses the well studied hard variant of stable marriage where preferences are allowed to have ties and to be incomplete. As a result, we prove that deciding the existence of a stable matching in our model is NP-complete. Complementing this negative result we present a polynomial-time algorithm for the above decision problem in a significant class of instances where the preferences are asymmetric. We also present a linear programming formulation whose feasibility fully characterizes the existence of stable matchings in this special case. Finally, we use our model to study a long standing open problem regarding the existence of cyclic 3D stable matchings. In particular, we prove that the problem of deciding whether a fixed 2D perfect matching can be extended to a 3D stable matching is NP-complete, showing this way that a natural attempt to resolve the existence (or not) of 3D stable matchings is bound to fail.

Many-to-one matchings without substitutability

We study the existence of stable matchings in the many-to-one college admission problem when there are no restrictions on college preferences. We show that the existence of a stable allocation is strongly tied to students having “harmonious preferences” over their sets of acceptable college choices. In other words, without any assumption on college preferences, a stable matching exists for any college admission problem if and only if there is no subset of students with misaligned college rankings.

Matching with aggregate externalities

Certain aggregate externalities, like those due to knowledge and public goods, do not change very much in response to changes in two individuals’ actions. Thus, individuals rationally regard the level of the externality as fixed in their negotiations with each other. We leverage this observation to develop a general framework for the existence of stable matchings in moderately sized one-to-one matching games, and we characterize intuitive restrictions on preferences that are sufficient for existence