Set Of Stable Matchings Research Articles

On the Set of Stable Matchings in a Bipartite Graph

The topic of stable matchings (marriages) in bipartite graphs gained popularity beginning from the appearance of the classical Gale and Shapley work. In this paper, a detailed review of selected and other related statements in this field that describe structured, polyhedral, and algorithmic properties of such objects and their sets accompanied by short proofs is given.

On the Set of Stable Matchings in a Bipartite Graph

On the Set of Stable Matchings in a Bipartite Graph

Unique stable matchings

In this paper we consider the issue of a unique prediction in one-to-one two-sided matching markets, as defined by Gale and Shapley (1962), and we prove the following: TheoremLet P be a one-to-one two-sided matching market and letP⁎be its associated normal form, a (weakly) smaller matching market with the same set of stable matchings that can be obtained using procedures introduced inIrving and Leather (1986)andBalinski and Ratier (1997). The following three statements are equivalent:(a)P has a unique stable matching.(b)Preferences onP⁎are acyclic, as defined byChung (2000).(c)InP⁎every market participant's preference list is a singleton.

Stable Matchings with Restricted Preferences: Structure and Complexity

In the stable marriage (SM) problem, there are two sets of agents—traditionally referred to as men and women—and each agent has a preference list that ranks (a subset of) agents of the opposite sex. The goal is to find a matching between men and women that is stable in the sense that no man-woman pair mutually prefers each other to their assigned partners. In a seminal work, Gale and Shapley [ 16 ] showed that stable matchings always exist and described an efficient algorithm for finding one. Irving and Leather [ 24 ] defined the rotation poset of an SM instance and showed that it determines the structure of the set of stable matchings of the instance. They further showed that every finite poset can be realized as the rotation poset of some SM instance. Consequently, many problems—such as counting stable matchings and finding certain “fair” stable matchings—are computationally intractable (NP-hard) in general. In this article, we consider SM instances in which certain restrictions are placed on the preference lists. We show that three natural preference models— k -bounded, k -attribute, and (k 1 , k 2 ) -list—can realize arbitrary rotation posets for constant values of k . Hence, even in these highly restricted preference models, many stable matching problems remain intractable. In contrast, we show that for any fixed constant k , the rotation posets of k -range instances are highly restricted. As a consequence, we show that exactly counting and uniformly sampling stable matchings, finding median, sex-equal, and balanced stable matchings, are fixed-parameter tractable when parameterized by the range of the instance. Thus, these problems can be solved in polynomial time on instances of the k-range model for any fixed constant k.

The lattice of worker-quasi-stable matchings

In a many-to-one matching model, we study the set of worker-quasi-stable matchings when firms' choice functions satisfy substitutability. Worker-quasi-stability is a relaxation of stability that allows blocking pairs involving a firm and an unemployed worker. We show that this set has a lattice structure and define a Tarski operator on this lattice that models a re-equilibration process and has the set of stable matchings as its fixed points.

Affinely representable lattices, stable matchings, and choice functions

Birkhoff’s representation theorem (Birkhoff, Duke Math J 3(3):443–454, 1937) defines a bijection between elements of a distributive lattice and the family of upper sets of an associated poset. Although not used explicitly, this result is at the backbone of the combinatorial algorithm by Irving et al. (J ACM 34(3):532-543, 1987) for maximizing a linear function over the set of stable matchings in Gale and Shapley’s stable marriage model (Gale and Shapley, Am Math Monthly 69(1):9–15 1962). In this paper, we introduce a property of distributive lattices, which we term as affine representability, and show its role in efficiently solving linear optimization problems over the elements of a distributive lattice, as well as describing the convex hull of the characteristic vectors of the lattice elements. We apply this concept to the stable matching model with path-independent quota-filling choice functions, thus giving efficient algorithms and a compact polyhedral description for this model. To the best of our knowledge, this model generalizes all those for which similar results were known, and our paper is the first that proposes efficient algorithms for stable matchings with choice functions, beyond classical extensions of the Deferred Acceptance algorithm.

Farsighted objections and maximality in one-to-one matching problems

We characterize the set of stable matchings when individuals are farsighted and when they choose their objections optimally along a farsighted objection path. We use a solution concept called maximal farsighted set (MFS), which is an adaptation of the concepts developed in Dutta and Vohra (2017) and Dutta and Vartiainen (2020) to one-to-one matching problems. MFS always exists, but it need not be unique. There is a unique largest MFS that contains all other, which is equal to the largest consistent set of Chwe (1994). This implies that the largest consistent set embodies the idea of maximality in one-to-one matching problems.

Cycles to compute the full set of many-to-many stable matchings

In a many-to-many matching model in which agents’ preferences satisfy substitutability and the law of aggregate demand, we present an algorithm to compute the full set of stable matchings. This algorithm relies on the idea of “cycles in preferences” and generalizes the algorithm presented in Roth and Sotomayor (1990) for the one-to-one model.

Restabilization Process in Matching Markets with Workers Proposing

This paper is focused on the analysis, in the framework of lattice theory, of the matchings obtained from restabilization (after disruption) of stable matchings. When the disruption is due to entry workers or closure of firms the unemployed workers make offers to firms. The stable matching obtained is the firms-worst stable matching of the set of stable matchings that the firms weakly prefer to the initial stable matching (i.e., before being disrupted by changes in the population). More precisely, their position within the lattice of stable matchings is shown.

Stability and Median Rationalizability for Aggregate Matchings

We develop the theory of stability for aggregate matchings used in empirical studies and establish fundamental properties of stable matchings including the result that the set of stable matchings is a non-empty, complete, and distributive lattice. Aggregate matchings are relevant as matching data in revealed preference theory. We present a result on rationalizing a matching data as the median stable matching.

School choice: Nash implementation of stable matchings through rank-priority mechanisms

We consider school choice problems (Abdulkadiroğlu and Sönmez, 2003) where students are assigned to public schools through a centralized assignment mechanism. We study the family of so-called rank-priority mechanisms, each of which is induced by an order of rank-priority pairs. Following the corresponding order of pairs, at each step a rank-priority mechanism considers a rank-priority pair and matches an available student to an unfilled school if the student and the school rank and prioritize each other in accordance with the rank-priority pair. The Boston or immediate acceptance mechanism is a particular rank-priority mechanism. Our first main result is a characterization of the subfamily of rank-priority mechanisms that Nash implement the set of stable matchings (Theorem 1). Our second main result is a strong impossibility result: under incomplete information, no rank-priority mechanism implements the set of stable matchings (Theorem 2).

A note on the lattice structure for matching markets via linear programming

Given two stable matchings in a many-to-one matching market with $ q $-responsive preferences, by manipulating the objective function of the linear program that characterizes the stable matching set, we compute the least upper bound and greatest lower bound between them.

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.

Improving solution times for stable matching problems through preprocessing

We present new theory, heuristics, and algorithms for preprocessing instances of the Stable Marriage problem with Ties and Incomplete lists (SMTI) and the Hospitals/Residents problem with Ties (HRT). Instances of these problems can be preprocessed by removing from the preference lists of some agents entries such that the set of stable matchings is not affected. Removing such entries reduces the problem size, creating smaller models that can be more easily solved by integer programming (IP) solvers. The new theorems are the first to describe when preference list entries can be removed from instances of HRT when ties are present on both sides, and also extend existing results on preprocessing instances of SMTI. A number of heuristics, as well as an IP model and a graph-based algorithm, are presented to find and perform this preprocessing. Experimental results show that our new graph-based algorithm achieves a 44% reduction in the average running time to find a maximum weight stable matching in real-world instances of SMTI compared to existing preprocessing techniques, and 80% compared to not using preprocessing. We also show that, when solving MAX-HRT instances with ties on both sides, our new techniques can reduce runtimes by up to 55%.

Two-sided strategy-proofness in many-to-many matching markets

We study the existence of group strategy-proof stable rules in many-to-many matching markets under responsiveness of agents’ preferences. We show that when firms have acyclical preferences over workers the set of stable matchings is a singleton, and the worker-optimal stable mechanism is a stable and group strategy-proof rule for firms and workers. Furthermore, acyclicity is the minimal condition guaranteeing the existence of stable and strategy-proof mechanisms in many-to-many matching markets.

The revealed preference theory of stable matchings with one-sided preferences

We investigate the testable implications of the theory of stable matchings in two-sided matching markets with one-sided preferences. Our main result connects the revealed preference analysis to the well-known lattice structure of the set of stable matchings, and tests the rationalizability of a data set by analyzing the joins and meets of matchings.

Stable Matchings in the Marriage Model with Indifferences

For the marriage model with indifferences, we define an equivalence relation over the stable matching set. We identify a sufficient condition, the closing property, under which we can extend results of the classical model (without indifferences) to the equivalence classes of the stable matching set. This condition allows us to extend the lattice structure over classes of equivalences and the rural hospital theorem.

Optimizing for Distributional Goals in School Choice Problems

I investigate three goals of school choice: welfare, encouraging neighborhood schools, and diversity. I use optimization problems to find the best stable and incentive compatible match for any combination of these objectives. These problems assume there is a continuum of students and school seats, which allows me to describe the incentive compatibility conditions in a tractable form. I prove that the set of stable matchings is generically continuous in the distribution of students and the school capacities, which implies that the characterization of the possible stable matches in the continuum model approximates the set of stable matches in a matching market with a large, but finite, number of students. I then apply my framework to data from Boston Public Schools. If the mechanism conditions on demographics, the improvement (relative to the status quo) in student welfare is equivalent to moving 291 students (out of 3,479) to schools one rank higher in their preference lists. In contrast, if the mechanism does not condition on demographics, the welfare improvement is equivalent to moving only 25.1 students to schools one rank higher. Improvements in the distributional goals can be made (e.g., increasing enrollment in neighborhood schools by 50%) without reducing welfare or diversity. This paper was accepted by Gabriel Weintraub, revenue management and market analytics.

Evolution and Rawlsian social choice in matching

This paper considers the marriage problem under dynamic rematching. It is shown that if players who obtain higher payoffs are less likely to experiment with non-best response behavior, then matchings selected in the long run will belong to the set of Rawlsian stable matchings – the set of stable matchings which maximize the payoff of the worst off player. Conversely, alternative behavioral rules will fail to select Rawlsian stable matchings in some environments. This constitutes an evolutionary axiomatization of Rawlsian stable matchings in terms of the behavioral rules that give rise to them.

Binary Operations for the Lattice Structure in a Many-to-Many Matching Model

The lattice structure of the set of stable matchings in many-to-many matching model is well known in literature. If preferences of the agents are substitutable, this result can be obtained by fixed-point methods, for that purpose an algorithm for finding a fixed-point matching is defined. Since the fixed-point set equals the set of stable matchings, the latter has a lattice structure too. In this paper, we consider a many-to-many matching market where the preferences of firms satisfy substitutability and the law of aggregate demand, and workers have responsive preferences. In this many-to-many matching market, we explicitly compute for any pair of stable matchings the least upper bound and the greatest lower bound, without using fixed-point methods.