Stable Matching Research Articles

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.

A general equilibrium model for multi-passenger ridesharing systems with stable matching

This paper proposes a general equilibrium model for multi-passenger ridesharing systems, in which interactions between ridesharing drivers, passengers, platforms, and transportation networks are endogenously captured. Stable matching is modeled as an equilibrium problem in which no ridesharing driver or passenger can reduce his/her ridesharing disutility by unilaterally switching to another matching sequence.This paper is one of the first studies that explicitly integrates the ridesharing platform’s multi-passenger matching problem into the model. By integrating matching sequence with hyper-network, ridesharing-passenger transfers are avoided in a multi-passenger ridesharing system. Moreover, the matching stability between the ridesharing drivers and passengers is extended to address the multi-OD multi-passenger case in terms of matching sequence. The paper provides a proof for the existence of the proposed general equilibrium.A sequence-bush algorithm is developed for solving the multi-passenger ridesharing equilibrium problem. This algorithm is capable to handle complex ridesharing constraints implicitly. Results illustrate that the proposed sequence-bush algorithm outperforms general-purpose solver, and provides insights into the equilibrium of the joint stable matching and route choice problem. Numerical experiments indicate that ridesharing trips are typically longer than average trip lengths. Sensitivity analysis suggests that a properly designed ridesharing unit price is necessary to achieve network benefits, and travelers with relatively lower values of time are more likely to participate in ridesharing.

K-Best Egalitarian Stable Marriages for Task Assignment

In a two-sided market with each agent ranking individuals on the other side according to their preferences, such as location or incentive, thestable marriageproblem calls to find aperfect matchingamong the two sides such that no pair of agents prefers each other to their assigned matches. Recent studies show that the number of solutions can be large in practice. Yet the classic solution by the Gale-Shapley (GS) algorithm isoptimalfor agents on the one side andpessimalfor those on the other side. Some algorithms find a stable marriage that optimizes a measure of the cumulative satisfaction of all agents, such asegalitarian cost.However, in many real-world circumstances, a decision-maker needs to examine a set of solutions that are stable and attentive to both sides and choose among them based on expert knowledge. With such a disposition, it is necessary to identify a set of high-quality stable marriages and provide transparent explanations for any reassigned matches to the decision-maker. In this paper, we provide efficient algorithms that find thek-best stable marriages byegalitarian cost.Our exhaustive experimental study using real-world data and realistic preferences demonstrates the efficacy and efficiency of our solution.

A counterexample of size 20 for the problem of finding a 3-dimensional stable matching with cyclic preferences

A counterexample of size 20 for the problem of finding a 3-dimensional stable matching with cyclic preferences

Marriage and Roommate

This paper has two objectives. One is to give a linear time algorithm that solves the stable roommates problem (i.e., obtains one stable matching) using the stable marriage problem. The idea is that a stable matching of a roommate instance [Formula: see text] is a stable matching (that however must satisfy a certain condition) of some marriage instance [Formula: see text]. [Formula: see text] is obtained just by making two copies of [Formula: see text], one for the men’s table and the other for the women’s table. The second objective is to investigate the possibility of reducing the roommate problem to the marriage problem (with one-to-one correspondence between their stable matchings) in polynomial time. For a given [Formula: see text], we construct the rotation POSET [Formula: see text] of [Formula: see text] and then we “halve” it to obtain [Formula: see text], by which we can forget the above condition and can use all the closed subsets of [Formula: see text] for all the stable matchings of [Formula: see text]. Unfortunately this approach works (runs in polynomial time) only for restricted instances.

Geometric characterization of the persistence of 1D maps

We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining a collection of sorted lists together with its persistence diagram.

Computing relaxations for the three-dimensional stable matching problem with cyclic preferences

Constraint programming has proven to be a successful framework for determining whether a given instance of the three-dimensional stable matching problem with cyclic preferences (3dsm-cyc) admits a solution. If such an instance is satisfiable, constraint models can even compute its optimal solution for several different objective functions. On the other hand, the only existing output for unsatisfiable 3dsm-cyc instances is a simple declaration of impossibility. In this paper, we explore four ways to adapt constraint models designed for 3dsm-cyc to the maximum relaxation version of the problem, that is, the computation of the smallest part of an instance whose modification leads to satisfiability. We also extend our models to support the presence of costs on elements in the instance, and to return the relaxation with lowest total cost for each of the four types of relaxation. Empirical results reveal that our relaxation models are efficient, as in most cases, they show little overhead compared to the satisfaction version.

Learning Equilibria in Matching Markets with Bandit Feedback

Large-scale, two-sided matching platforms must find market outcomes that align with user preferences while simultaneously learning these preferences from data. Classical notions of stability (Gale and Shapley, 1962; Shapley and Shubik, 1971) are, unfortunately, of limited value in the learning setting, given that preferences are inherently uncertain and destabilizing while they are being learned. To bridge this gap, we develop a framework and algorithms for learning stable market outcomes under uncertainty. Our primary setting is matching with transferable utilities, where the platform both matches agents and sets monetary transfers between them. We design an incentive-aware learning objective that captures the distance of a market outcome from equilibrium. Using this objective, we analyze the complexity of learning as a function of preference structure, casting learning as a stochastic multi-armed bandit problem. Algorithmically, we show that “optimism in the face of uncertainty,” the principle underlying many bandit algorithms, applies to a primal-dual formulation of matching with transfers and leads to near-optimal regret bounds. Our work takes a first step toward elucidating when and how stable matchings arise in large, data-driven marketplaces.

Experimental Evaluation of AGV Dispatching Methods in an Agent-Based Simulation Environment and a Digital Twin

A critical part of Automated Material Handling Systems (AMHS) is the task allocation and dispatching strategy employed. In order to better understand and investigate this component, we here present an extensive experimental evaluation of three different approaches with randomly generated, as well as custom designed, environment configurations. While previous studies typically focused on use cases based on highly constrained navigation capabilities (e.g., overhead hoist transport systems), our evaluation is built around highly mobile, free-ranging vehicles, i.e., Autonomous Mobile Robots (AMR) that are gaining popularity in a broad range of applications. Consequently, our experiments are conducted using a microscopic agent-based simulation, instead of the more common discrete-event simulation model. Dispatching methods often are built around the assumption of the asynchronous evaluation of an event-based model, i.e., vehicles trigger a cascade of individual dispatching decisions, e.g., when reaching intersections. We find that this does not translate very well to a fleet of highly mobile systems that can change direction at any time. With this in mind, we present formulations of well known dispatching approaches that are better suited for a synchronous evaluation of the dispatching decisions. The formulations are based on the Stable Marriage Problem (SMP) and the Linear Sum Assignment Problem (LSAP). We use matching and assignment algorithms to compute the actual dispatching decisions. The selected algorithms are evaluated in a multi-agent simulation environment. To integrate a centralised fleet management, a digital twin concept is proposed and implemented. By this approach, the fleet management is independent of the implementation of the specific agents, allowing to quickly adapt to other simulation-based or real application scenarios. For the experimental evaluation, two new performance measures related to the efficiency of a material handling system are proposed, Travel Efficiency and Throughput Effort. The experimental evaluation indicates that reassignment mechanisms in the dispatching method can help to increase the overall efficiency of the fleet. We did not find significant differences in absolute performance in terms of throughput rate. Additionally, the difference in performance between SMP- and LSAP-based dispatching with reassignment seems negligible. We conclude with a discussion, where we consider potential confounding factors and relate the findings to previously reported results found in the literature.

Constrained school choice: an experimental QRE analysis

The theoretical literature on public school choice proposes centralized mechanisms that assign children to schools on the basis of parents’ preferences and the priorities children have for different schools. The related experimental literature analyzes in detail how various mechanisms fare in terms of welfare and stability of the resulting matchings, yet often provides only aggregate statistics of the individual behavior that leads to these outcomes (i.e., the degree to which subjects tell the truth in the induced simultaneous move game). In this paper, we show that the quantal response equilibrium (QRE) adequately describes individual behavior and the resulting matching in three constrained problems for which the immediate acceptance mechanism and the student-optimal stable mechanism coincide. Specifically, the comparative statics of the logit-QRE with risk-neutral and expected-payoff-maximizing agents capture the directional changes of subject behavior and the prevalence of the different stable matchings when cardinal payoffs (i.e., relative preference intensities) are modified in the experiment.

Online doctor-patient dynamic stable matching model based on regret theory under incomplete information

With the development of the Internet era, online medical community is increasingly favored by patients and health care workers. How to make a reasonable doctor-patient matching and meet patients' relevant needs through online medical platform is the key to realizing effective allocation of medical resources. This paper proposes an online doctor-patient dynamic stable matching model to deeply investigate the online doctor-patient matching problem under incomplete information. Firstly, we assign the score to each doctor based on the patient's attribute expectations and the doctor's evaluation information. Considering the difficulty of obtaining complete information in real life and different attribute representations, we propose incomplete information filling methods based on collaborative filtering algorithm. To make the filling result more reasonable, we introduce probabilistic linguistic information entropy to process the incomplete information. Secondly, considering psychological behavior, we use regret theory to calculate patients' satisfaction with doctors. Finally, with the definitions of rational matching and stable matching, we propose a doctor-patient dynamic stable matching model. This model can solve the problems like a temporary appointment and private matching and ensure the stability of matching. In addition, the model proposed in this paper considers not only patients' but also the doctors' needs. The case of online doctor-patient matching is based on “haodf.com”, and through comparative analysis can prove the effectiveness and superiority of this method.

Marital preferences and stable matching in cultural evolution

Building on previous literature that examines the influence of intergenerational transmission in cultural evolution, we highlight the importance of the marriage market in the determination of cultural homogeneity (“melting pot”) versus heterogeneity (“diversity”). To do so, we characterize cultural evolutionary processes under different distributions of marital preferences and stable matching schemes. In our setting, cultural substitutability (Bisin and Verdier, 2001) is neither sufficient nor necessary for cultural heterogeneity. We introduce a new concept, elasticity of cultural substitution, to capture the degree of increase in vertical socialization efforts in response to minority population decline. With perfect or inelastic vertical transmission in homogamous families, cultural heterogeneity is sustained only if all proposers are homophilic or all members of a cultural group are homophilic. With imperfect vertical transmission in homogamous families, the presence of heterophilic agents may destabilize cultural heterogeneity, and the proportion of heterophilic agents and elasticity of cultural substitution determine whether cultural heterogeneity can be sustained. We discuss the model's implications for the long-lasting impact of temporary gender imbalance on cultural evolution as well as the cultural assimilation and preservation of minorities and immigrants under distinct governmental and religious attitudes toward intermarriage.

Integrated preference argumentation and applications in consumer behaviour analyses

Consumer preference studies in economics rest heavily on the behavioural interpretation of preference especially in the form of Revealed Preference Theory (RPT). Viewing purchasing decisions as a kind of human reasoning, in this paper we are interested in generalising behaviourism to preference-based argumentation where existing frameworks are arguably governed by the opposing mentalistic interpretation of preference. Concretely our contributions are three-fold. First we re-construct and unify two main approaches to RPT in argumentation terms by developing a Revealed Preference Argumentation (RPA) framework. We show that key RPT-based consumer analyses such as different rationality checks of a consumer behaviour and various kinds of such behaviour's extrapolations can be translated to computational tasks in RPA. RPA is in turn subsumed by the second contribution - an Integrated Preference Argumentation (IPA) framework which fully integrates mentalism and behaviourism. In particular we show that RPA is just a special case of IPA with only revealed preference, while existing preference-based argumentation frameworks can be viewed as IPA frameworks with only stated preference. As the third contribution, we develop a complete set of IPA algorithms and establish their correctness and termination for a general class of IPA frameworks. For demonstration we implement the algorithms in Prolog to obtain an IPA reasoning engine and test the engine with various RPT-based consumer behaviour analyses. In sum, it is argued that the current paper provides not only a theory of Integrated Preference Argumentation, but also a related development tool for future applications of argumentation to behavioural economics - a largely unexplored area so far though Dung used the stable marriage problem in microeconomic game theory to partly motivate the argumentation theory.

Stable and Efficient Resource Allocation with Contracts

Consider indivisible-object allocation with contracts, such as college admissions, where contracts specify majors. Can a designer guarantee a stable and (student) efficient matching? I show that contracts put stability and efficiency at odds; a necessary condition to ensure these properties is student-lexicographic priorities—schools must rank contracts from “second-tier” students consecutively. I present the weakest restriction guaranteeing stability and efficiency, and characterize necessary and sufficient conditions for any mechanism within a general class to deliver a stable and efficient matching in an incentive compatible manner. I apply this result to two well-known mechanisms: deferred acceptance and top trading cycles. (JEL C78, D82, D86, I23)

Stable Matching with Mistaken Agents

Stable Matching with Mistaken Agents

Theory of brain complexity and marital behaviors: The application of complexity science and neuroscience to explain the complexities of marital behaviors.

The extant theories on the quality and stability of marital relationships have some difficulties in explaining some of the complexities of marital behaviors. The present article is an initial attempt to explain the complexities of marital behaviors based on the science of complexity and neuroscience. This article proposes a new theoretical framework relying on this simple argument that marital behaviors, as one of the most complex human behaviors, are the product of one's brain's complex adaptive system (CAS). Hence, to understand the complexities of marital behaviors, a movement toward familiarity with the brain's CAS involved in marital behaviors needs to be started. The article presents the theory of brain complexity and marital behaviors (BCM) and outlines its assumptions, concepts, and propositions. Then, BCM is compared with the extant theories on happy and stable marriage, and finally, it was concluded by discussing the testability and the potential application of the theory. This article might inspire interdisciplinary studies of marital relationships, complex systems, and neuroscience and may have considerable practical implications.

Finding popular branchings in vertex-weighted directed graphs

Popular matchings have been intensively studied recently as a relaxed concept of stable matchings. By applying the concept of popular matchings to branchings in directed graphs, Kavitha et al. introduced popular branchings. Let G=(VG,EG) be a directed graph, where each vertex has preferences over its incoming arcs, and B and B′ be branchings in G. A vertex v∈VG prefers B to B′ if v prefers its incoming arc of B to that of B′, where having an arbitrary incoming arc is preferred to having no incoming arc. We say that B is more popular than B′ if the number of vertices preferring B to B′ is greater than the number of vertices preferring B to B′. A branching B is called a popular branching if there is no branching more popular than B. Kavitha et al. proposed an algorithm for finding a popular branching when the preferences of each vertex are given by a strict partial order. The correctness of this algorithm is proved by utilizing classical theorems on the duality of weighted arborescences. In this paper, we generalize popular branchings to weighted popular branchings in vertex-weighted directed graphs, in the same manner as weighted popular matchings by Mestre. We give an algorithm for finding a weighted popular branching for the case where the preferences of each vertex are given by a total preorder and the weights satisfy certain conditions. Our algorithm is based on Kavitha et al.'s algorithm, while it includes elaborated procedures resulting from the vertex-weights. Its correctness also builds upon the duality of weighted arborescences.

Task Offloading, Caching and Matching in Ultra-Dense Relay Networks

In recent years, ultra-dense relay network (UDRN) has become a hot research topic. Mobile edge computing (MEC) can improve the performance of intelligent interactive applications in terms of time delay and energy consumption. In traditional MEC, MEC server in the base station (BS) is responsible for all the computing in the network. However, the MEC deployed in the BS may lead to huge computation burden on the BS and heavy traffic load through the network. Deploying MEC server in the relays of the network, which helps computing and caching for the users, will lower the burden of the BS, reduce the demand for traffic transmission through the network and improve the quality of service for users. This paper studies the problem of task caching, offloading and stable matching (SM) in UDRNs with MEC deployed in the relays, which is formulated as a multi-variable integer programming problem. To enable task caching and offloading, we propose an improved branch-and-bound algorithm and a low-complexity suboptimal algorithm to obtain the optimal and suboptimal schemes, respectively. Then, according to the principle of minimizing individual costs, the mutual preference lists between the users and relays are established. Based on the mutual preference lists, we propose an improved SM algorithm to obtain the user-relay SM results. Simulation results show that the proposed algorithms can bring considerable performance gains compared with conventional algorithms.

Review of the Theory of Stable Matchings and Contract Systems

A review of works devoted to the theory of stable matchings or, more generally, of stable networks of contracts is given. A set (network) of contracts is called stable if no coalition has an available contract that gives all coalition members strictly more than the proposed set. In a special case, this concept was introduced in 1962 by Gale and Shapley and has since gone a long way in its development both theoretically (theorems, structures, and algorithms) and in the field of applications in economics, physics, biology, and mathematics.