College Admissions Problem Research Articles

The law of large numbers for large stable matchings

In many empirical studies of a large two-sided matching market (such as in a college admissions problem), the researcher performs statistical inference under the assumption that they observe a random sample from a large matching market. In this paper, we consider a setting in which the researcher observes either all or a nontrivial fraction of outcomes from a stable matching. We establish a concentration inequality for empirical matching probabilities assuming strong correlation among the colleges’ preferences while allowing students’ preferences to be fully heterogeneous. Our concentration inequality yields laws of large numbers for the empirical matching probabilities and other statistics commonly used in empirical analyses of a large matching market. To illustrate the usefulness of our concentration inequality, we prove consistency for estimators of conditional matching probabilities and measures of positive assortative matching.

EdgeSFG: A matching game mechanism for service function graph deployment in industrial edge computing environment

Network function virtualization-enabled industrial edge computing is a promising service paradigm. In industrial edge computing (IEC), virtual service functions (VSFs) are used to implement basic functionalities and deliver concrete services. To improve the service response speed, VSFs are organized into service function graphs (SFGs) to process service data parallelly. The main challenge is to select proper edge servers for SFG deployment and minimize latency. However, finding an optimal solution to the SFG deployment problem is an NP-hard problem. This paper proposes a matching-game-based algorithm to deploy a large number of concurrent SFGs in an IEC environment. In the proposed algorithm, the deployment of concurrent SFGs is formulated as a college admission problem where SFGs propose to edge servers expecting a low service latency, and then the edge servers decide whether to accept the SFG proposals according to the possible service gain. A distributed matching procedure is introduced to solve the matching game problem between the SFGs and edge servers. The simulations are conducted to verify the SFGs and evaluate the performance of the proposed algorithm in concurrent SFG deployment. Simulation results show that proposed algorithm performs well in reducing average service latency and improving service acceptance rate.

Siting Renewable Energy Facilities Using a Matching Algorithm: A Case Study in Japan

Finding an optimal match between installation sites and renewable energy (RE) facilities while ensuring that private initiatives meet local socio-environmental needs is a significant albeit complicated task. Different sites may need diverse considerations, such as landscape conservation, while information on the true local preferences and costs of RE facilities is unknown to the planner, causing information asymmetry and inefficiency. This study explores how a matching model can be utilised for empirically planning RE siting using an illustrative case study. It employs the so-called ‘college admission problem’ of the matching model. The matching algorithm enables the matching of sites and RE specifications, reflecting the true preferences of local people regarding facility siting. The matching result would ensure the most desirable choice for local people, as adopting the ‘student-optimal matching’ algorithm generates desirable matching patterns for the locals among the stable matching patterns.

Stability and immunity to capacity manipulation in large matching markets

Stable mechanisms are not strategy-proof in matching markets; however, they work well in practice. We study the college admission problem where colleges have completely heterogeneous preferences, and there is a continuum of students. We find an upper bound for the colleges’ capacities vector such that colleges have the incentive to report their capacities truthfully in any stable mechanism.

Stable matching and protocol-free equilibrium

We consider a non-cooperative decentralized matching game as sequential bargaining. A protocol-free equilibrium is a subgame perfect equilibrium which is robust to a choice of a bargaining protocol. We show that a matching is generated in a protocol-free equilibrium if and only if it is stable. The bargaining game can be applied to a wide class of matching problems including marriage problems, college admissions problems, matching with contracts, assignment games, and cooperative NTU games.

Improving the Chilean College Admissions System

In “Improving the Chilean College Admissions System,” Rios, Larroucau, Parra, and Cominetti describe the design and implementation of a new system to solve the Chilean college admissions problem. The authors develop an algorithm that (i) obtains all applicant/program pairs that can be part of a stable allocation when preferences are not strict and when all students tied in the last seat of a program (if any) must be allocated and (ii) efficiently incorporates affirmative action, which is part of the system to correct the inefficiencies that arise from having double-assigned students. By unifying the regular admission with affirmative action, the solution proposed and implemented by the authors has improved the allocation of approximately 2.5% of students assigned every year since 2016, helping to improve the overall efficiency of the system.

A simple sufficient condition for a unique and student-efficient stable matching in the college admissions problem

Consider the college admissions problem. Let us say that (student and college) preferences are student-oriented iff whenever two students disagree about the ranking of two colleges, each one of the two students is ranked higher by the college he prefers than the other student. We show that when preferences are oriented there is a unique stable matching, and that no other matching, stable or not, is weakly preferred by every student.

Expandable Preference Lists: A Mechanism for College Admissions in a Large Market

Although Gale-Shapley’s student optimal stable matching mechanism (SOSM) has desirable properties, it is infeasible for college admissions in a large market because it is impossible to ask every student to provide a complete ranking of thousands of colleges. Regarding college admissions problems in large-scale contexts, this paper introduces a new matching mechanism that allows students to provide incomplete but expandable preference lists (EPLs). In contrast to the standard one-shot (direct-revelation) or multirun (sequential-like) designs, the EPL mechanism matches students and colleges in a continuous period. Before the matching period begins, every college provides an algorithm to give a complete strict preference list over all students to the clearinghouse, whereas every student only needs to provide an incomplete initial preference list. During the matching period, (1) every student can expand her own preference list but cannot withdraw or change the order of colleges in her already submitted preference list; (2) the clearinghouse produces and updates the virtual (not final) matching results according to SOSM; and (3) a student can confirm (finalize) her own “best match” without the need to wait until the matching period ends. Theoretical analysis suggests that the EPL mechanism not only inherits much of the desirable stability and incentive properties of the SOSM but is also partially obviously strategy-proof. This paper proposes that by allowing interaction among participants, we can design matching mechanisms with more desirable properties.

Entering classes in the college admissions model

This note reveals a characteristic of stable matchings in the college admissions problem and provides structural insights and a unified treatment for several results on entering classes in this model, including the famous “Rural Hospital Theorem”. We also show that the worst student determines the entire entering class.

An efficient implementation of the Gale and Shapley “propose-and-reject” algorithm

We consider a version of the Hospitals/Residents problem which was first defined in 1962 by Gale and Shapley [9] under the name College Admissions Problem. In particular, we consider the Firms/Candidates problem, where each Firm wishes to hire at least one Candidate and each Candidate can be finally assigned to a single Firm. We present an efficient implementation of the Gale and Shapley propose-and-reject algorithm when applied to the case of the Firms/Candidates problem.

Subjective Homophily and the Fixtures Problem

The Stable Fixtures problem (Irving and Scott (2007)) is a generalized matching model that nests the well-known Stable Roommates, Stable Marriage, and College Admissions problems as special cases. This paper extends a result of the Stable Roommates problem to demonstrate that a class of homophilic preferences with an appealing psychological interpretation is sufficient to ensure that starting from an arbitrary matching, a decentralized process of allowing the sequential matching of randomly chosen blocking pairs will converge to a pairwise-stable matching with probability one. Strategic implications of this class of preferences are examined and further possible generalizations and directions for future research are discussed.

College admissions with entrance exams: Centralized versus decentralized

We study a college admissions problem in which colleges accept students by ranking students' efforts in entrance exams. Students' ability levels affect the cost of their efforts. We solve and compare equilibria of “centralized college admissions” (CCA) where students apply to all colleges and “decentralized college admissions” (DCA) where students only apply to one college. We show that lower ability students prefer DCA whereas higher ability students prefer CCA. Many predictions of the theory are supported by a lab experiment designed to test the theory, yet we find a number of differences that render DCA less attractive than CCA compared to the equilibrium benchmark.

A Hybrid Heuristic for QoS Aware Matching of User's Job and Virtual Machines in Cloud Environment

Cloud computing is a technological advancement that provides services in the form of utility on a pay-per-use basis. As the cloud market is expanding, numerous service providers are joining the cloud platform with their services. This creates an indecision amongst the users to choose an appropriate service provider especially when the cloud provider provisions diverse type of virtual machines. The problem becomes more challenging when the user has different jobs requiring specific quality of service. To address the aforementioned problem, this article applies a hybrid heuristic using College Admission Problem and Analytical Hierarchical Process for stable matching of the users' job with the cloud's virtual machines. The case study depicts the effectiveness of the proposed model.

A New Algorithm for Student-Optimal Matching: A Framework for Management Institution's Admission in India

College admission problem is a matching problem with colleges offering seats (admission offer) to students. Each college has its own preference list of students and each student has also its list of preferred colleges. Many stable matching can be achieved. One of these matching is college-optimal, and one is student-optimal. Existing application of centralized matching offers static solution, since features of market places are static. In the context of proposed application for post graduate admission processes at management institutions in India, outside-market exit option by student is allowed till the start of the course, which in turn leads to further confirmation of wait-listed status. Possibility of outside-market exit option makes it difficult to apply deferred acceptance algorithm as one time application. This paper proposes a matching algorithm and a framework which allow outside-market exit option to have less impact on both parties. The matching obtained by existing process is not a stable matching, since it works as deferred acceptance procedure terminated in finite time, approximately 60 days with larger time-window for non-acceptance. The clearing house, of the proposed framework, can obtain stable matching on 8th day only, and remaining time period will be reserved for the settlement due to outside-market exit option.

Matching problem for primary and secondary signals in dual-phase TPC detectors

The problem of matching primary and secondary light signals, belonging to the same event, is presented in the context of dual-phase time projection chambers. In large scale detectors the secondary light emission could be delayed up to order of milliseconds, which, combined with high signal rates, could make the matching of the signals challenging. A possible approach is offered in the framework of the Stable Marriage and the College Admission problem, for both of which solutions are given by the Gale–Shapley algorithm.

Limited choice in college admissions: An experimental study

This paper investigates a college admissions problem in which students' choice set is limited by colleges' strategic actions, such as early decision programs in the US or simultaneous entrance examinations in Asia, in order to prevent multiple applications. We call this action the conflicting strategy and conduct laboratory experiments to explore the problem. We argue that colleges' prestige and students' uncertainty about admissions outcomes are two important factors. Our model and experimental results demonstrate that when uncertainty is high and the prestige difference is low, a lower-ranked college can obtain more desired students through the conflicting strategy. Moreover, through this strategy, students with stronger preferences for a higher-ranked college have a higher chance to be accepted, which could result in an increase of students' aggregate welfare.

Mutually best matches

We study iterated formation of mutually best matches (IMB) in college admissions problems. When IMB produces a non-wasteful matching, the matching has many good properties like Pareto optimality and stability. Moreover, in this case IMB selects the unique core allocation and truth-telling is a Nash equilibrium for students. If preferences satisfy a single peakedness condition, or have a single crossing property, then IMB is guaranteed to produce a non-wasteful matching. These properties guarantee also that the Deferred Acceptance algorithm (DA) and the Top Trading Cycles algorithm (TTC) produce the same matching as IMB. We compare these results with some well-known results about when DA is Pareto optimal, or when DA and TTC produce the same matching.

The College Admissions Problem with Endogenous Choice of Admissions Policy Quality

This study analyzes the role of an admissions policy, a stochastic relationship between student caliber and a college`s noisy evaluations of them. The college first announces an admissions policy and its quality, following which a frictional college admissions problem occurs. The college can be better informed of student caliber by adopting a high-quality admissions policy. The change increases the marginal student type, admissions rates, and college utility. The cutoff score, however, varies arbitrarily. At equilibrium, the college sets admissions policy quality lower than the socially optimal level, resulting in an inefficiently large pool of applicants.

An extendable stable matching algorithm of a kind of bipartite graph

Real-life examples of assignment problems, like college admission, the labor market for medical interns, and sorority rush, can be formulated as bipartite matching. The college admission problem is one of many-to-one matching in two-sided matching models. We can obtain a bipartite graph by viewing all colleges and students as two disjointed sets of vertices, and joining an edge between their corresponding vertices, if the student enrolled this college. Thus, the college admission problem can be formulated as finding a stable extensive matching in the corresponding bipartite graph. In this paper, by using the definition of the stable extensive matching, we give an algorithm with O(m3n3) for the college admission problem, which is different from the Gale and Shapley algorithm. Moreover, the algorithm can be viewed as a generalization of Hungarian Algorithm.

Integer programming methods for special college admissions problems

We develop integer programming (IP) solutions for some special college admission problems arising from the Hungarian higher education admission scheme. We focus on four special features, namely the solution concept of stable score-limits, the presence of lower and common quotas, and paired applications. We note that each of the latter three special feature makes the college admissions problem NP-hard to solve. Currently, a heuristic based on the Gale---Shapley algorithm is being used in the Hungarian application. The IP methods that we propose are not only interesting theoretically, but may also serve as an alternative solution concept for this practical application, and other similar applications. We finish the paper by presenting a simulation using the 2008 data of the Hungarian higher education admission scheme.