Implementation of college admission rules
Tarık Kara1997, Economic Theory
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
We consider both Nash and strong Nash implementation of various matching rules for college admissions problems. We show that all such rules are supersolutions of the stable rule. Among these rules the "lower bound" stable rule is implementable in both senses. The "upper bound" Pareto and individually rational rule is strong Nash implementable yet it is not Nash implementable. Two corollaries of interest are the stable rule is the minimal (Nash or strong Nash) implementable solution that is Pareto optimal and individually rational, and the stable rule is the minimal (Nash or strong Nash) implementable extension of any of its subsolutions. JEL Classification Numbers: C78, D78. * We wish to thank Professor William Thomson for his efforts in supervision as well as his useful suggestions. We are grateful to the participants in his reading class, workshops at Bilkent University, University of Rochester, and in particular Jeffrey Banks, Stephen Ching, Bhaskar Dutta, Rangarajan Sundaram and an anonymous referee for their helpful comments. Correspondence to: T. Kara 1 Gale and Shapley [4] introduces the college admissions problems. Gale and Sotomayor [53, Roth [15, 16, 17, 18], Roth and Sotomayor [19] study various aspects of the college admissions problems.
Related papers
Simple Mechanism to Implement the Core of College Admission Problems
Games and Economic Behavior
This paper provindes three simple mechanisms to implement allocations in the core of matching markets. We analyse some sequential mechanisms which mimic matching procedures for many-to-one real life matching markets.
Simple Mechanisms to Implement the Core of College Admissions Problems
Games and Economic Behavior, 2000
This paper provides three simple mechanisms to implement allocations in the core of matching markets. \Ve analyze sorne sequential rnechanisms which mimic matching procedures for many-to-one reallife matching markets. \Ve show that only cor,e allocations should be attained when agents act strategically faced váth these mechanisms. Two mechanisms implement the core correspondence in SPE, whereas the third implements the students• optimal stable solution.
Median Stable Matching for College Admissions
International Journal of Game Theory, 2006
We give a simple and concise proof that so-called generalized median stable matchings are well-defined for college admissions problems. Furthermore, we discuss the fairness properties of median stable matchings and conclude with two illustrative examples of college admissions markets, the lattices of stable matchings, and the corresponding generalized median stable matchings.
College admissions with ties and common quotas: Integer programming approach
European Journal of Operational Research, 2021
Admission to universities is organised in a centralised scheme in Hungary. In this paper we investigate two major specialities of this application: ties and common quotas. A tie occur when some students have the same score at a programme. If not enough seats are available for the last tied group of applicants at a programme then there are three reasonable policies used in practice: 1) all must be rejected, as in Hungary 2) all can be accepted, as in Chile 3) a lottery decides which students are accepted from this group, as in Ireland. Even though student-optimal stable matchings can be computed efficiently for each of the above three cases, we developed (mixed) integer programming (IP) formulations for solving these problems, and compared the solutions obtained by the three policies for a real instance of the Hungarian application from 2008. In the case of Hungary common quotas arise from the faculty quotas imposed on their programmes and from the national quotas set for state-financed students in each subject. The overlapping structure of common quotas makes the computational problem of finding a stable solution NP-hard, even for strict rankings. In the case of ties and common quotas we propose two reasonable stable solution concepts for the Hungarian and Chilean policies. We developed (mixed) IP formulations for solving these stable matching problems and tested their performance on the large scale real instance from 2008 and also for one from 2009 under two different assumptions. We demonstrate that the most general case is also solvable in practice by IP technique.
College Admissions Problem with Ties and Flexible Quotas
SSRN Electronic Journal, 2014
We study a variant of the classical college admissions problem in which applicants have strict preferences over careers, but careers may present ties in their preferences over students. When careers have a fixed admission quota and a tie occurs in the last vacancy, a tie-break rule is required to complete the matching process. In this paper we consider a di↵erent situation in which tie-breaking is not allowed to discriminate among students, and instead careers are required to adjust their quotas to accommodate all the applicants in a tie. We extend the notion of stable assignment to this setting and we describe algorithms to compute them. We investigate the optimality properties of the solutions, and we show that the resulting mechanisms are neither monotone nor strategy-proof. We show that the welfare of students is increased when flexible quotas and a student-optimal assignment are combined. As a case study we consider real instances for the college admissions problem in Chilean universities, and we argue that a student-optimal assignment is preferable in this case.
Three remarks on the many-to-many stable matching problem
Mathematical Social Sciences, 1999
We propose a general definition of stability, setwise-stability, and show that it is a stronger requirement than pairwise-stability and core. We also show that the core and the set of pairwise-stable matchings may be non-empty and disjoint and thus setwise-stable matchings may not exist. For many labor markets the effects of competition can be characterized by requiring only pairwise-stability. For such markets we define substitutability and we prove the existence of pairwise-stable matchings. The restriction of our proof to the College Admission Model is simple and short and provides an alternative proof for the existence of stable matchings for this model.
Capacity Planning in Stable Matching: An Application to School Choice
arXiv (Cornell University), 2021
We introduce the problem of jointly increasing school capacities and finding a student-optimal assignment in the expanded market. Due to the impossibility of efficiently solving the problem with classical methods, we generalize existent mathematical programming formulations of stability constraints to our setting, most of which result in integer quadratically-constrained programs. In addition, we propose a novel mixed-integer linear programming formulation that is exponentially large on the problem size. We show that its stability constraints can be separated by exploiting the objective function, leading to an effective cutting-plane algorithm. We conclude the theoretical analysis of the problem by discussing some mechanism properties. On the computational side, we evaluate the performance of our approaches in a detailed study, and we find that our cutting-plane method outperforms our generalization of existing mixed-integer approaches. We also propose two heuristics that are effective for large instances of the problem. Finally, we use the Chilean school choice system data to demonstrate the impact of capacity planning under stability conditions. Our results show that each additional seat can benefit multiple students and that we can effectively target the assignment of previously unassigned students or improve the assignment of several students through improvement chains. These insights empower the decision-maker in tuning the matching algorithm to provide a fair applicationoriented solution.
Implementation in generalized matching problems
Journal of Mathematical Economics, 1996
We search for (Nash) implementable solutions on a class of one-to-one matching problems which includes both the housing market (Shapley and Scarf 1974) and marriage problems (Gale and Shapley 1962). We show that the core correspondence is implementable. We furthermore show that any solution that is Pareto efficient, individually rational, and implementable is a supersolution of the core correspondence. That is, the core correspondence is the minimal solution that is Pareto efficient, individually rational, and implementable. A corollary of independent interest in the context of the housing market is that the core correspondence is the only single-valued solution that is Pareto-efficient, individually rational, and implementable.
Paths to stability for college admissions with budget constraints
International Journal of Game Theory, 2016
We study two-sided matching problem considered in Abizada (Theor Econ 11(2), 735-756, 2016), where one side (colleges) can make monetary transfers (offer stipends) to the other (students) subject to budget constraints. Colleges have strict preferences over sets of students and value money only to the extent that it allows them to enroll better or additional students. A student can attend at most one college and receive a stipend from it. Each student has preferences over college-stipend bundles. Although in the presence of budget constraints, the conditions that are essential for most of the results on stability in the literature fail, Abizada (Theor Econ 11(2), 735-756, 2016) shows that for this model a pairwise stable allocation always exists. In this paper, we show that starting from an arbitrary allocation, there is a sequence of allocations, each allocation being obtained from the previous one by "satisfying" a blocking pair, such that the final allocation is pairwise stable.
The College Admissions Problem with Uncertainty
2006
We consider a college admissions problem with uncertainty. Unlike Gale and Shapley (1962), we realistically assume that (i) students’ college application choices are nontrivial because applications are costly, (ii) college rankings of students are noisy and thus uncertain at the time of application, and (iii) matching between colleges and students takes place in a decentralized setting. We analyze an equilibrium model where two ranked colleges set admissions standards for student caliber signals, and students, knowing their calibers, decide where to apply to. Do the best students try to attend the best colleges? While application noise works against this, we show that weaker students may apply more ambitiously than stronger ones, further overturning it. But we prove that a unique equilibrium with assortive matching of student caliber and college quality exists provided application costs are small and the capacity of the lesser college is not too small. We also provide equilibrium co...
Related papers
The stability of the equilibrium outcomes in the admission games induced by stable matching rules
International Journal of Game Theory, 2008
A stable matching rule is used as the outcome function for the Admission game where colleges behave straightforwardly and the students' strategies are given by their preferences over the colleges. We show that the college-optimal stable matching rule implements the set of stable matchings via the Nash equilibrium (NE) concept. For any other stable matching rule the strategic behavior of the students may lead to outcomes that are not stable under the true preferences. We then introduce uncertainty about the matching selected and prove that the natural solution concept is that of NE in the strong sense. A general result shows that the random stable matching rule, as well as any stable matching rule, implements the set of stable matchings via NE in the strong sense. Precise answers are given to the strategic questions raised.
Random Matching In the College Admissions Problem
Economic Theory, 2008
Integer programming formulations for college admissions with ties
Pázmány Péter Catholic University, 2018
When two students with the same score are competing for the last slot at a university programme in a central admission scheme then different policies may apply across countries. In Ireland only one of these students is admitted by a lottery. In Chile both students are admitted by slightly violating the quota of the programme. Finally, in Hungary none of them is admitted, leaving one slot empty. We describe the solution by the Hungarian policy with various integer programing formulations and test them on a real data from 2008 with around 100,000 students. The simulations show that the usage of binary cutoff-score variables is the most efficient way to solve this problem when using IP technique. We also compare the solutions obtained on this problem instance by different admission policies. Although these solutions are possible to compute efficiently with simpler methods based on the Gale-Shapley algorithm, our result becomes relevant when additional constraints are implied or more complex goals are aimed, as it happens in Hungary where at least three other special features are present: lower quotas for the programmes, common quotas and paired applications for teachers studies.
Integer Programming Methods for Special College Admissions Problems
Lecture Notes in Computer Science, 2014
are circulated to promote discussion and provoque comments. Any references to discussion papers should clearly state that the paper is preliminary. Materials published in this series may subject to further publication.
School Choice: Nash Implementation of Stable Matchings through Rank-Priority Mechanisms
SSRN Electronic Journal, 2017
We consider school choice problems (Abdulkadiroglu 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 (i.e., fair) matchings (Theorem 1). We show that our characterization also holds for "sub-implementation" and "sup-implementation" (Corollaries 3 and 4). Our second main result is a strong impossibility result: under incomplete information, no rank-priority mechanism implements the set of stable matchings (Theorem 2).
Student Course Allocation with Constraints
2019
Real-world matching scenarios, like the matching of students to courses in a university setting, involve complex downward-feasible constraints like credit limits, time-slot constraints for courses, basket constraints (say, at most one humanities elective for a student), in addition to the preferences of students over courses and vice versa, and class capacities. We model this problem as a many-to-many bipartite matching problem where both students and courses specify preferences over each other and students have a set of downward-feasible constraints. We propose an Iterative Algorithm Framework that uses a many-to-one matching algorithm and outputs a many-to-many matching that satisfies all the constraints. We prove that the output of such an algorithm is Pareto-optimal from the student-side if the many-to-one algorithm used is Pareto-optimal from the student side. For a given matching, we propose a new metric called the Mean Effective Average Rank (MEAR), which quantifies the goodn...
The Compromise Algorithm in the Swedish PhD Admissions Problem
2016
What mechanism should be designed to allocate PhD applicants to universities in Sweden? We introduce the Swedish PhD admissions problem, and it is influenced by the college admissions problem (Gale and Shapley 1962) and the student placement problem (Balinski and Sonmez 1999). In order to “solve” this problem, we design a novel mechanism, namely the compromise algorithm. We propose three theorems from this algorithm, i.e. equivalence theorems. The equivalence theorems specify the equivalence relations among stability, worse and responsiveness. Additionally, we find a positive result that the number of fields determines the strategy-proofness of the algorithm; meanwhile, the student optimal stable matching and the university optimal stable matching can be treated as special cases of our model when we restrict the number of fields. Generally, the compromise algorithm generates a stable matching that falls in between the student optimal stable matching and the university optimal stable...
Nash implementation of matching rules
Journal of Economic Theory, 1996
We consider the Nash implementation of Pareto optimal and individually rational solutions in the context of matching problems. We show that all such rules are supersolutions of the stable rule. Among these solutions, we show that thè`l ower bound'' stable rule and the``upper bound'' Pareto and individually rational rule are Nash implementable. The proofs of these results are by means of a recent technique developed by Danilov [2]. Two corollaries of interest are the stable rule is the minimal implementable solution that is Pareto optimal and individually rational and the stable rule is the minimal Nash implementable extension of any of its subsolutions.
Strategyproof and fair matching mechanism for ratio constraints
Autonomous Agents and Multi-Agent Systems
We introduce a new type of distributional constraints called ratio constraints, which explicitly specify the required balance among schools in two-sided matching. Since ratio constraints do not belong to the known well-behaved class of constraints called M-convex set, developing a fair and strategyproof mechanism that can handle them is challenging. We develop a novel mechanism called quota reduction deferred acceptance (QRDA), which repeatedly applies the standard DA by sequentially reducing artificially introduced maximum quotas. As well as being fair and strategyproof, QRDA always yields a weakly better matching for students compared to a baseline mechanism called artificial cap deferred acceptance (ACDA), which uses predetermined artificial maximum quotas. Finally, we experimentally show that, in terms of student welfare and nonwastefulness, QRDA outperforms ACDA and another fair and strategyproof mechanism called Extended Seat Deferred Acceptance (ESDA), in which ratio constrai...
The stable admissions polytope
Mathematical Programming, 2000
The stable admissions polytope-the convex hull of the stable assignments of the university admissions problem-is described by a set of linear inequalities. It depends on a new characterization of stability and arguments that exploit and extend a graphical approach that has been fruitful in the analysis of the stable marriage problem. Key words. stable assignment-stable marriage-two-sided market-polytopes-graphs-many-to-one matching 1. Stable assignments An admissions problem or admissions game (, q) is specified by a directed graph defined over a grid , and positive integers q, as follows. There are two distinct, finite M.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.