Choice Function-Based Two-Sided Markets: Stability, Lattice Property, Path Independence and Algorithms

Fleiner, Tamás; Jankó, Zsuzsanna

doi:10.3390/a7010032

Cited by 17 publications

(23 citation statements)

References 11 publications

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“…As we mentioned in the Introduction, our results presented in Sections 3 and 4 are deducible from some general theorems on substitutable choice functions by Kelso and Crawford [18] and Roth [23], as Fleiner and Jankó [13] pointed out. The selection of the colleges can be described by their choice functions.…”

Section: General Arguments With Choice Functionsmentioning

confidence: 89%

“…Moreover, in Section 4 we prove that the applicant-oriented versions provide the minimal stable score-limits (therefore they are the best possible solutions for the applicants), whilst the college-oriented versions provide maximal stable score-limits (therefore, they are the worst possible solutions for the applicants). We note that the above results are deducible from some general theorems on substitutable choice functions by Kelso and Crawford [18] and Roth [23], as it was very recently demonstrated by Fleiner and Jankó [13]. We describe these arguments in detail at the end of Section 4.…”

Section: Introductionmentioning

confidence: 90%

“…Furthermore, Fleiner and Jankó [13] gave new results on the structure of stable matchings that applies for L-stable and H-stable score limits as well. They noticed that the choice function of the colleges under L-stability satisfy the path-independence property, that is for any set of applicants…”

Section: General Arguments With Choice Functionsmentioning

confidence: 99%

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College admissions with stable score-limits

Bíró

Kiselgof

2013

Cent Eur J Oper Res

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A common feature of the Hungarian, Irish, Spanish and Turkish higher education admission systems is that the students apply for programmes and are ranked according to their scores. Students who apply for a programme with the same score are tied. Ties are broken by lottery in Ireland, by objective factors in Turkey (such as date of birth) and other precisely defined rules in Spain. In Hungary, however, an equal treatment policy is used, students applying for a programme with the same score are all accepted or rejected together. In such a situation there is only one decision to make, whether or not to admit the last group of applicants with the same score who are at the boundary of the quota. Both concepts can be described in terms of stable score-limits. The strict rejection of the last group with whom a quota would be violated corresponds to the concept of H-stable (i.e. higher-stable) score-limits that is currently used in Hungary. We call the other solutions based on the less strict admission policy as L-stable (i.e. lower-stable) score-limits. We show that the natural extensions of the Gale-Shapley algorithms produce stable score-limits, moreover, the applicant-oriented versions result in the lowest score-limits (thus optimal for students) and the college-oriented versions result in the highest score-limits with regard to each concept. When comparing the applicant-optimal H-stable and L-stable score-limits we prove that the former limits are always higher for every college. Furthermore, these two solutions provide upper and lower boundaries for any solution arising from a tie-breaking strategy. Finally we show that both the H-stable and the L-stable applicant-proposing score-limit algorithms are manipulable.

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Section: General Arguments With Choice Functionsmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 90%

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College admissions with stable score-limits

Bíró

Kiselgof

2013

Cent Eur J Oper Res

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“…A stable set of score-limits always exists, and may be found using a generalised Gale-Shapley algorithm. Moreover the applicant proposing version leads to an applicant-optimal solution, where each of the score-limit at each college is as small as possible, and a similar statement applies for the college proposing-version (see [5], [8] and [12] for details).…”

Section: Stable Score-limits With Tiesmentioning

confidence: 99%

“…Finally, either we need to minimise the sum of score-limits with the objective function or to use conditions (10), (11), (12) and (13) to ensure that the decrease of any positive score-limit in the solution would cause the violation of a quota.…”

Section: Stable Score-limits With Ties and Lower Quotasmentioning

confidence: 99%

Integer Programming Methods for Special College Admissions Problems

Bíró

McBride

2014

Combinatorial Optimization and Applications

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Abstract. 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 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.

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Cutoff stability under distributional constraints with an application to summer internship matching

2022

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We introduce a new two-sided stable matching problem that describes the summer internship matching practice of an Australian university. The model is a case between two models of Kamada and Kojima on matchings with distributional constraints. We study three solution concepts, the strong and weak stability concepts proposed by Kamada and Kojima, and a new one in between the two, called cutoff stability. Kamada and Kojima showed that a strongly stable matching may not exist in their most restricted model with disjoint regional quotas. Our first result is that checking its existence is NP-hard. We then show that a cutoff stable matching exists not just for the summer internship problem but also for the general matching model with arbitrary heredity constraints. We present an algorithm to compute a cutoff stable matching and show that it runs in polynomial time in our special case of summer internship model. However, we also show that finding a maximum size cutoff stable matching is NP-hard, but we provide a Mixed Integer Linear Program formulation for this optimisation problem.

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Choice Function-Based Two-Sided Markets: Stability, Lattice Property, Path Independence and Algorithms

Cited by 17 publications

References 11 publications

College admissions with stable score-limits

College admissions with stable score-limits

Integer Programming Methods for Special College Admissions Problems

Cutoff stability under distributional constraints with an application to summer internship matching

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