Many-to-Many Matching Problem with Quotas

Transcription

1 Many-to-Many Matching Proble with Quotas Mikhail Freer and Mariia Titova February 2015 Discussion Paper Interdisciplinary Center for Econoic Science 4400 University Drive, MSN 1B2, Fairfax, VA Tel: Fax: ICES Website: ICES RePEc Archive Online at:

2 Many-to-Many Matching Proble with Quotas Mikhail L. Freer Mariia O. Titova Abstract In this paper, we present a solution to the any-to-any atching proble with quotas. In our setting, we have students who select an exact nuber of courses (exact quota q s ), and courses that ust adit at least q c students (lower quota). We present a generalization of the deferred acceptance echanis introduced in Gale and Shapley (1962), which returns the best pairwise-stable atching for courses, which is also unifor: every course has approxiately the sae nuber of students. Keywords: Two-sided atching, stability 1 Introduction The original atching proble was first articulated by Gale and Shapley (1962),who also proposed a echanis for finding a stable solution in one-to-one and any-to-one cases. Since then, the one-to-one and any-to-one atching echanis has been applied in nuerous settings, e.g., siple exchange arket by Shapley and Scarf (1974), kidney exchange by Roth et al. (2004), labor arket by Peranson and Roth (1997), and any others. Despite these advances, applications like the labor arket, exchange arket, and school choice process often present a any-to-any atching proble, to which a solution has yet to be proposed. In this paper, we present a generalization of the deferred acceptance (DA) algorith to the case of any-to-any atching with different preferences and quotas. Our exaple is acoursechoiceproble.inoursetting,studentshaveaquotaq S of courses that they ust take. Courses, in turn, have a lower quota q C such that each course ust accept at least q C students. We focus on a atching a unifor nuber of students to each course, such that every course has ore or less the sae nuber of students enrolled. This ay not be possible in every case; however, our algorith is sufficiently universal that it will always return a stable atching even if a unifor atching is ipossible. Departent of Econoics and Interdisciplinary Center for Econoic Science, George Mason University, Fairfax, VA, USA. E-ail: freer@gu.edu. National Research University Higher School of Econoics, Moscow, Russia. E-ail: otitova@gail.co. 1

3 2 The Model Suppose we have a set of students S = {S 1,...,S n },andasetofcoursesc = {C 1,...,C }. Each student ust choose exactly q S courses, while each course ust adit at least q C students. Students ay have (possibly different) preferences over the set of courses. Such preferences can be represented by a linear order, and all courses have (possibly different) linear order preferences over the set of students. To begin, we will attept to obtain a atching with an approxiately equal nuber of students attending each course, which we call unifor distribution. If no unifor stable atching exists, our echanis will nonetheless return a stable atching, provided one exists for a given profile of preferences. The average nuber of people in each course will be k = nq S, where n is the nuber of students, and is the nuber of courses. Clearly, this nuber is not necessarily an integer. Let us assue that p apple nq S apple p, where j nqs k p = 2 N, l nqs p = 2 N. Definition 1. Matching µ is a apping which assigns exactly q S different courses to each student in S and at least q C different students to each course in C. We denote the set of courses assigned by atching µ to the student s i and the set of students assigned to the course c j as µ(s i ) and µ(c j ),respectively. Definition 2. Blocking pair is a student s i and course c j,whoprefereachothertosoeof their partners in the current atching, i.e. s i cj s for soe s 2 µ(c j ) and c j si c for soe c 2 µ(s i ). Definition 3. The atching is (pairwise) stable, if there are no blocking pairs. A theory of stability for atching probles and connection between different types of stability was developed by Echenique and Oviedo (2006). The sae interpretation can be to the quotas we consider in the current paper. In line with Sotoayor (1999), we use the concept of pairwise-stability, rather than setwise-stability; thus, in principle, the atching can be outside the core. 2

4 3 The Mechanis Firstly, we calculate k, p, andp. The goal is for each course to enroll either p or p students, and for each student to be aditted to exactly q S courses. The algorith Step 1: Each course proposes to accept the first p students in its preference list. Each student accepts no ore than q S proposals according to his/her preferences, rejecting the rest. Step k: Each course that has z<pstudents proposes to accept p z students it has not yet proposed to. Each student accepts no ore than q S proposals according to his/her preferences, rejecting the others. The algorith stops when every course that has not reached the axiu quota p has proposed adission to every student. Reark 1. The echanis is also applicable for the odel with upper quotas for courses, if the upper quota for courses is no less that p. Reark 2. The echanis can be applied if and only if the length of every course preference list is at least p, eaning that for every course at least p students are acceptable. More properties of the atching obtained by this algorith are discussed in the section below. 4 Results Our ain result is pairwise stability: Theore 1. The atching returned by echanis is stable, whether lower quotas p are filled, or not. Proof. Suppose there is a blocking pair (s j,c i ),suchthatc i /2 µ(s j ),andthereisc k : c i sj c k and c k 2 µ(s j ). Then two options are possible: 1) c i never proposed to s j,sos j is worse than all students currently enrolled in course c i,so(s j,c i ) is not a blocking pair; 2) c i proposed to s j,buts j rejected. It eans, that s j preferred other q S courses and (s j,c i ) is not a blocking pair. Acoursethatdidnotfillitslowerquotaalsocannotbeinablockingpair,sinceithas proposed adission to all of the students, and at least n p +1 students traded it for a ore preferable course. Theore 2. Stable atching returned by the echanis is the best "unifor" 1 stable atching for courses. 1 The coparison is possible over the set of µ 0 : µ(c i ) = µ 0 (c i ) for all c i 2 C. 3

5 Proof. Assue that 9µ 0 which is a better atching for c k : 9s i,s j 2 S s.t. s i /2 µ(c k ), s i 2 µ 0 (c k ),buts i ck s j, s j 2 µ(c k ) Then, either the course did not propose adission to s i (contradiction to the echanis) or the student rejected the proposal (µ 0 is not stable). If in µ(c k ) the upper quota p is not filled, then all students were proposed adission, and soe of the rejected this course, so ) µ 0 is not stable. Theore 3. If our algorith returns a atching µ with at least one unfilled quota p, then no stable atching, with every college getting p or p students, exists. Proof. Suppose c i didn t fill the lower quota in µ. Alsosupposethereexistsastableatching µ 0, where the college c i fills p. Then, all students in µ 0 (c i ) are strictly better than one student in µ(c i ) (epty space), which contradicts the fact that µ is the best possible stable atching for courses, i.e. the supreu of the lattice of stable atchings. These three theores iply that even if the lower quota p is not filled, we still get a stable atching. Since in practice p is usually significantly greater than q C,ourechanisprovides the ost unifor distribution of students in each course possible in a stable atching. It can be seen fro the echanis that we do not let courses propose adission to ore than p students at any tie. The next section illustrates that doing so when soe courses did not fill their quota q C does not result in any other stable atchings. 4.1 Changing the upper quota p Proposition 1. If course c j has less than q C students after all proposals with p>q C,there is no stable atching with any p 0 p. Proof. Course proposed adission to all students fro its profile and at least n q C +1 students rejected it. So with all p 0 p at least n q c +1students will reject the offer fro c j. The reason is that they will get no less proposals than in the case with p quotas, since because all courses have ore attepts to propose places for the students. Proposition 2. If course c j has less than q C students after all proposals with p is no stable atching with any p 0 < p. q C,there Proof. If p 0 < p, thennoatchingsexistduetothewaywedefinedp. Corollary. Fro the two propositions above it follows that if at least one course does not reach the quota q C with p, then there is no stable atching with any other p 0. In other words, by restricting the nuber of course proposals by p in our echanis, we do not lose any relevant atchings; if no atching is returned, then none exists for a given profile of preferences. 4

6 5 Conclusion In this paper, we introduced a generalization of the deferred acceptance algorith for a any-to-any atching proble with quotas. Our results suggest that if a pairwise-stable atching exists, our algorith will achieve it, and that this atching is as unifor as possible. By providing a echanis, we also prove sufficient conditions for a stable atching: any atching returned by our echanis will be stable, whether or not all quotas are satisfied. In the end, our desire to construct a unifor atching does not significantly affect our results: we show that changing the upper quota for courses does not directly affect the existence of a unifor atching. If one course did not have enough students enrolled in a unifor setting, changing the upper quota would not help it. Although the echanis returns Pareto efficient 2, and pairwise-stable atching, the echanis is not strategy-proof. It is consistent with previous results: in probles with quotas, stability and strategy-proofness cannot be obtained siultaneously. Manipulation under the deferred acceptance algorith is not difficult, as was shown in Chen and Sönez (2006). Nonetheless, anipulation does not appear frequently in the any-to-one atching proble and does not significantly iprove the outcoe of an individual. One ore arguent against considering the strategy-proofness is suggested by Peranson and Roth (1999): less than 0.1% of individuals in the National Resident Matching Progra can iprove their outcoe by strategic behavior. References Chen, Y. and Sönez, T. (2006). School choice: an experiental study. Journal of Econoic theory, 127(1): Echenique, F. and Oviedo, J. (2006). A theory of Stability in Many-to-Many Matching Markets. Theoretical Econoics, 1(2): Gale, D. and Shapley, L. S. (1962). College Adissions and the Stability of Marriage. The Aerican Matheatical Monthly, 69(1):9 15. Peranson, E. and Roth, A. E. (1997). The effects of the change in the NRMP atching algorith. JAMA, 278(9): Peranson, E. and Roth, A. E. (1999). The Redesign of the Matching Market for Aerican Physicians: Soe Engineering Aspects of Econoic Design. Aerican Econoic Review, 89(4): Roth, A. E., Sönez, T., and Ünver, M. U. (2004). Kidney Exchange. The Quarterly Journal of Econoics, pages With respect to a fixed nuber of students per course. 5

7 Shapley, L. and Scarf, H. (1974). On Cores and Indivisibility. Journal of atheatical econoics, 1(1): Sotoayor, M. (1999). Three Rearks on the Many-to-Many Stable Matching Proble. Matheatical Social Sciences, 38(1):