June 28, 2012
Abstract
In this paper we introduce a new class of matching problems which mimics tuition exchanges programs used by colleges in US as a benet to their faculty members. The most important benet of participating to the tuition exchange program is that colleges strengthen their compen-sation package to their faculty and sta at a very nominal cost. Participating colleges nd The Tuition Exchange can serve as a strong incentive for top job candidates to accept their oers.
Hence, the tuition exchange programs help level the playing eld for small colleges in hiring and retaining promising faculty. In tuition exchange programs, each college ranks its own faculty members according to the length of the employment of the faculty. Based on this ranking each college determines the set of eligible dependents of faculty who can participate the scholarship program. Then, the eligible students (dependents) are awarded with scholarship according to the preferences of colleges over eligible students, preferences of eligible students over colleges and the number of available slot in each college. The main concern for each colleges is maintaining a balance between the number of students certied as eligible by that institution (exports) and the number of scholarships awarded to students certied as eligible by other member colleges enrolling at that institution (imports). We propose a new mechanism, two sided top trading mechanism (2S-TTC), which is a variant of well-known top trading cycle mechanism . To our knowledge this is the rst time such that a variant of TTC mechanism is used in a market in which both sides (colleges and students) are strategic. We show that 2S-TTC mechanism selects balanced matching which is not dominated by another balanced matching. Moreover, it cannot be manipulated by students and it respects the internal rankings of colleges. We also show that it is the unique mechanism holding these features.
∗We thank John Duy for initial discussions.
†Address: The University of Texas at Austin, Department of Economics ; e-mail: umutdur@gmail.com; web page:
https://sites.google.com/site/umutdur/
‡Address: Boston College, Department of Economics, 140 Commonwealth Ave., Chestnut Hill, MA, 02467; e-mail:
unver@bc.edu; web page: http://www2.bc.edu/~unver
1
On the structural characteristics of the Stable Marriage polytope
∗Pavlos Eirinakis1, Dimitrios Magos2, Ioannis Mourtos1, Panayiotis Miliotis1
1Department of Management Science & Technology,
Athens University of Economics and Business, 76 Patission Ave., 104 34 Athens, Greece email: {peir,mourtos,miliotis}@aueb.gr
2Department of Informatics, Technological Educational Institute of Athens
Ag. Spyridonos Str., 12210 Egaleo, Greece email: dmagos@teiath.gr
Abstract
TheStable Marriage problem asks for a matching of men to women that is stable under given preferences.
It has been observed that some man-woman pairs have the property that although they are non-stable (i.e., they participate in no solution), they cannot be removed from the preference lists; such a removal would alter the set of solutions. However, these pairs have not been characterized yet. Likewise, some of the fundamental characteristics of the Stable Marriage polytope have not been established. In the current work, we show that these two seemingly distant open issues are closely related. We identify the pairs with the above-mentioned property and present a polynomial algorithm for producing them. This is accomplished by using the partial order defined onrotations, representable by therotation-poset graphG, and itstransitive reduction G−. Utilizing that result, we derive the dimension of the Stable Marriage polytopePand all alternative minimal linear descriptions.
More specifically, we establish that the dimension of P equals the number of nodes ofG (i.e., the number of rotations). Further, we establish the minimal equation system and show that non-removable non-stable pairs induce some of the facets of P.The remaining facets of P are also identified with the use of the graph G−. Hence, we obtain a minimal linear description ofP.In fact, we derive all alternative such descriptions as different inequalities may define the same facet and several equations may take each other’s place in the minimal equation system.
∗This research has been co-funded by the European Union (European Social Fund) and Greek national resources under the framework of the “Archimedes III: Funding of Research Groups in TEI of Athens” project of the “Education & Lifelong Learning” Operational Programme.
1
An Equilibrium Analysis of the Probabilistic Serial Mechanism
Özgün Ekici
yOnur Kesten
zMay, 2012
Abstract
The prominent mechanism of the recent literature in the assignment problem is the probabilistic serial (PS). Under PS, the truthful (preference) pro…le always constitutes an ordinal Nash Equilibrium, inducing a random assignment that satis…es the appealing ordinal e¢ ciency and envy-freeness properties. We show that both properties may fail to be satis…ed by a random assignment induced in an ordinal Nash Equilibrium where one or more agents are non-truthful. Worse still, the truthful pro…le may not constitute a Nash Equilibrium, and every non-truthful pro…le that constitutes a Nash Equilibrium may lead to a random assignment which is not ordinally e¢ cient, not even weakly envy-free, and which admits an ex-post ine¢ cient decomposition. A strong ordinal Nash Equilibrium may not exist, but when it exists, any pro…le that constitutes a strong ordinal Nash Equilibrium induces the random assignment induced under the truthful pro…le. The results of our equilibrium analysis of PS call for caution when implementing it in small assignment problems.
JEL Classi…cation Numbers: C70, D61, D63
Keywords: random assignment, probabilistic serial, equilibrium, Nash Equilibrium, ordinal Nash Equilibrium, strong ordinal Nash Equilibrium, ordinal e¢ ciency, envy-freeness.
We would like to thank Utku Ünver and Morimutsu Kurino for useful discussions and comments.
yÖzye¼gin University, Istanbul, Turkey. Email: ozgun.ekici@ozyegin.edu.tr
zTepper School of Business, Carnegie Mellon University, PA 15213, USA. Email: oke-sten@andrew.cmu.edu.