scholarly journals School choice under partial fairness

2019 ◽  
Vol 14 (4) ◽  
pp. 1309-1346 ◽  
Author(s):  
Umut Dur ◽  
A. Arda Gitmez ◽  
Özgür Yilmaz
Keyword(s):  
School Choice ◽  
Stable Matching ◽  
Choice Problem ◽  
Stable Matchings ◽  
Characterization Result ◽  
Partial Stability ◽  
Incentive Property ◽  
Student Exchange ◽  
Efficient Matching ◽  
Weak Axiom

We generalize the school choice problem by defining a notion of allowable priority violations. In this setting, a weak axiom of stability (partial stability) allows only certain priority violations. We introduce a class of algorithms called the student exchange under partial fairness (SEPF). Each member of this class gives a partially stable matching that is not Pareto dominated by another partially stable matching (i.e., constrained efficient in the class of partially stable matchings). Moreover, any constrained efficient matching that Pareto improves upon a partially stable matching can be obtained via an algorithm within the SEPF class. We characterize the unique algorithm in the SEPF class that satisfies a desirable incentive property. The extension of the model to an environment with weak priorities enables us to provide a characterization result that proves the counterpart of the main result in Erdil and Ergin (2008).

Stable Matching with Double Infinity of Workers and Firms

2018 ◽  
Vol 19 (2) ◽  
Author(s):  
Matías Fuentes ◽  
Fernando Tohmé
Keyword(s):  
Stable Matching ◽  
Pareto Optimal ◽  
Stable Matchings ◽  
Large Market ◽  
Countably Infinite

Abstract In this paper we analyze the existence of stable matchings in a two-sided large market in which workers are assigned to firms. The market has a continuum of workers while the set of firms is countably infinite. We show that, under certain reasonable assumptions on the preference correspondences, stable matchings not only exist but are also Pareto optimal.

Near-Feasible Stable Matchings with Couples

2018 ◽  
Vol 108 (11) ◽  
pp. 3154-3169 ◽  
Author(s):  
Thành Nguyen ◽  
Rakesh Vohra
Keyword(s):  
Medical Students ◽  
Stable Matching ◽  
Teaching Hospitals ◽  
Matching Problem ◽  
Stable Matchings ◽  
Student Preferences

The National Resident Matching program seeks a stable matching of medical students to teaching hospitals. With couples, stable matchings need not exist. Nevertheless, for any student preferences, we show that each instance of a matching problem has a “nearby” instance with a stable matching. The nearby instance is obtained by perturbing the capacities of the hospitals. In this perturbation, aggregate capacity is never reduced and can increase by at most four. The capacity of each hospital never changes by more than two. (JEL C78, D47, I11, J41, J44)

scholarly journals Adapting Stable Matchings to Evolving Preferences

2020 ◽  
Vol 34 (02) ◽  
pp. 1830-1837 ◽  
Author(s):  
Robert Bredereck ◽  
Jiehua Chen ◽  
Dušan Knop ◽  
Junjie Luo ◽  
Rolf Niedermeier
Keyword(s):  
Computational Complexity ◽  
Computational Cost ◽  
Modern Society ◽  
Stable Matching ◽  
Stable Marriage ◽  
Fixed Parameter Tractability ◽  
Stable Matchings ◽  
Changing Environments ◽  
Fixed Parameter ◽  
Comprehensive Picture

Adaptivity to changing environments and constraints is key to success in modern society. We address this by proposing “incrementalized versions” of Stable Marriage and Stable Roommates. That is, we try to answer the following question: for both problems, what is the computational cost of adapting an existing stable matching after some of the preferences of the agents have changed. While doing so, we also model the constraint that the new stable matching shall be not too different from the old one. After formalizing these incremental versions, we provide a fairly comprehensive picture of the computational complexity landscape of Incremental Stable Marriage and Incremental Stable Roommates. To this end, we exploit the parameters “degree of change” both in the input (difference between old and new preference profile) and in the output (difference between old and new stable matching). We obtain both hardness and tractability results, in particular showing a fixed-parameter tractability result with respect to the parameter “distance between old and new stable matching”.

scholarly journals Coalitions and cliques in the school choice problem

2015 ◽  
Vol 8 (5) ◽  
pp. 801-823
Author(s):  
Sinan Aksoy ◽  
Adam Azzam ◽  
Chaya Coppersmith ◽  
Julie Glass ◽  
Gizem Karaali ◽  
...  
Keyword(s):  
School Choice ◽  
Choice Problem

scholarly journals Pairwise Stable Matching in Large Economies

Econometrica ◽  
2021 ◽  
Vol 89 (6) ◽  
pp. 2929-2974 ◽  
Author(s):  
Michael Greinecker ◽  
Christopher Kah
Keyword(s):  
General Class ◽  
Stable Matching ◽  
High Dimensional ◽  
Transferable Utility ◽  
Stable Matchings ◽  
Matching Problems ◽  
Joint Distributions ◽  
Special Cases ◽  
Matching Models ◽  
Stability Notions

We formulate a stability notion for two‐sided pairwise matching problems with individually insignificant agents in distributional form. Matchings are formulated as joint distributions over the characteristics of the populations to be matched. Spaces of characteristics can be high‐dimensional and need not be compact. Stable matchings exist with and without transfers, and stable matchings correspond precisely to limits of stable matchings for finite‐agent models. We can embed existing continuum matching models and stability notions with transferable utility as special cases of our model and stability notion. In contrast to finite‐agent matching models, stable matchings exist under a general class of externalities.

scholarly journals The structure of priority in the school choice problem

2019 ◽  
Vol 35 (3) ◽  
pp. 361-381
Author(s):  
Conal Duddy
Keyword(s):  
School Choice ◽  
Walking Distance ◽  
Extended Formulation ◽  
Choice Problem ◽  
A Priority ◽  
Priority Ordering

AbstractIn a school choice problem, each school has a priority ordering over the set of students. These orderings depend on criteria such as whether a student lives within walking distance or has a sibling at the school. A priority ordering provides a ranking of students but nothing more. I argue that this information is sufficient when priority is based on merit but not when priority is based on criteria such as walking distance. I propose an extended formulation of the problem wherein a ‘priority matrix’, indicating which criteria are satisfied by each student-school pair, replaces the usual priority orderings.

scholarly journals Stability and Median Rationalizability for Aggregate Matchings

Games ◽  
2021 ◽  
Vol 12 (2) ◽  
pp. 33
Author(s):  
Federico Echenique ◽  
SangMok Lee ◽  
Matthew Shum ◽  
M. Bumin Yenmez
Keyword(s):  
Distributive Lattice ◽  
Empirical Studies ◽  
Revealed Preference ◽  
Stable Matching ◽  
Stable Matchings ◽  
Preference Theory ◽  
Fundamental Properties ◽  
Revealed Preference Theory

We develop the theory of stability for aggregate matchings used in empirical studies and establish fundamental properties of stable matchings including the result that the set of stable matchings is a non-empty, complete, and distributive lattice. Aggregate matchings are relevant as matching data in revealed preference theory. We present a result on rationalizing a matching data as the median stable matching.

scholarly journals Mallows permutations as stable matchings

2020 ◽  
pp. 1-25
Author(s):  
Omer Angel ◽  
Alexander E. Holroyd ◽  
Tom Hutchcroft ◽  
Avi Levy
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Stable Matching ◽  
Countable Family ◽  
Stable Matchings ◽  
Uncountable Family ◽  
Edge Crossing ◽  
Vertex Set ◽  
Infinite Intervals ◽  
The Law

Abstract We show that the Mallows measure on permutations of $1,\dots ,n$ arises as the law of the unique Gale–Shapley stable matching of the random bipartite graph with vertex set conditioned to be perfect, where preferences arise from the natural total ordering of the vertices of each gender but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely, every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph $K_{{\mathbb Z},{\mathbb Z}}$ falls into one of two classes: a countable family $(\sigma _n)_{n\in {\mathbb Z}}$ of tame stable matchings, in which the length of the longest edge crossing k is $O(\log |k|)$ as $k\to \pm \infty $ , and an uncountable family of wild stable matchings, in which this length is $\exp \Omega (k)$ as $k\to +\infty $ . The tame stable matching $\sigma _n$ has the law of the Mallows permutation of ${\mathbb Z}$ (as constructed by Gnedin and Olshanski) composed with the shift $k\mapsto k+n$ . The permutation $\sigma _{n+1}$ dominates $\sigma _{n}$ pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.

scholarly journals Controlled School Choice with Soft Bounds and Overlapping Types

2017 ◽  
Vol 58 ◽  
pp. 153-184 ◽  
Author(s):  
Ryoji Kurata ◽  
Naoto Hamada ◽  
Atsushi Iwasaki ◽  
Makoto Yokoo
Keyword(s):  
School Choice ◽  
Stable Matching ◽  
Extended Model ◽  
The Public ◽  
Distributional Constraints ◽  
Deferred Acceptance ◽  
Strategy Proof ◽  
Stability Definition ◽  
Model Computer ◽  
Straightforward Extension

School choice programs are implemented to give students/parents an opportunity to choose the public school the students attend. Controlled school choice programs need to provide choices for students/parents while maintaining distributional constraints on the composition of students, typically in terms of socioeconomic status. Previous works show that setting soft-bounds, which flexibly change the priorities of students based on their types, is more appropriate than setting hard-bounds, which strictly limit the number of accepted students for each type. We consider a case where soft-bounds are imposed and one student can belong to multiple types, e.g., “financially-distressed” and “minority” types. We first show that when we apply a model that is a straightforward extension of an existing model for disjoint types, there is a chance that no stable matching exists. Thus we propose an alternative model and an alternative stability definition, where a school has reserved seats for each type. We show that a stable matching is guaranteed to exist in this model and develop a mechanism called Deferred Acceptance for Overlapping Types (DA-OT). The DA-OT mechanism is strategy-proof and obtains the student-optimal matching within all stable matchings. Furthermore, we introduce an extended model that can handle both type-specific ceilings and floors and propose a extended mechanism DA-OT* to handle the extended model. Computer simulation results illustrate that DA-OT outperforms an artificial cap mechanism where we set a hard-bound for each type in each school. DA-OT* can achieve stability in the extended model without sacrificing students’ welfare.

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