Finding a stable matching under type-specific minimum quotas

Abstract

In matching problems with minimum and maximum type-specific quotas, there may not exist a stable (i.e., fair and non-wasteful) assignment (Ehlers et al., 2014). This paper investigates the structure of schools' priority rankings which guarantees stability. First, we show that there always exists a fair and non-wasteful assignment if for each type of students, schools have common priority rankings over a certain number of bottom students. Next, we show that the pairwise version of this condition characterizes the maximal domain of two schools' priority rankings over same type students to guarantee the existence of stable assignments. To prove the existence theorem, we propose a new mechanism Deferred Acceptance with Precedence Lists (DAPL), which is feasible, non-wasteful, strictly PL-fair and group strategy-proof for any priority rankings. Strict PL-fairness is weaker than fairness, but DAPL satisfies fairness under our sufficient condition. We also show that there is no strategy-proof mechanism that Pareto dominates DAPL whenever the outcome of DAPL is Pareto dominated by a stable assignment. * This is originally Chapter 3 of my Ph.D. thesis at Harvard University. I am very grateful to my advisors Eric Maskin, Scott Kominers and Edward Glaeser for their constant guidance. I am grateful to Lars

Strategyproof and fair matching mechanism for ratio constraints

“…Even though this is a strong assumption, it is a necessary condition for the existence of a feasible matching in our model. The same assumption is widely used in existing works [2,15,18,43]. Moreover, to guarantee the existence of a feasible matching, ratio must be at…”

“…A standard market deals with maximum quotas, which are capacity limits that cannot be exceeded. However, many real-world matching markets are subject to a variety of distributional constraints, including regional maximum quotas, which restrict the total number of students assigned to a set of schools [25], minimum quotas, which guarantee that a certain number of students are assigned to each school [7,15,17,20,41,42], and diversity constraints, which enforce that a school satisfies a balance between different types of students, typically in terms of socioeconomic status [13,19,28,31,43].…”

Matching with floor constraints

“…Fragiadakis et al (2016) take the impossibility result of Ehlers et al (2014) and propose two mechanisms that offer different trade‐offs between fairness and non‐wastefulness. Tomoeda (2018) adopts the stability notion introduced by Ehlers et al (2014) and finds sufficient conditions on hospital preferences such that a stable matching exists.…”

Affinely representable lattices, stable matchings, and choice functions

“…Although successful, the marriage model does not capture features that have become of crucial importance both inside and outside academia. For instance, there is growing attention to models that can increase diversity in school cohorts (Nguyen and Vohra, 2019;Tomoeda, 2018). Such constraints cannot be represented in the original model, or even its one-to-many or many-to-many generalizations, since admission decisions with diversity concerns cannot be captured by a strict preference list.…”