Designing Matching Mechanisms under General Distributional Constraints

Abstract

In this paper, we consider two-sided, many-to-one matching problems where agents in one side of the market (schools) impose some distributional constraints (e.g., a maximum quota for a set of schools), and develop a strategyproof mechanism that can handle a very general class of distributional constraints. We assume distributional constraints are imposed on a vector, where each element is the number of contracts accepted for each school. The only requirement we impose on distributional constraints is that the family of vectors that satisfy distributional constraints must be hereditary, which means if a vector satisfies the constraints, any vector that is smaller than it also satisfies them. When distributional constraints are imposed, a stable matching may not exist. We develop a strategyproof mechanism called Adaptive Deferred Acceptance mechanism (ADA), which is nonwasteful and "more fair" than a simple nonwasteful mechanism called the Serial Dictatorship mechanism (SD) and "less wasteful" than another simple fair mechanism called the Artificial Cap Deferred Acceptance mechanism (ACDA). We show that we can apply this mechanism even * Goto, Kurata, Yokoo: Department of Electrical Engineering and Computer Science, Kyushu University, Fukuoka 819-0395, Japan. Kojima: Department of Economics, Stanford University, Stanford, CA, 94305, United States. Tamura: Department of Mathematics, Keio University, Yokohama 223-8522, Japan.Emails: goto@agent.inf.kyushu-u.ac.jp, fkojima@stanford.edu, kurata@agent.inf.kyushu-u.ac.jp, aki-tamura@math.keio.ac.jp, yokoo@inf.kyushu-u.ac.jp. Fanqi Shi provided excellent research assistance. Tamura and Yokoo acknowledge the financial support from JSPS Kakenhi Grant Number 24220003. Kojima acknowledges the financial support from the National Research Foundation through its Global Research Network Grant (NRF-2013S1A2A2035408) as well as the Sloan Foundation.1 if the distributional constraints do not satisfy the hereditary condition by applying a simple trick, assuming we can find a vector that satisfy the distributional constraints efficiently. Furthermore, we demonstrate the applicability of our model in actual application domains. JEL Classification: C78, D61, D63

Attendance Boundary Policies and the Limits to Combating School Segregation

“…Controlled school choice is a recent attempt at incorporating political aims, e.g. desegregation (Cantillon, 2017) by using quotas (Kamada and Kojima, 2015;Goto et al, 2017). Such policies therefore represent an alternative to changing attendance boundaries.…”

Recent Developments in Matching with Constraints

“…7 Another possible direction of research is to study more general feasibility constraints. Kamada and Kojima (2016b) allow for a broader class of upper-bound constraints than those presented here (Goto et al (2016b) also study that class of constraints). This class of constraints, however, excludes lowerbound (floor) constraints, which are studied by Ehlers et al (2014), Goto et al (2016a), and Fragiadakis and Troyan (2016), among others.…”

“…Goto et al (2016b) propose a mechanism that is strategy-proof for doctors in the same environment as Kamada and, but the outcome of their mechanism is not weakly stable.8 Kojima, Tamura and Yokoo (2015) explore this connection through the theory of discrete convex analysis.…”

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…”

“…Developing a non-trivial strategyproof and fair mechanism that can handle ratio constraints is theoretically interesting and challenging. In existing works, it is shown that if constraints belong to a well-behaved class (which is called M-convex set), then a mechanism called generalized DA, which is based on DA, is strategyproof and fair [18,30]. As we discuss later, ratio constraints do not belong to this class.…”