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

Controlled School Choice with Soft Bounds and Overlapping Types

“…Alternatively, Goto, Kojima, Kurata, Tamura, and Yokoo (2016) present a general framework for handling more unrestricted class of constraints (beyond M -concavity) and a class of strategy-proof mechanism that is different from the generalized DA mechanism. Our DA-OT is not an instance of their mechanism; it is an instance of the generalized DA mechanism.…”

Stable project allocation under distributional constraints

“…[39]). Strategy-proofness can also be satisfied by modified variants of the deferred-acceptance mechanism for the case of lower quotas, as suggested also for the Japanese resident allocations [28,29,25]. However, if we allow ties and we consider goals such as rank-minimisation then our mechanism becomes manipulable.…”

“…Distributional constraints are present in many two-sided matching markets. In the Japanese resident allocation the government wants to ensure that the doctors are evenly distributed across the country, and to achieve this they imposed lower quotas on the number of doctors allocated in each region [28,29,25]. Distributional objectives can also appear in school choice programs, where the decision makers want to control the socio-ethnical distribution of the students [2,15,20,21].…”

A Rule for Committee Selection with Soft Diversity Constraints

“…We now present an axiom that avoids scenarios where a candidate may feel that she deserves the place of a lesser ranked candidate. The axiom is adapted from the literature on stable matching with distributional constraints Goto et al, 2017;Kojima et al, 2014;Ehlers et al, 2014;Kamada and Kojima, 2015).…”