Stable Matchings with Choice Correspondences Under Acyclicity

TL;DR

This paper investigates the existence of stable matchings with choice correspondences under weakened acyclicity conditions, introducing new algorithms and stability notions for complex matching markets.

Contribution

It extends existing frameworks by weakening assumptions, introduces a Grow or Discard Algorithm, and proposes new stability concepts for many-to-many and one-to-one markets.

Findings

Stable matchings exist under substitutability and acyclicity conditions.

A constructive Grow or Discard Algorithm finds stable matchings.

New stability notion for one-to-one markets under binary acyclicity.

Abstract

We study the existence of stable matchings when agents have choice correspondences instead of preference relations. We extend the framework of \cite{chambers2017choice} by weakening the path independence assumption. For many-to-many markets, we show that stable matchings exist when choice correspondences satisfy substitutability and a new general acyclicity condition. We provide a constructive proof using a Grow or Discard Algorithm that iteratively expands or eliminates contracts until a strongly maximal individually rational set is reached. We provide an algorithm to obtain stable matchings in which rejected contracts are not permanently discarded, distinguishing our approach significantly from standard DAA-type algorithms. For one-to-one markets, we introduce a replacement-based notion of stability and provide an algorithm that constructs stable matchings when choice correspondences satisfy binary acyclicity, a property weaker than path independence. JEL classification: C62, C78, D01, D47 Keywords: choice correspondences, substitutability, general acyclicity, many-to-many matching, matching with contracts, Grow or Discard algorithm, replacement stability, binary acyclicity.