Unique Stable Matchings

TL;DR

This paper investigates conditions for the uniqueness of stable matchings in two-sided markets, establishing equivalences involving market normal forms, preference acyclicity, and singleton preference lists.

Contribution

It proves that market normal form, preference acyclicity, and singleton preferences are equivalent conditions for unique stable matchings in two-sided markets.

Findings

Unique stable matching characterized by preference acyclicity.

Normal form market preserves stable matchings.

Preferences being singletons implies uniqueness.

Abstract

In this paper we consider the issue of a unique prediction in one to one two sided matching markets, as defined by Gale and Shapley (1962), and we prove the following. Theorem. Let P be a one-to-one two-sided matching market and let P be its associated normal form, a (weakly) smaller matching market with the same set of stable matchings, that can be obtained using procedures introduced in Irving and Leather (1986) and Balinski and Ratier (1997). The following three statements are equivalent (a) P has a unique stable matching. (b) Preferences on P* are acyclic, as defined by Chung (2000). (c) In P* every market participant's preference list is a singleton.