Unique stable matchings

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.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).