Uniqueness of stable matchings without large indifference sets

Prove that in matching markets with aligned preferences and measurable agent populations, the set of stable matchings is a singleton when there is no positive-measure subset of agents who are indifferent among multiple potential partners (i.e., when the model excludes large groups of indifferent agents).

Background

The paper shows that stable matchings may be non-unique when there is a positive measure of agents who are indifferent among several potential matches, providing an explicit counterexample with two distinct stable matchings that differ in welfare implications. The authors then advance a conjecture that uniqueness can be recovered by ruling out such large indifference sets.

They prove uniqueness in one-dimensional spatial markets (R) with non-atomic measures when the signed imbalance measure changes sign a finite number of times, via a constructive algorithm (Proposition 1). They also discuss structural properties in higher dimensions (R^d), noting that uniqueness can hold under certain optimal transport cost structures. The conjecture aims to generalize uniqueness beyond the specific one-dimensional case by imposing a condition that excludes large sets of indifferent agents.