A characterization of the von Neumann and Morgenstern stable set in matching markets

2026-07-01

AI Summary

This study addresses the computational complexity of von Neumann–Morgenstern (vNM) stable sets in one-to-one matching markets, which arises from the definition of domination. By generalizing the classical Decomposition Lemma to arbitrary internally stable matching pairs, the work uncovers an intrinsic connection between internal stability and the cyclic structure of the market, and constructs a reduced environment in which all undominated outcomes are concentrated. Building on this reduction, it establishes an equivalence between vNM stable sets and the core of the simplified market, thereby proving their unique existence and providing an efficient constructive algorithm for their computation. This paper presents the first structural characterization and practical algorithmic solution for vNM stable sets in such settings.

Abstract

This paper studies the structure and computation of von Neumann-Morgenstern (vNM) stable sets in one-to-one matching markets. While pairwise stability and corewise stability coincide under strict preferences and provide a well-understood benchmark, vNM stability is defined through dominance relations among sets of matchings and remains considerably more difficult to characterize. A key contribution of the paper is a generalization of the classical Decomposition Lemma. We show that the structural decomposition traditionally used to compare stable matchings extends to any pair of matchings belonging to the same internally stable set. This result reveals a previously unexplored connection between internal stability and the cycle structure underlying matching markets. Building on this characterization, we identify the relationships that are relevant for dominance-based stability and derive a reduced environment that concentrates all undominated outcomes. Our main result shows that the vNM stable set is unique and admits a simple characterization in terms of the core of this reduced environment. The characterization provides both structural insight and a constructive procedure for computing the vNM stable set using standard matching theoretic tools.

Problem

• von Neumann-Morgenstern stable set
• matching markets
• dominance relations
• internal stability
• core

Innovation

• von Neumann-Morgenstern stable set
• matching markets
• Decomposition Lemma
• internal stability
• core