Instability in Stable Marriage Problem: Matching Unequally Numbered Men and Women

TL;DR

This paper investigates how the classic stable marriage problem behaves when the two groups being matched are of unequal sizes, revealing that small size differences can significantly affect the matching outcomes.

Contribution

It introduces a generalized stable marriage model with unequal group sizes and analyzes the impact on the Gale-Shapley algorithm's solutions.

Findings

Small deviations in group sizes cause large changes in matchings.

Unequal sizes lead to inevitable singles in the matching.

Theoretical and simulation results confirm the instability effects.

Abstract

The Stable Marriage Problem is to find a one-to-one matching for two equally sized sets of agents. Due to its widespread applications in the real world, especially the unique importance to the centralized match maker, a very large number of questions have been extensively studied in this field. This article considers a generalized form of stable marriage problem, where different numbers of men and women need to be matched pairwise and the emergence of single is inevitable. Theoretical analysis and numerical simulations confirm that even small deviations from equal number of two sides can have a large impact on matching solution of Gale-Shapley Algorithm. These results provide insights to many of the real-world applications when matching two sides with unequal number.