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

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.

The Stable Marriage Problem: an Interdisciplinary Review from the Physicist's Perspective

“…How to define the energy for these people? There are several ways to do this, here we shall adopt the convention used in [39] where the energy of a single is defined as 1 + N for men and 1 + M for women. For N ≤ M , all men will be married; while for N > M, there will be M married men and N − M single men.…”

Competition May Increase Social Utility in Bipartite Matching Problem

“…In particular, for a completely random wish list, that is when ω = 0, the average energy of men is log(N) + 0.522, which is far less than the average energy of women, which is N/[log(N) + 0.522], so the active side takes a huge advantage in the matching. However, a recent research [21] shows that in a random bipartite matching, if one woman is removed from the matching, the average energy of men will become N/log(N), and the average energy of women will become log(N). The happiness of the positive and passive sides is completely reversed.…”

“…Here we thoroughly study the impact of competition on bipartite matching, and instead of intuitive result that competition may reduce the well-being of the society, our result shows that proper competition can make society happier. On the other hand, a recent research [21] shows that random bipartite matching is very unstable because reducing only one woman will lead to a dramatic change in the matching result. However, unstable situations are rarely observed in real life, even though the numbers of matching parties in reality are often different.…”

Coordinating Manipulation in Real-time Interactive Mechanism of College Admission: Agent-Based Simulations

“…Experimental evidence shows that the performance of the parallel choice mechanism is better than the immediate acceptance mechanism [12]. e parallel choice mechanism resembles the deferred acceptance mechanism [2], which is first proposed by Gale and Shapley [1] and widely adopted in two-sided matching [13,14]. Deferred acceptance mechanism is proved to have some nice properties such as strategyproof, stability, and elimination of justified envy [2,13,14].…”