Random Stable Matchings

TL;DR

This paper investigates the likelihood of stable matchings existing in random graphs with various structures, providing numerical results and conjectures on their asymptotic behavior.

Contribution

It offers the first numerical analysis of the probability of stable matchings in random graphs and formulates conjectures based on observed patterns.

Findings

Probability decays algebraically on graphs with connectivity Θ(n)

Probability decays exponentially on regular grids

Probability converges to a positive value on finite connectivity Erdös-Rényi graphs

Abstract

The stable matching problem is a prototype model in economics and social sciences where agents act selfishly to optimize their own satisfaction, subject to mutually conflicting constraints. A stable matching is a pairing of adjacent vertices in a graph such that no unpaired vertices prefer each other to their partners under the matching. The problem of finding stable matchings is known as stable marriage problem (on bipartite graphs) or as stable roommates problem (on the complete graph). It is well-known that not all instances on non-bipartite graphs admit a stable matching. Here we present numerical results for the probability that a graph with $n$ vertices and random preference relations admits a stable matching. In particular we find that this probability decays algebraically on graphs with connectivity $\Theta(n)$ and exponentially on regular grids. On finite connectivity Erd\"os-R\'enyi graphs the probability converges to a value larger than zero. Based on the numerical results and some heuristic reasoning we formulate five conjectures on the asymptotic properties of random stable matchings.