On random exchange-stable matchings

TL;DR

This paper analyzes the properties of exchange-stable matchings in bipartite and non-bipartite settings, providing asymptotic expectations, variance bounds, and probability estimates for the existence of such matchings.

Contribution

It establishes the asymptotic expected number of exchange-stable matchings and bounds their standard deviation, revealing the rarity of instances with many such matchings.

Findings

Expected number of e-stable matchings asymptotic to rac{\u221a{\u03c0 n}}{2} for two-sided case

Expected number of e-stable matchings asymptotic to e^{1/2} for one-sided case

Probability that no matching is both stable and e-stable is exponentially high

Abstract

Consider the group of $n$ men and $n$ women, each with their own preference list for a potential marriage partner. The stable marriage is a bipartite matching such that no unmatched pair (man, woman) prefer each other to their partners in the matching. Its non-bipartite version, with an even number $n$ of members, is known as the stable roommates problem. Jose Alcalde introduced an alternative notion of exchange-stable, one-sided, matching: no two members prefer each other's partners to their own partners in the matching. Katarina Cechl\'arov\'a and David Manlove showed that the e-stable matching decision problem is $NP$-complete for both types of matchings. We prove that the expected number of e-stable matchings is asymptotic to $\left(\frac{\pi n}{2}\right)^{1/2}$ for two-sided case, and to $e^{1/2}$ for one-sided case. However, the standard deviation of this number exceeds $1.13^n$, ($1.06^n$ resp.). As an obvious byproduct, there exist instances of preference lists with at least $1.13^n$ ($1.06^n$ resp.) e-stable matchings. The probability that there is no matching which is stable and e-stable is at least $1-e^{-n^{1/6+o(1)}}$, ($1-O(2^{-n/2})$ resp.).