The random stable roommates problem typically has no solution

TL;DR

This paper proves that for large random instances of the stable roommates problem, the probability of having a stable matching approaches zero, confirming a long-standing conjecture and quantifying the rarity of solutions.

Contribution

It rigorously proves that random stable roommates instances almost surely lack solutions as the number of participants grows large.

Findings

Probability of stable matching is at most n^{-1/17} for large n

Confirms Gusfield and Irving's 1989 conjecture

Shows stable matchings are extremely rare in large random instances

Abstract

Assume that $n = 2k$ potential roommates each have an ordered preference of the $n-1$ others. A stable matching is a perfect matching of the $n$ roommates in which no two unmatched people prefer each other to their matched partners. In their seminal 1962 stable marriage paper, Gale and Shapley noted that not every instance of the stable roommates problem admits a stable matching. In the case when the preferences are chosen uniformly at random, Gusfield and Irving predicted in 1989 that there is no stable matching with high probability for large $n$. We prove this conjecture and show that for $n$ sufficiently large, the probability there is a stable matching is at most $n^{-1/17}$.