The random stable roommates problem typically has no solution

TL;DR

The paper resolves a long-standing conjecture on the random stable roommates problem by showing that, for large $n$, the existence of a stable matching is extremely unlikely, with $\mathbb{P}(X\ge1) \le n^{-1/17}$. The authors introduce a novel conditional approach: fix a stable matching $\Pi$ and analyze the stability probabilities of other matchings $\Pi_1$ through the symmetric-difference cycles, using a careful decomposition into three rare-event types ($\mathcal{D}_1$, $\mathcal{D}_2$, $\mathcal{D}_3$) and a quasirandomness event on the preference lists. They develop a new conditional framework that yields precise first- and higher-moment bounds for multiple stable matchings, including disjoint and overlapping cycle configurations, and show that disjoint cycles typically produce many stable companions of $\Pi$, forcing a polynomially small probability of any stable matching existing overall. By optimizing parameters and leveraging a Lambert $W$-based calculation, they derive the explicit exponent $1/17$, thereby establishing a polynomial decay in the existence probability and providing the first vanishing upper bound. This work thus completes Gusfield–Irving's conjecture and sharpens the understanding of stability in random instances with potential implications for related matching models.

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 $n^{-1/17}$.

Theorems & Definitions (44)

• Theorem 1
• Lemma 2: Pittel, P:93
• Lemma 3
• Lemma 4
• Lemma 5
• proof : Proof of Theorem \ref{['th:no-stable-matching']}
• Lemma 6
• Lemma 7
• proof
• Lemma 9