Unbalanced Random Matching Markets with Partial Preferences

TL;DR

The paper resolves a fundamental question in random stable matching: how long must agents’ partial preference lists be to guarantee a perfect stable matching with high probability, in both balanced and imbalanced markets. It establishes a sharp threshold on the list length $d$, showing balanced markets require $d_0 \sim \ln^2 n$, while unbalanced markets with imbalance $\alpha=o(1)$ require $d_0 = \ln n \cdot \ln \left( \frac{1+\alpha}{\alpha + 1/(n(1+\alpha))} \right)$; above this threshold all stable matchings are perfect w.h.p., and below it none are. The authors develop novel tools—probability games on rejection chains, a balls-in-bins coupling for DA analyses, and a sandwiching lemma across three random models $M$, $M^{\mathcal{C}}$, and $M^{\mathcal{J}}$—to obtain tight upper and lower bounds and to study the effect of competition on proposer ranks (the Stark-competition effect). They also extend the analysis to unbalanced markets and provide a lower bound on the expected rank of stable partners under partial preferences, highlighting the practical implications for market design when preferences are truncated. Overall, the work clarifies how imbalance and partial information interact to shape the existence and quality of stable matchings in large random markets.

Abstract

Properties of stable matchings in the popular random-matching-market model have been studied for over 50 years. In a random matching market, each agent has complete preferences drawn uniformly and independently at random. Wilson (1972), Knuth (1976) and Pittel (1989) proved that in balanced random matching markets, the proposers are matched to their $\ln n$th choice on average. In this paper, we consider markets where agents have partial (truncated) preferences, that is, the proposers only rank their top $d$ partners. Despite the long history of the problem, the following fundamental question remained unanswered: \emph{what is the smallest value of $d$ that results in a perfect stable matching with high probability?} In this paper, we answer this question exactly -- we prove that a degree of $\ln^2 n$ is necessary and sufficient. That is, we show that if $d < (1-ε) \ln^2 n$ then no stable matching is perfect and if $d > (1+ ε) \ln^2 n$, then every stable matching is perfect with high probability. This settles a recent conjecture by Kanoria, Min and Qian (2021). We generalize this threshold for unbalanced markets: we consider a matching market with $n$ agents on the shorter side and $n(α+1)$ agents on the longer side. We show that for markets with $α=o(1)$, the sharp threshold characterizing the existence of perfect stable matching occurs when $d$ is $\ln n \cdot \ln \left(\frac{1 + α}{α+ (1/n(α+1))} \right)$. Finally, we extend the line of work studying the effect of imbalance on the expected rank of the proposers (termed the ``stark effect of competition''). We establish the regime in unbalanced markets that forces this stark effect to take shape in markets with partial preferences.

Theorems & Definitions (27)

• Theorem
• Theorem
• Theorem 1
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• Theorem 4
• Definition 5
• Lemma 7
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• proof