On constrained matchings, stable under random preferences

TL;DR

This paper studies the existence and number of stable matchings in a bipartite setting with random admissibility of pairs, revealing phase transitions depending on the probability parameter p.

Contribution

It extends previous work by analyzing the impact of random admissibility on stable matchings, identifying thresholds for their existence and quantity.

Findings

Expected number of stable matchings tends to 0 if p<log^2(n)/n.

A stable matching exists whp if p>(9/4)log^2(n)/n.

Number of unmatched individuals grows as a fractional power of n if p<log^2(n)/n.

Abstract

Colloquially, there are two groups, $n$ men and $n$ women, each man (woman) ranking women (men) as potential marriage partners. A complete matching is called stable if no unmatched pair prefer each other to their partners in the matching. If some pairs are not admissible, then such a matching may not exist, but a properly defined partial stable matching exists always, and all such matchings involve the same, equi-numerous, groups of men and women. Earlier we proved that, for the complete, random, preference lists, with high probability (whp) the total number of complete stable matchings is, roughly, of order $n^{1/2}$, at least. Here we consider the case that the preference lists are still complete, but a generic pair (man,woman) is admissible with probability $p$, independently of all other $n^2-1$ pairs. It is shown that the expected number of complete stable matchings tends to $0$ if, roughly, $p<\tfrac{\log^2 n}{n}$ and to infinity if $p>\tfrac{\log^2 n}{n}$. We show that whp: (a) there exists a complete stable matching if $p>(9/4)\tfrac{\log^2 n}{n}$, (b) the number of unmatched men and women is bounded if $p> \tfrac{\log^2n}{n}$, and (c) this number grows as a fractional power of $n$ for $p<\tfrac{\log^2 n}{n}$.