Random Almost-Popular Matchings

TL;DR

This paper investigates the existence of approximately popular matchings in a setting where individuals have preferences over items, establishing bounds on the ratio of items to people that determine the likelihood of such matchings existing.

Contribution

The paper introduces bounds on the ratio of items to people that guarantee the existence of epsilon-popular matchings with high probability, extending previous results for the zero-popularity case.

Findings

Upper bound: epsilon-popular matchings exist with high probability if alpha(1-e^{-1/alpha}) > 1-epsilon.

Lower bound: epsilon-popular matchings exist with low probability if alpha(1-e^{-(1+e^{1/alpha})/alpha}) < 1-2epsilon.

Abstract

For a set $A$ of $n$ people and a set $B$ of $m$ items, with each person having a preference list that ranks all items from most wanted to least wanted, we consider the problem of matching every person with a unique item. A matching $M$ is called $\epsilon$-popular if for any other matching $M'$, the number of people who prefer $M'$ to $M$ is at most $\epsilon n$ plus the number of those who prefer $M$ to $M'$. In 2006, Mahdian showed that when randomly generating people's preference lists, if $m/n > 1.42$, then a 0-popular matching exists with $1-o(1)$ probability; and if $m/n < 1.42$, then a 0-popular matching exists with $o(1)$ probability. The ratio 1.42 can be viewed as a transition point, at which the probability rises from asymptotically zero to asymptotically one, for the case $\epsilon=0$. In this paper, we introduce an upper bound and a lower bound of the transition point in more general cases. In particular, we show that when randomly generating each person's preference list, if $\alpha(1-e^{-1/\alpha}) > 1-\epsilon$, then an $\epsilon$-popular matching exists with $1-o(1)$ probability (upper bound); and if $\alpha(1-e^{-(1+e^{1/\alpha})/\alpha}) < 1-2\epsilon$, then an $\epsilon$-popular matching exists with $o(1)$ probability (lower bound).