Random Popular Matchings with Incomplete Preference Lists

Abstract

For a set A of n people and a set B of m items, with each person having a preference list that ranks some items in order of preference, we consider the problem of matching every person with a unique item. A matching M is popular if for any other matching $M'$, the number of people who prefer M to $M'$ is not less than the number of those who prefer $M'$ to M. For given n and m, consider the probability of existence of a popular matching when each person’s preference list is independently and uniformly generated at random. Previously, Mahdian showed that when people’s preference lists are strict (containing no ties) and complete (containing all items in B), if $\alpha = m/n > \alpha _*$, where $\alpha _* \approx 1.42$ is the root of equation $x^2 = e^{1/x}$, then a popular matching exists with probability $1-o(1)$; and if $\alpha < \alpha _*$, then a popular matching exists with probability o(1), i.e. a phase transition occurs at $\alpha _*$. In this paper, we investigate phase transitions in more general cases when people’s preference lists are not complete. In particular, we show that in the case that each person has a preference list of length k, if $\alpha > \alpha _k$, where $\alpha _k \ge 1$ is the root of equation $x e^{-1/2x} = 1-(1-e^{-1/x})^{k-1}$, then a popular matching exists with probability $1-o(1)$; and if $\alpha < \alpha _k$, then a popular matching exists with probability o(1).

A full version of the paper is available at https://arxiv.org/abs/1609.07288.