Random Popular Matchings with Incomplete Preference Lists

TL;DR

This paper studies the probability of the existence of popular matchings in random incomplete preference lists, revealing phase transitions depending on the ratio of items to people and list length.

Contribution

It extends previous work on complete lists to incomplete lists with fixed length, identifying new phase transition thresholds for the existence of popular matchings.

Findings

Phase transition occurs at a specific ratio 2 for complete lists.

For incomplete lists of fixed length , a similar phase transition is identified.

The threshold  depends on the list length and is characterized by a specific equation.

Abstract

Given a set $A$ of $n$ people and a set $B$ of $m \geq n$ items, with each person having a list that ranks his/her preferred items in order of preference, we want to match every person with a unique item. A matching $M$ is called 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 the case that people's preference lists are strict but not complete. We show that in the case where every person has a preference list with length of a constant $k \geq 4$, a similar phase transition occurs at $\alpha_k$, where $\alpha_k \geq 1$ is the root of equation $x e^{-1/2x} = 1-(1-e^{-1/x})^{k-1}$.