Likelihood of the Existence of Average Justified Representation

TL;DR

This paper analyzes the probability of the existence of committees satisfying average justified representation (AJR) in approval-based multi-winner elections under the Erdős–Rényi model, revealing phase transitions in existence probability based on approval probability.

Contribution

It provides a complete characterization of the existence of AJR committees in the Erdős–Rényi model with constant candidates as voters grow large, identifying phase transition points.

Findings

Existence of AJR committees exhibits phase transitions at two critical probabilities.

When approval probability is outside the phase transition interval, AJR committees almost surely exist.

Within the phase transition interval, the probability of existence is bounded away from 0 and 1.

Abstract

We study the approval-based multi-winner election problem where $n$ voters jointly decide a committee of $k$ winners from $m$ candidates. We focus on the axiom \emph{average justified representation} (AJR) proposed by Fernandez, Elkind, Lackner, Garcia, Arias-Fisteus, Basanta-Val, and Skowron (2017). AJR postulates that every group of voters with a common preference should be sufficiently represented in that their average satisfaction should be no less than their Hare quota. Formally, for every group of $\lceil\ell\cdot\frac{n}{k}\rceil$ voters with $\ell$ common approved candidates, the average number of approved winners for this group should be at least $\ell$. It is well-known that a winning committee satisfying AJR is not guaranteed to exist for all multi-winner election instances. In this paper, we study the likelihood of the existence of AJR under the Erd\H{o}s--R\'enyi model. We consider the Erd\H{o}s--R\'enyi model parameterized by $p\in[0,1]$ that samples multi-winner election instances from the distribution where each voter approves each candidate with probability $p$ (and the events that voters approve candidates are independent), and we provide a clean and complete characterization of the existence of AJR committees in the case where $m$ is a constant and $n$ tends to infinity. We show that there are two phase transition points $p_1$ and $p_2$ (with $p_1\leq p_2$) for the parameter $p$ such that: 1) when $p<p_1$ or $p>p_2$, an AJR committee exists with probability $1-o(1)$, 2) when $p_1<p<p_2$, an AJR committee exists with probability $o(1)$, and 3) when $p=p_1$ or $p=p_2$, the probability that an AJR committee exists is bounded away from both $0$ and $1$.