Likelihood of the Existence of Average Justified Representation

TL;DR

This work analyzes the existence of average justified representation (AJR) in approval-based multi-winner elections under the Erdős--Rényi bipartite model with fixed parameters $m$ and $k$. By developing an intermediate threshold characterization and leveraging a polyhedron/PMV approach, the authors identify two phase-transition points, $p_1^* = \frac{1}{k}$ and $p_2^*$ (defined via $k(2x(1-x)^k+kx^2(1-x)^{k-1})=1$), that separate regimes where AJR committees exist with high probability, do not exist with high probability, and where the probability is bounded away from both 0 and 1. The analysis shows that for $\ell\ge 2$, cohesive groups automatically satisfy AJR once they exist, making $\ell=1$ the critical obstruction; the corner cases at $p_1^*$ and $p_2^*$ are handled with a polyhedral method yielding $\Theta(1)$ probabilities. The results validate and refine prior empirical observations, clarify differences between AJR and other JR variants, and open avenues to study AJR under broader random models and proportionality measures with potential implications for fairness in committee selection.

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 Erdős--Rényi model. We consider the Erdős--Rényi 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 $1$.

Figures (5)

• Figure 1: An example where AJR committees do not exist. In this example, each black circle stands for a voter. The colored rectangles stand for candidates. If a circle is in a rectangle, it means that this voter approves this candidate. In this example, $n=12$ and $m=4$, and we are to select a winning committee of size three (i.e., $k=3$). In this case, there are four $1$-cohesive groups, each corresponding to one side of the large square. Selecting any three of these four candidates will lead to some group's average satisfaction being only $0.5$, which is less than the requirement $1$.
• Figure 2: Diagram of relationships among different variants of JR. Arrows indicate implications.
• Figure 3: The values of $p_1^\ast$ and $p_2^\ast$ for $k=2,\ldots,10$.
• Figure 4: Illustration of construction group $V$. Numbers in the chunk represent the number of approved candidates in $W$ for each chunk of voters.
• Figure 5: Values of $U(T)$ with different $k$

Theorems & Definitions (56)

• Example 1.1
• Definition 2.1: Average Justified Representation (AJR)
• Theorem 2.2
• Proposition 2.3
• Lemma 3.1
• Proposition 4.1
• Proposition 4.2
• Proposition 4.3
• Proposition 4.4
• proof : Proof (last part of Proposition \ref{['prop:ell=1']})