Likelihood of the Existence of Average Justified Representation

2025-10-21

AI Summary

This paper investigates the existence of committees satisfying Average Justified Representation (AJR) in approval-based multiwinner elections. Under the Erdős–Rényi random model—with a fixed number of candidates and the number of voters tending to infinity—it establishes the first complete characterization of the phase transition behavior for AJR existence. Specifically, there exist two critical approval probability thresholds $p_1 < p_2$: AJR committees exist with probability $1 - o(1)$ when $p < p_1$ or $p > p_2$, with probability $o(1)$ when $p_1 < p < p_2$, and at the critical points $p = p_1, p_2$, the existence probability converges to non-degenerate constants in $(0,1)$. This double phase transition reveals AJR’s robust feasibility in both sparse and dense approval regimes, while identifying a structural obstruction in the intermediate regime. The analysis integrates random graph modeling, probabilistic methods, and combinatorial arguments, yielding the first precise asymptotic characterization of the feasibility boundary for fair representation.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Knowledge Representation and Reasoning: Preferences
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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 $lceilellcdotfrac{n}{k} ceil$ 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 ErdH{o}s--R'enyi model. We consider the ErdH{o}s--R'enyi model parameterized by $pin[0,1]$ 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 $p_1leq p_2$) 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 $1$.

Problem

• Analyzing existence probability of fair representation in multi-winner elections
• Characterizing phase transitions for AJR committees under random preferences
• Determining threshold probabilities where justified representation becomes likely

Innovation

• Analyzes AJR existence via Erdős–Rényi random model
• Characterizes phase transitions with two critical probabilities
• Determines AJR likelihood as voter count approaches infinity