An Analysis of Random Elections with Large Numbers of Voters

TL;DR

This paper analyzes the likelihood of cyclic structures in large elections' margin graphs, revealing that more cyclic configurations become less probable as the number of voters increases.

Contribution

It introduces a detailed probabilistic analysis of margin graphs in large elections, focusing on the prevalence of cycles using advanced mathematical tools.

Findings

More cyclic margin graphs are less likely with increasing voters

Application of the central limit theorem to election margin analysis

Use of graph homology and linear algebra in voting theory

Abstract

In an election in which each voter ranks all of the candidates, we consider the head-to-head results between each pair of candidates and form a labeled directed graph, called the margin graph, which contains the margin of victory of each candidate over each of the other candidates. A central issue in developing voting methods is that there can be cycles in this graph, where candidate $\mathsf{A}$ defeats candidate $\mathsf{B}$, $\mathsf{B}$ defeats $\mathsf{C}$, and $\mathsf{C}$ defeats $\mathsf{A}$. In this paper we apply the central limit theorem, graph homology, and linear algebra to analyze how likely such situations are to occur for large numbers of voters. There is a large literature on analyzing the probability of having a majority winner; our analysis is more fine-grained. The result of our analysis is that in elections with the number of voters going to infinity, margin graphs that are more cyclic in a certain precise sense are less likely to occur.