Estimating the Margin of Victory of an Election using Sampling

TL;DR

This paper introduces efficient sampling algorithms to estimate the margin of victory in elections, which is crucial for election robustness and post-election processes, across various voting rules including some NP-hard cases.

Contribution

It formalizes the $(c, \, \epsilon, \, \delta)$--Margin of Victory problem and provides sampling-based solutions for multiple voting rules, even where computing the exact margin is NP-hard.

Findings

Efficient algorithms for estimating election margins under various voting rules.

Sampling methods work for NP-hard margin computation cases like maximin and Copeland$^{\alpha}$.

Provides bounds on estimation accuracy with high probability.

Abstract

The margin of victory of an election is a useful measure to capture the robustness of an election outcome. It also plays a crucial role in determining the sample size of various algorithms in post election audit, polling etc. In this work, we present efficient sampling based algorithms for estimating the margin of victory of elections. More formally, we introduce the \textsc{$(c, \epsilon, \delta)$--Margin of Victory} problem, where given an election $\mathcal{E}$ on $n$ voters, the goal is to estimate the margin of victory $M(\mathcal{E})$ of $\mathcal{E}$ within an additive factor of $c MoV(\mathcal{E})+\epsilon n$. We study the \textsc{$(c, \epsilon, \delta)$--Margin of Victory} problem for many commonly used voting rules including scoring rules, approval, Bucklin, maximin, and Copeland$^{\alpha}.$ We observe that even for the voting rules for which computing the margin of victory is NP-Hard, there may exist efficient sampling based algorithms, as observed in the cases of maximin and Copeland$^{\alpha}$ voting rules.