[Paper Review] Estimating the Margin of Victory of an Election using Sampling

Palash Dey, Y. Narahari|arXiv (Cornell University)|May 4, 2015

Game Theory and Voting Systems21 references19 citations

TL;DR

This paper introduces a sampling-based approach to estimate the margin of victory (MoV) in elections with high probability and low sample complexity. It presents efficient randomized algorithms for common voting rules—including Copelandα, maximin, and scoring rules—achieving a (1−O(1/log m), ε, δ)-approximation with sample complexity independent of the number of voters n.

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 extsc{$(c, ε, δ)$--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 $c MoV(\mathcal{E})+εn$. We study the extsc{$(c, ε, δ)$--Margin of Victory} problem for many commonly used voting rules including scoring rules, approval, Bucklin, maximin, and Copeland$^α.$ 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$^α$ voting rules.

Motivation & Objective

• Address the challenge of estimating the margin of victory (MoV) in large-scale elections when full vote observation is infeasible.
• Formulate the (c,ε,δ)-Margin of Victory problem as a hybrid approximation that balances additive and multiplicative error bounds.
• Develop sampling-based algorithms that minimize sample complexity while ensuring high-probability accuracy for various voting rules.
• Demonstrate that even NP-hard MoV computation problems admit efficient sampling solutions for key voting rules like Copelandα and maximin.
• Provide theoretical guarantees on approximation quality and sample complexity for practical applications such as post-election audits and opinion polling.

Proposed method

• Define the (c,ε,δ)-MoV problem as estimating MoV within additive error c·MoV + εn with probability ≥1−δ.
• Introduce a surrogate quantity Γ(ℰ) that approximates the margin of victory and is computable via sampling.
• Use randomized sampling to estimate pairwise dominance counts Dℰ(x,y) between candidates, with error bounded by ε′n.
• Construct an estimator Γ̄(ℰ) from sampled dominance counts, proving it concentrates around Γ(ℰ) with high probability.
• For Copelandα, scale Γ̄(ℰ) by a factor of 4(log m + 1)/(2 log m + 3) to derive the final MoV estimate M̄.
• Prove that the resulting estimate M̄ is a (1−O(1/log m), ε, δ)-approximation of the true MoV with high probability.

Experimental results

Research questions

• RQ1 Can we estimate the margin of victory in elections using only a small, randomly sampled subset of votes?
• RQ2 What is the minimum sample complexity required to achieve a (c,ε,δ)-approximation of the margin of victory across different voting rules?
• RQ3 For voting rules where MoV computation is NP-hard (e.g., Copelandα), can sampling-based methods still provide efficient and accurate estimates?
• RQ4 How does the number of candidates m affect the approximation quality of the sampling-based MoV estimator?
• RQ5 Can the proposed framework be extended to leverage social network structures among voters to further reduce sample complexity?

Key findings

• The proposed sampling-based algorithm achieves a (1−O(1/log m), ε, δ)-approximation for the margin of victory in Copelandα elections.
• The sample complexity of the algorithm is independent of the number of voters n, making it scalable for large-scale elections.
• For maximin and Copelandα voting rules, efficient sampling algorithms exist despite the NP-hardness of exact MoV computation.
• The estimator Γ̄(ℰ) is shown to concentrate within 4ε′n of the true Γ(ℰ), with high probability, using O(1/ε′²) samples per pairwise comparison.
• The final MoV estimate M̄ satisfies |M̄ − MoV| ≤ (1−O(1/log m))·MoV + εn with probability at least 1−δ.
• The framework provides a practical solution for post-election audits and polling by enabling sample complexity to be determined before sampling, based on estimated MoV.