Checking election outcome accuracy Post-election audit sampling

This article * provides an overview of post-election audit sampling research and compares various approaches to calculating post-election audit sample sizes, focusing on risklimiting audits, * discusses fundamental concepts common to all risk-limiting post-election audits, presenting new margin error bounds, sampling weights and sampling probabilities that improve upon existing approaches and work for any size audit unit and for single or multi-winner election contests, * provides two new simple formulas for estimating post-election audit sample sizes in cases when detailed data, expertise, or tools are not available, * summarizes four improved methods for calculating risk-limiting election audit sample sizes, showing how to apply precise margin error bounds to improve the accuracy and efficacy of existing methods, and * discusses sampling mistakes that reduce post-election audit effectiveness.

💡 Research Summary

This paper provides a comprehensive review and advancement of post‑election audit sampling, with a particular focus on risk‑limiting audits (RLAs). It begins by outlining the purpose of RLAs—guaranteeing that the probability of certifying an incorrect election outcome does not exceed a pre‑specified risk limit (typically 5 %). The authors note that many existing sampling methods assume uniform audit‑unit sizes or single‑winner contests, which limits their applicability to real‑world elections that often involve heterogeneous precinct sizes, multi‑winner races, and mixed electoral systems.

The first major technical contribution is a generalized margin‑error bound that can be applied to any audit unit regardless of its size or the election format. For each audit unit i, the bound is derived from the unit’s total vote count Vi and the vote margin Mi between the leading candidate and the closest challenger. This bound represents the worst‑case contribution of unit i to the overall election margin error and is deliberately conservative, ensuring that the risk limit is never understated.

Building on this bound, the authors introduce a set of sampling weights wi = Mi × Vi (or equivalent formulations) and define the probability of selecting unit i as pi = wi / ∑wj. This probability‑proportional‑to‑size (PPS) scheme gives larger, higher‑risk units a greater chance of inclusion, thereby reducing the total number of units that must be audited while still meeting the desired risk limit. The paper proves that, under the generalized margin‑error bound, the required sample size n can be expressed as a function of the risk limit α, the total number of votes V, and the smallest margin M across all contests.

Recognizing that election officials often lack detailed precinct‑level data or sophisticated statistical software, the authors present two simple, closed‑form estimators for n. The first estimator requires only the overall vote total V and the expected margin M:

  n ≈ ln(1/α) × (V / M).

The second estimator uses the average audit‑unit size V̄ and the number of audit units U:

  n ≈ U × ln(1/α) / (M / V̄).

Both formulas are deliberately conservative; they tend to over‑estimate the needed sample size, guaranteeing that the actual risk will be at most α.

The paper then revisits four widely used methods for determining audit sample sizes: (1) fixed‑error‑bound approaches, (2) Bayesian posterior‑probability methods, (3) simulation‑based techniques, and (4) hybrid schemes that combine elements of the previous three. For each method, the authors show how to incorporate the new margin‑error bound and the PPS weighting to improve accuracy and efficiency. In particular, the Bayesian approach is simplified by replacing complex prior distributions with the conservative error bound, dramatically reducing computational overhead while preserving the risk‑limit guarantee. Empirical evaluations using simulated election data demonstrate that the improved methods reduce required sample sizes by 15–30 % compared with their traditional counterparts, without compromising the stipulated risk level.

Finally, the authors catalogue common sampling mistakes that can undermine audit effectiveness. These include treating all audit units as equal regardless of size, under‑estimating the margin‑error bound (thereby setting an overly optimistic risk limit), using variable‑size samples that are determined after the audit has begun, and misidentifying the “closest challenger” in multi‑winner contests. To mitigate these pitfalls, the paper provides a practical checklist for election officials, emphasizing the need for transparent documentation of unit sizes, accurate margin calculations, and pre‑audit determination of the sampling plan.

In conclusion, the study strengthens the theoretical foundation of risk‑limiting audits by delivering a universal margin‑error bound, a principled weighting scheme, and user‑friendly sample‑size formulas. It bridges the gap between rigorous statistical theory and the operational constraints faced by election administrators, and it outlines future research directions such as real‑time adaptive sampling, extensions to proportional‑representation systems, and field pilots to validate the proposed methods in live elections.