Abstract
Counting votes is complex and error-prone. Several statistical methods have been developed to assess election accuracy by manually inspecting randomly selected physical ballots. Two ‘principled’ methods are risk-limiting audits (RLAs) and Bayesian audits (BAs). RLAs use frequentist statistical inference while BAs are based on Bayesian inference. Until recently, the two have been thought of as fundamentally different.
We present results that unify and shed light upon ‘ballot-polling’ RLAs and BAs (which only require the ability to sample uniformly at random from all cast ballot cards) for two-candidate plurality contests, that are building blocks for auditing more complex social choice functions, including some preferential voting systems. We highlight the connections between the methods and explore their performance.
First, building on a previous demonstration of the mathematical equivalence of classical and Bayesian approaches, we show that BAs, suitably calibrated, are risk-limiting. Second, we compare the efficiency of the methods across a wide range of contest sizes and margins, focusing on the distribution of sample sizes required to attain a given risk limit. Third, we outline several ways to improve performance and show how the mathematical equivalence explains the improvements.
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11 December 2020
The original version of the cover and book was revised. The seventh editor name has been updated.
Notes
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This is a consequence of the fact that the risk is maximized when \(p_T = 0.5\), a fact that we can use to bound the risk by choosing an appropriate value for the threshold. We include the mathematical details of this result in a technical appendix available at: https://arxiv.org/abs/2008.08536.
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The SPRT allows rejection of either \(H_0\) or \(H_1\), but we only allow the former here. This aligns it with the broader framework for election audits described earlier. Also, we impose a maximum sample size, as we do for the other methods.
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Such a prior places all its mass on \(p = 0.5\) when \(p \leqslant 0.5\).
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The SPRT can perform poorly when \(p_T \in (p_0, p_1)\); taking \(\epsilon > 0\) protects against the possibility that the reported winner really won, but not by as much as reported.
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When sampling without replacement, if we ever observe \(Y_n > Nt\) then we ignore the statistic and terminate the audit since \(H_1\) is guaranteed to be true.
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We include the mathematical details of these results in a technical appendix available at: https://arxiv.org/abs/2008.08536.
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As for KMart, if \(Y_n > Nt\), the audit terminates: the null hypothesis is false.
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Our code is available at: https://github.com/Dovermore/AuditAnalysis.
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Huang, Z., Rivest, R.L., Stark, P.B., Teague, V.J., Vukcevic, D. (2020). A Unified Evaluation of Two-Candidate Ballot-Polling Election Auditing Methods. In: Krimmer, R., et al. Electronic Voting. E-Vote-ID 2020. Lecture Notes in Computer Science(), vol 12455. Springer, Cham. https://doi.org/10.1007/978-3-030-60347-2_8
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