Abstract
Risk-limiting audits (RLAs), an ingredient in evidence-based elections, are increasingly common. They are a rigorous statistical means of ensuring that electoral results are correct, usually without having to perform an expensive full recount—at the cost of some controlled probability of error. A recently developed approach for conducting RLAs, SHANGRLA, provides a flexible framework that can encompass a wide variety of social choice functions and audit strategies. Its flexibility comes from reducing sufficient conditions for outcomes to be correct to canonical ‘assertions’ that have a simple mathematical form.
Assertions have been developed for auditing various social choice functions including plurality, multi-winner plurality, super-majority, Hamiltonian methods, and instant runoff voting. However, there is no systematic approach to building assertions. Here, we show that assertions with linear dependence on transformations of the votes can easily be transformed to canonical form for SHANGRLA. We illustrate the approach by constructing assertions for party-list elections such as Hamiltonian free list elections and elections using the D’Hondt method, expanding the set of social choice functions to which SHANGRLA applies directly.
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Notes
- 1.
Any social choice function that is a scoring rule—that assigns ‘points’ to candidates on each ballot, sums the points across ballots, and declares the winner(s) to be the candidate(s) with the most ‘points’—can be audited using SHANGRLA, as can some social choice functions that are not scoring rules, such as super-majority and IRV.
- 2.
It can be more efficient to sample ballots in ‘rounds’ rather than singly; SHANGRLA can accommodate any valid test of the assorter nulls.
- 3.
Another source of complexity is the opportunity for voters to select, exclude, or prioritise individual candidates within the party.
- 4.
Below, in discussing assorters, we use the term ‘entity’ more abstractly. For instance, when voters may rank a subset of entities, the assorters may translate ranks into scoring functions in a nonlinear manner, as in [2]—we do not detail that case here.
- 5.
Constructing such a set is outside the scope of this paper; we suspect there is no general method. Moreover, there may be social choice functions for which there is no such set.
- 6.
Note that \(g(b) = 0\) for any invalid ballot b, based on previous definitions.
- 7.
Note that \(h(b) = 1/2\) if ballot b has no valid vote in the contest.
- 8.
If the votes \(b_j\) are bounded above by s and below by zero, then a bound (not necessarily the sharpest) on g is given by taking just the votes that contribute negative values to g, setting all of those votes to s, and setting the other votes to 0:
$$\begin{aligned} a = \sum _{j : a_j < 0} a_j s. \end{aligned}$$ - 9.
The description is based on the (German only) official information from Hesse, see https://wahlen.hessen.de/kommunen/kommunalwahlen-2021/wahlsystem, last accessed 24.07.2021.
- 10.
TestNonnegMean.initial_sample_size() from https://github.com/pbstark/SHANGRLA/blob/main/Code/assertion_audit_utils.py, last accessed 24.07.2021.
- 11.
https://www.europarl.europa.eu/RegData/etudes/BRIE/2019/637966/EPRS_BRI(2019)637966_EN.pdf, last accessed 24.07.2021.
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Blom, M. et al. (2021). Assertion-Based Approaches to Auditing Complex Elections, with Application to Party-List Proportional Elections. In: Krimmer, R., et al. Electronic Voting. E-Vote-ID 2021. Lecture Notes in Computer Science(), vol 12900. Springer, Cham. https://doi.org/10.1007/978-3-030-86942-7_4
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