[Paper Review] Assertion-Based Approaches to Auditing Complex Elections, with Application to Party-List Proportional Elections

Michelle Blom, Jurlind Budurushi+5 more|arXiv (Cornell University)|Jul 25, 2021

Electoral Systems and Political ParticipationSocial Sciences6 references9 citations

TL;DR

This paper presents a systematic method to translate linear assertions about vote tallies into canonical assorter forms for SHANGRLA, enabling risk-limiting audits (RLAs) of complex party-list proportional elections such as those using the D’Hondt and Hamiltonian free list methods. The key contribution is a general framework that transforms linear inequalities involving vote transformations into SHANGRLA-compatible assertions, enabling efficient, statistically rigorous audits without full recounts.

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.

Motivation & Objective

• To address the lack of systematic methods for generating assertions in risk-limiting audits (RLAs) for complex social choice functions like party-list proportional elections.
• To enable efficient, statistically sound audits of proportional representation systems such as D’Hondt and Hamiltonian free list elections, which are often considered too complex or expensive to audit.
• To develop a general transformation from linear inequalities involving vote totals into canonical assorter forms compatible with the SHANGRLA framework.
• To demonstrate that assertions for proportional systems can be both necessary and sufficient for correctness, ensuring audit reliability.
• To expand the scope of SHANGRLA to include complex, real-world electoral systems used in jurisdictions like Hesse, Germany.

Proposed method

• Formulate sufficient conditions for correct election outcomes as sets of linear inequalities involving vote totals and seat allocations.
• Transform these linear assertions into canonical form using a general mathematical procedure that maps inequalities to assorters with bounded values.
• Define proto-assorters based on vote contributions (e.g., per-party vote totals divided by seat indices) and apply a normalization to ensure values lie in [0,1] with mean 1/2 under the null hypothesis.
• Derive final assorters by adjusting for bounds and applying the SHANGRLA normalization formula: h(b) = (g(b) - a) / (b - a), where a and b are lower and upper bounds.
• Apply statistical testing to each assorter in parallel to determine if the mean exceeds 1/2, rejecting the null hypothesis if evidence is strong enough.
• Ensure that all assertions being true implies the reported outcome is correct, while allowing for efficient sampling via sequential testing.

Experimental results

Research questions

• RQ1 How can linear assertions about vote tallies in proportional elections be systematically transformed into canonical assorter forms for SHANGRLA?
• RQ2 Can risk-limiting audits be effectively applied to D’Hondt and Hamiltonian free list elections using assertion-based methods?
• RQ3 What conditions ensure that a set of assertions is both sufficient and efficient for auditing complex proportional systems?
• RQ4 How do multi-candidate and multi-vote ballots affect the construction of valid, efficient assorters in proportional elections?
• RQ5 To what extent can the SHANGRLA framework be extended to audit systems with non-scoring, non-monotonic, or complex seat-allocation rules?

Key findings

• The paper successfully derives the first assertion-based risk-limiting audit method for Hamiltonian free list elections, enabling statistical verification of proportional outcomes.
• It provides the first systematic approach to constructing SHANGRLA-compatible assorters for D’Hondt-style highest averages methods using linear inequalities on vote-to-seat ratios.
• The method ensures that if all assertions are true, the reported election outcome is correct, and the audit can terminate early with high confidence using minimal manual ballot inspection.
• For multi-candidate voting, the framework generalizes the assorter construction to handle multiple votes per ballot, with the proto-assorter gA,B(b) = bA/d(WA) - bB/d(LB), where bA and bB are total votes for parties A and B.
• The normalized assorter hA,B(b) is derived as hA,B(b) = (bA*d(LB)/d(WA) - bB + m)/(2m), which reduces to the single-vote case when m=1.
• The approach is shown to be both necessary and sufficient for correctness in the tested proportional systems, supporting efficient auditing even when margins are small.