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Graph-Based Audits for Meek Single Transferable Vote Elections

Edouard Heitzmann

TL;DR

This work tackles the challenge of auditing algorithmic STV elections, where chronology-sensitive rules make traditional RLAs difficult. It proposes a graph-based RLA framework that fixes a subgraph $G$ within the space of all election sequences $\Omega$ and certifies the outcome by showing the true path remains inside $G$, focusing on the winner set rather than exact elimination order. The authors develop a detailed formalism around the Meek STV variant, introduce instant keep factors, discrepancy assorters, and layered audit graphs, and provide variance-bounding techniques for sparse samples. Through case studies of Scotland, Portland, and Australia, they demonstrate that Meek STV can be auditable with modest sampling (ASN) even in large elections, while highlighting challenges such as elimination clouds and irregular states. The framework offers a scalable, chronology-agnostic approach to validating STV outcomes and outlines concrete directions for extending audit graphs, handling higher-degree states, and exploring hybrids with WIGM.

Abstract

In the context of election security, a Risk-Limiting Audit (RLA) is a statistical framework that uses a minimal partial recount of the ballots to guarantee that the results of the election were correctly reported. A generalized RLA framework has remained elusive for algorithmic election rules such as the Single Transferable Vote (STV) rule, because of the dependence of these rules on the chronology of eliminations and elections leading to the outcome of the election. This paper proposes a new graph-based approach to audit these algorithmic election rules, by considering the space of all possible sequences of elections and eliminations. If we fix a subgraph of this universal space ahead of the audit, a sufficient strategy is to verify statistically that the true election sequence does not leave the fixed subgraph. This makes for a flexible framework to audit these elections in a chronology-agnostic way.

Graph-Based Audits for Meek Single Transferable Vote Elections

TL;DR

This work tackles the challenge of auditing algorithmic STV elections, where chronology-sensitive rules make traditional RLAs difficult. It proposes a graph-based RLA framework that fixes a subgraph within the space of all election sequences and certifies the outcome by showing the true path remains inside , focusing on the winner set rather than exact elimination order. The authors develop a detailed formalism around the Meek STV variant, introduce instant keep factors, discrepancy assorters, and layered audit graphs, and provide variance-bounding techniques for sparse samples. Through case studies of Scotland, Portland, and Australia, they demonstrate that Meek STV can be auditable with modest sampling (ASN) even in large elections, while highlighting challenges such as elimination clouds and irregular states. The framework offers a scalable, chronology-agnostic approach to validating STV outcomes and outlines concrete directions for extending audit graphs, handling higher-degree states, and exploring hybrids with WIGM.

Abstract

In the context of election security, a Risk-Limiting Audit (RLA) is a statistical framework that uses a minimal partial recount of the ballots to guarantee that the results of the election were correctly reported. A generalized RLA framework has remained elusive for algorithmic election rules such as the Single Transferable Vote (STV) rule, because of the dependence of these rules on the chronology of eliminations and elections leading to the outcome of the election. This paper proposes a new graph-based approach to audit these algorithmic election rules, by considering the space of all possible sequences of elections and eliminations. If we fix a subgraph of this universal space ahead of the audit, a sufficient strategy is to verify statistically that the true election sequence does not leave the fixed subgraph. This makes for a flexible framework to audit these elections in a chronology-agnostic way.
Paper Structure (20 sections, 63 equations, 8 figures, 6 tables)

This paper contains 20 sections, 63 equations, 8 figures, 6 tables.

Table of Contents

  1. Introduction.
  2. A Helpful Case Study.
  3. Mathematical Formalism.
  4. Experimental Design.
  5. Graph-Based Audits.
  6. Variance Bounds for Sparse Samples.
  7. Audit Graphs.
  8. Instant Keep Factors.
  9. Local Null Audits.
  10. General Setup.
  11. Degree-0 Audits.
  12. Degree-1 Audits.
  13. Degree-2 Audits.
  14. Average Sampling Numbers.
  15. Experimental Results.
  16. ...and 5 more sections

Figures (8)

  • Figure 1: The audit graphs for an auditable margin of $40$ votes in Ward 9.
  • Figure 2: The Universal Graph $\Omega$ in Example \ref{['eg:P_6']}.
  • Figure 3: The universal graph $\Omega$ in Example \ref{['eg:P_9']}
  • Figure 4: The audit tree for an auditable margin of 40 votes.
  • Figure 5: ASN as a function of Least Auditable Margin $A$ when $\alpha=.05$.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.5
  • Definition 2.1
  • Example 2.3
  • Example 2.4
  • Definition 2.6
  • Definition 2.7
  • Definition 2.9
  • Definition 3.1
  • ...and 3 more