Efficient Lower Bounding of Single Transferable Vote Election Margins

TL;DR

This work tackles the difficult problem of determining margins of victory in STV elections by refining lower-bound computation. Building on BST-19, the authors introduce transfer-path reasoning, a displacement-based lower bound, and an order-dominance rule to prune redundant prefixes, alongside incorporating ConcreteSTV upper bounds and improved MINLP-based manipulation bounds via DistanceTo$^R_{STV}$. The resulting margin-stv algorithm consistently yields tighter lower bounds and faster runtimes across real-world contests, often enabling exact-margin determinations for small elections and improving auditing guidance. The practical impact lies in more efficient risk-limiting audits and robust evaluation of election integrity, with future work aimed at generalizing margins and enriching prefix information. Overall, the paper advances combinatorial optimization methods for STV by combining tighter transfer analyses, displacement costs, and structural equivalence to prune the search space more effectively.

Abstract

The single transferable vote (STV) is a system of preferential proportional voting employed in multi-seat elections. Each ballot cast by a voter is a (potentially partial) ranking over a set of candidates. The margin of victory, or simply 'margin', is the smallest number of ballots that need to be manipulated to alter the set of winners. Knowledge of the margin of an election gives greater insight into both how much time and money should be spent on auditing the election, and whether uncovered mistakes throw the election result into doubt -- requiring a costly repeat election -- or can be safely ignored without compromising the integrity of the result. Lower bounds on the margin can also be used for this purpose, in cases where exact margins are difficult to compute. There is one existing approach to computing lower bounds on the margin of STV elections, while there are multiple approaches to finding upper bounds. In this paper, we present improvements to this existing lower bound computation method for STV margins. The improvements lead to increased computational efficiency and, in many cases, to the algorithm computing tighter (higher) lower bounds.

Figures (3)

• Figure 1: Number of contests in each category. 'Best Solution' means that the method was one of the methods that obtained the largest margin lower bound (out of all methods on that contest). 'Optimal Solution' means that the method returned a margin lower bound that was within 1 ballot of the provided upper bound on the margin. '& Fastest' means that in addition the method returned a solution in the shortest time (or within 1 second; average of 3 runs).
• Figure 2: For each method, we plot the percentage of election contests $i$ where the runtime of that method is within $x \geq 1$ seconds of the fastest method for that contest, $f_i$, and the resulting lower bound found is no worse than that found by $f_i$.
• Figure 3: For each method, we plot the percentage of election contests $i$ where the runtime of that method is within $x \geq 1$% of that of the fastest method on that contest, $f_i$, and the resulting lower bound found is no worse than that found by $f_i$.

Theorems & Definitions (12)

• Example 2.1
• Definition 2.1: STV Election
• Definition 2.2: Margin
• Definition 2.3: Election order
• Example 3.1
• Example 3.2
• Example 3.3
• Example 4.1
• Example 4.2
• Example 4.3