3+ Seat Risk-Limiting Audits for Single Transferable Vote Elections

TL;DR

This work addresses risk-limiting audits for multiwinner STV elections with three or more seats, contingent on at least one first-round winner. It develops an assertion-based RLA framework within the SHANGRLA formalism, introducing the IQX assertion and a dual-loop search to form full or partial audits. Empirical evaluation on 513 Scottish local elections demonstrates meaningful auditability, with partial audits often_verifying most winners and full audits achievable in a substantial fraction, albeit less frequently as seats rise. The results highlight both the practicality of partial auditing in complex STV elections and the need for further refinements to handle larger-seat contests efficiently.

Abstract

Constructing efficient risk-limiting audits (RLAs) for multiwinner single transferable vote (STV) elections is a challenging problem. An STV RLA is designed to statistically verify that the reported winners of an election did indeed win according to the voters' expressed preferences and not due to mistabulation or interference, while limiting the risk of accepting an incorrect outcome to a desired threshold (the risk limit). Existing methods have shown that it is possible to form RLAs for two-seat STV elections in the context where the first seat has been awarded to a candidate in the first round of tabulation. This is called the first winner criterion. We present an assertion-based approach to conducting full or partial RLAs for STV elections with three or more seats, in which the first winner criterion is satisfied. Although the chance of forming a full audit that verifies all winners drops substantially as the number of seats increases, we show that we can quite often form partial audits that verify most, and sometimes all, of the reported winners. We evaluate our method on a dataset of over 500 three- and four-seat STV elections from the 2017 and 2022 local council elections in Scotland.

Figures (4)

• Figure 1: Outline of the outer-loop of the Dual-Loop STV Audit method for an STV election $E = (\mathcal{C}, \mathcal{B}, Q, N)$, with reported winners $\mathcal{W}$, reported losers $\mathcal{L}$, and reported transfer values $\tau^R_w$ for each winner $w \in \mathcal{W}$, $\bm{\tau}^R = [\tau^R_w | \forall w \in \mathcal{W}]$. This outer-loop calls a procedure that takes a candidate transfer value lower bound vector and generates new candidates in its neighbourhood, Neighbours$_{LB}$.
• Figure 2: Construction of a candidate full and/or partial audit for an STV election $E = (\mathcal{C}, \mathcal{B}, Q, N)$ given: reported losers $\mathcal{L}$; assumed winners on first preferences $W'$; assumed lower and upper bounds on the transfer values of candidates in $W'$, $\bm{\underline{\tau}}$ and $\bm{\overline{\tau}}$; assertions used to verify that the candidates in $W'$ won on first preferences, $\mathcal{A}_{\textsf{IQ}}$; and the set of currently unverified winners, $\mathcal{R}$.
• Figure 3: Inner-loop of the Dual-Loop audit method for an STV election $E = (\mathcal{C}, \mathcal{B}, Q, N)$, with: reported winners and losers $\mathcal{W}$, $\mathcal{L}$; reported transfer values $\tau^R_w$ for each $w \in \mathcal{W}$, $\bm{\tau}^R = [\tau^R_w | \forall w \in \mathcal{W}]$; assumed winners on first preferences $W'$; assumed lower bounds on transfer values for $W'$, $\bm{\underline{\tau}}$; assertions used to verify that the candidates in $W'$ won on first preferences, $\mathcal{A}_{\textsf{IQ}}$; and the set of unverified winners, $\mathcal{R}$. The inner-loop calls a procedure that takes a candidate transfer value upper bound vector and generates new candidates, Neighbours$_{UB}$.
• Figure 4: Side-by-side box plots showing the spread of ASNs (x-axes in log scale) of formed audits that verify varying numbers of winners for our data set of 252 3-seat (left) and 261 4-seat (right) STV contests. For each box plot: the vertical line in the middle of each 'box' shows the median ASN for each group, the edges of the boxes are the first and third quartiles, and the 'whiskers' extending out of each box show the minimum and maximum ASN.

Theorems & Definitions (2)

• definition thmcounterdefinition: STV Election
• definition thmcounterdefinition: Projection $\mathbf{\sigma_\mathcal{S}(\pi)}$