arxiv: 2605.12676 · v1 · submitted 2026-05-12 · 💰 econ.GN · q-fin.EC

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Ballot Exhaustion in Multiwinner Single Transferable Vote Elections

David McCune , E.E. Naber

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Pith reviewed 2026-05-14 20:11 UTC · model grok-4.3

classification 💰 econ.GN q-fin.EC

keywords ballot exhaustionsingle transferable votemultiwinner electionsScottish local governmentvoter representationelectoral outcomesballot completion models

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The pith

Raw ballot exhaustion rates in multiwinner STV overstate how often voters lose representation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines more than 5.4 million ballots from 1,070 Scottish local government elections to measure different forms of exhaustion in multiwinner single transferable vote. It reports that 27.9 percent of ballots exhaust by the final round, yet the corresponding weight exhaustion rate falls to only 7.1 percent because many exhausted ballots have already helped elect a candidate. Most such ballots belong to voters whose first choice or another high-ranked choice wins a seat, so representation is still achieved. The study further tests whether exhaustion changes seat allocations by extending partial ballots under several completion models and finds that a proportional model alters only 3.5 percent of seats. These patterns indicate that raw exhaustion figures exaggerate the practical loss of voter influence.

Core claim

In multiwinner STV, 27.9 percent of ballots exhaust by the end of counting, but only 7.1 percent of total vote weight exhausts because exhausted ballots frequently come from voters who already secured representation through their first or other top-ranked candidates. Under a proportional ballot-completion model, exhaustion alters outcomes in just 3.5 percent of seats, although a substantial share of winners still fail to reach quota after all eliminations.

What carries the argument

Formal definitions that separate raw exhausted ballots, weight exhaustion, non-first-choice exhausted ballots, and unrepresented exhausted ballots, applied to large-scale Scottish election data.

If this is right

Raw exhaustion rates of 27.9 percent correspond to weight exhaustion of only 7.1 percent.
Most exhausted ballots belong to voters who obtain representation through one of their ranked choices winning a seat.
Under the proportional completion model, exhaustion changes the winner in only 3.5 percent of seats.
A substantial number of elected candidates fail to reach quota even after all losing candidates are eliminated.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

STV may deliver more effective voter representation than raw exhaustion statistics suggest, which could inform debates over adopting the system in other multiwinner settings.
Empirical checks of how voters actually complete longer ballots would test whether the low impact on outcomes holds beyond the Scottish data.
Reforms aimed at reducing weight exhaustion might matter more for representation than efforts focused solely on minimizing raw ballot exhaustion.

Load-bearing premise

The proportional ballot-completion model accurately reflects how voters would complete their ballots if given more options.

What would settle it

Data from elections that present voters with longer candidate lists showing that actual ballot completions produce substantially higher shares of unrepresented exhausted ballots than the proportional model predicts.

Figures

Figures reproduced from arXiv: 2605.12676 by David McCune, E.E. Naber.

**Figure 1.** Figure 1: Box plots showing the three ballot exhaustion rates across all elections in the Scottish dataset. fractional reweighting.1 In this setting, it is natural to focus on aggregate notions such as nontransferable votes. By contrast, the availability of ballot-level preference-profile data in Scotland, together with fractional transfer rules, allows us to define and analyze exhaustion at the level of individual … view at source ↗

**Figure 2.** Figure 2: The percentage of ballots of a given length for a choice of n and S where n ∈ {6, 7, 8, 9} and S ∈ {3, 4}. For example, across all elections with n = 6 and S = 3, 16.8% of ballots have length one [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** Figure 3: (Left) The exhaustion rate of all elections versus the number of rounds. (Right) The weight exhaustion rates of all elections [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: The rate of ballot exhaustion by round across all elections which last at least k rounds [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

**Figure 5.** Figure 5: The rate of non-first-choice ballot exhaustion by round across all elections which last at least k rounds [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗

**Figure 6.** Figure 6: The rate of unrepresented ballot exhaustion by round across all elections which last at least k rounds [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗

**Figure 7.** Figure 7: The round-by-round percent of non-transferable ballots and non-transferable weight across all elections which run for at least k rounds [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗

**Figure 8.** Figure 8: The percentage of ballots exhausted grouped by when a seat is earned in 3-seat elections. The height of each bar gives the percentage of ballots exhausted when the kth seat is earned, and the orange shows the portion of ballots exhausted without being applied to the vote count of a winner when they won their seat [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗

**Figure 9.** Figure 9: The percentage of ballots exhausted grouped by when a seat is earned in 4-seat elections. The height of each bar gives the percentage of ballots exhausted when the kth seat is earned, and the orange shows the portion of ballots exhausted without being applied to the vote count of a winner when they won their seat [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗

**Figure 10.** Figure 10: Rejected ballot rates separated by election year. For 2007 we have elections only for the district of Glasgow City, and we group these elections with the 2012 elections. To make the figure more readable we omit one outlier election from 2017, a Dundee City council election with a rejected ballot rate of 12.3% [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗

read the original abstract

We study ballot exhaustion in multiwinner single transferable vote (STV) elections using a dataset of 1,070 Scottish local government elections comprising over 5.4 million ballots. While ballot exhaustion has been studied extensively in single-winner elections, comparatively little work examines exhaustion in the multiwinner setting. We introduce formal definitions of several types of exhaustion in STV elections, distinguishing between exhausted ballots, non-first-choice exhausted ballots, unrepresented exhausted ballots, and weight exhaustion. These definitions clarify important conceptual differences between ballots that cease to transfer and ballots that fail to contribute meaningfully to representation. Our empirical analysis shows that 27.9\% of ballots are exhausted by the final round of counting, although the corresponding weight exhaustion rate is only 7.1\%, indicating that many exhausted ballots have already contributed to the election of a candidate. Moreover, most exhausted ballots correspond to voters who achieve some form of representation, either because their first-ranked candidate wins or because a candidate ranked among their top choices is elected. These results suggest that raw exhaustion rates alone substantially overstate the extent to which voters lose their influence or fail to obtain representation under STV. We also investigate whether exhaustion can affect electoral outcomes by extending partial ballots under several completion models. Under extreme assumptions, exhaustion can potentially alter a substantial number of outcomes, but under a proportional ballot-completion model only 3.5\% of seats change. Finally, we show that a substantial number of winners fail to reach quota, even after the elimination of all losing candidates. These results help clarify the practical and normative significance of ballot exhaustion in real-world STV elections.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows raw exhaustion rates in multiwinner STV overstate lost influence, with direct data showing 27.9% ballots exhausted but only 7.1% weight lost and most still getting a top choice elected.

read the letter

Hey, this one on ballot exhaustion in multiwinner STV is worth your time. The central finding is that 27.9% of ballots exhaust by the end but only 7.1% of the vote weight does, and most exhausted ballots still see a first or top-ranked candidate win. That distinction between ballot exhaustion and weight exhaustion is the useful part, and the numbers come straight from the Scottish counting sequences without any fitting. They define several types of exhaustion clearly—exhausted ballots, non-first-choice exhausted, unrepresented exhausted, and weight exhaustion—which helps separate ballots that stop transferring from those that actually fail to deliver representation. The dataset is large and real: 1,070 local elections and over 5 million ballots, all from actual STV runs. The main claims hold up because they are computed directly from observed partial ballots and the real elimination order. The robustness checks across completion models are a plus for the secondary counterfactual on seat changes. The soft spot is the proportional completion model used for the 3.5% seat-change figure; it is plausible but still assumes how voters would finish their rankings, even though the paper tests extremes and reports the range. The observation that many winners fall short of quota is also interesting but feels a bit underdeveloped. This is for people working on electoral systems or STV reform who want concrete numbers rather than theory. The data work is clean and the distinctions are sharp enough that it deserves referee time.

Referee Report

0 major / 3 minor

Summary. The paper analyzes ballot exhaustion in multiwinner STV elections with a dataset of 1,070 Scottish local government elections and over 5.4 million ballots. It introduces formal definitions distinguishing exhausted ballots, non-first-choice exhausted ballots, unrepresented exhausted ballots, and weight exhaustion. Empirically, 27.9% of ballots exhaust by the final round while only 7.1% of vote weight exhausts, and most exhausted ballots correspond to voters who achieve representation because their first choice or another top-ranked candidate is elected. Counterfactual analysis under several ballot-completion models shows that exhaustion alters outcomes in only 3.5% of seats under a proportional model, and many winners fail to reach quota even after all eliminations.

Significance. If the results hold, the work supplies large-scale empirical grounding showing that raw ballot-exhaustion rates substantially overstate the loss of voter influence or representation under STV. The clear conceptual distinctions and direct, model-independent measurements from real election data are strengths. The findings have implications for electoral-system evaluation and normative assessments of proportional representation. The absence of fitted parameters for the primary statistics and the reporting of multiple completion models for robustness add credibility.

minor comments (3)

The introduction would benefit from a short paragraph situating the multiwinner definitions against prior single-winner exhaustion studies to sharpen the contribution.
Tables presenting the 27.9% ballot-exhaustion and 7.1% weight-exhaustion aggregates should include year-by-year or council-type breakdowns to demonstrate stability across the 1,070 elections.
The description of the proportional ballot-completion model would be clearer with an explicit algorithmic outline or pseudocode showing how remaining preferences are assigned proportionally.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript and for the positive assessment of its contributions. The recommendation for minor revision is noted. As the report contains no specific major comments requiring point-by-point response, we have prepared the following accordingly.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's primary results are direct empirical counts from 1,070 observed Scottish STV elections: ballot exhaustion (27.9%), weight exhaustion (7.1%), and representation status of exhausted ballots are computed by applying the actual STV counting sequence to the recorded partial ballots. These quantities require no fitted parameters, no self-referential definitions, and no load-bearing self-citations. The proportional ballot-completion model appears only in a secondary counterfactual on seat changes (3.5%), where the paper already reports results under multiple alternative models. All central claims are therefore self-contained against the external election records.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the accuracy of the introduced exhaustion definitions and the validity of the ballot completion models used in simulations; no free parameters are fitted to the main results.

axioms (1)

standard math Standard STV counting rules including Droop quota and sequential transfer methods as implemented in Scottish local elections
The analysis assumes the conventional implementation of STV rules without deriving them.

pith-pipeline@v0.9.0 · 5592 in / 1071 out tokens · 43422 ms · 2026-05-14T20:11:07.313208+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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