[Paper Review] Monotonicity Anomalies in Scottish Local Government Elections

David McCune, Adam Graham-Squire|arXiv (Cornell University)|May 28, 2023

Game Theory and Voting SystemsEconomics, Econometrics and Finance3 citations

TL;DR

This study analyzes 1,079 Scottish local government STV elections to empirically measure monotonicity anomalies—where increased voter support harms a candidate’s chance of election. It finds 62 elections with such anomalies, with rates of 1–4% per anomaly type, demonstrating that while rare, these paradoxes occur in real-world multiwinner STV elections for the first time in documented form.

ABSTRACT

Single Transferable Vote (STV) is a voting method used to elect multiple candidates in ranked-choice elections. One weakness of STV is that it fails multiple fairness criteria related to monotonicity and no show paradoxes. We analyze 1,079 local government STV elections in Scotland to estimate the frequency of such monotonicity anomalies in real-world elections, and compare our results with prior empirical and theoretical research about the rates at which such anomalies occur. In 62 of the 1079 elections we found some kind of monotonicity anomaly. We generally find that the rates of anomalies are similar to prior empirical research and much lower than what most theoretical research has found. The STV anomalies we find are the first of their kind to be documented in real-world multiwinner elections.

Motivation & Objective

• To empirically investigate the frequency of monotonicity anomalies in real-world multiwinner STV elections, particularly in contrast to theoretical predictions.
• To determine whether monotonicity anomalies—where increased support harms a candidate’s election chances—occur in actual STV elections, especially in Scotland’s local government elections.
• To compare empirical anomaly rates with prior theoretical and semi-empirical studies, which often predict much higher frequencies.
• To provide the first documented evidence of monotonicity anomalies in multiwinner STV elections using full ballot data.
• To assess the practical significance of these anomalies by analyzing the number of affected voters and the conditions under which anomalies arise.

Proposed method

• Collected and processed publicly available STV ballot data from 1,079 Scottish local government elections (30 single-winner, 1,049 multiwinner) conducted since 2007.
• Implemented custom Python code to systematically scan each election’s preference rankings for all types of monotonicity anomalies: upward, downward, no-show, and committee size anomalies.
• Applied formal definitions of monotonicity failure from social choice theory to detect cases where altering ballot rankings in a way that should help a candidate instead harms their outcome.
• Used a combination of exact computation and logical checks to verify anomaly conditions across all possible ballot permutations relevant to each anomaly type.
• Accounted for ballot truncation by analyzing only the ranked preferences present, acknowledging that incomplete ballots may reduce anomaly frequency compared to full-ballot models.
• Compared results across different election types (single vs. multiwinner) and levels of competitiveness to assess sensitivity of anomaly rates.

Figure 1. For a given $p$ , the percentage of three-candidate-close elections that demonstrate an anomaly. The black diamonds use elections which demonstrate any anomaly, while the red disks exclude elections demonstrating a committee size anomaly.

Experimental results

Research questions

• RQ1 How frequently do monotonicity anomalies occur in real-world multiwinner STV elections, as measured in actual Scottish local government elections?
• RQ2 Do the empirical rates of monotonicity anomalies in STV elections align with theoretical predictions, or are they significantly lower in practice?
• RQ3 What is the impact of ballot truncation on the frequency of monotonicity anomalies in STV elections?
• RQ4 Are monotonicity anomalies more common in competitive or closely contested elections, and how does this affect their real-world significance?
• RQ5 Can strategic voting to exploit monotonicity anomalies be practically achieved in STV elections, given the complexity and data requirements?

Key findings

• A total of 62 out of 1,079 Scottish STV elections exhibited at least one type of monotonicity anomaly, representing an overall anomaly rate of approximately 5.7%.
• The rate of upward monotonicity anomalies (where increased support harms a candidate) was 3.6%, downward anomalies 1.3%, and no-show anomalies 3.6%.
• The study found that anomaly rates in real-world STV elections are significantly lower than theoretical upper bounds, which often assume worst-case voter behavior and full ballots.
• Ballot truncation in Scottish elections likely suppresses anomaly frequency, as incomplete preferences reduce the number of potential paradoxical configurations.
• The researchers identified the first documented cases of monotonicity anomalies in multiwinner STV elections, providing empirical validation of theoretical concerns.
• Despite the presence of anomalies, the number of voters affected per anomaly is small, and the conditions required to trigger an anomaly are highly specific and unlikely to be exploited strategically in practice.

Figure 2. For a given $p$ , the percentage of two-candidate-close elections that demonstrate an anomaly. The black diamonds use elections which demonstrate any anomaly, while the red disks exclude elections demonstrating a committee size anomaly.