Proportionality in Practice: Quantifying Proportionality in Ordinal Elections

TL;DR

This paper develops quantitative measures to assess the proportionality of voting rules in ordinal elections, revealing that real-world data often diverges from theoretical proportionality axioms and evaluating multiple voting methods.

Contribution

It introduces new measures for proportionality assessment and applies them to real election data, challenging assumptions about the practical relevance of proportionality axioms.

Findings

Solid coalitions rarely occur in practice, questioning the importance of proportionality axioms.

Quantitative measures reveal differences in proportionality among voting rules.

Ballot truncation impacts the perceived proportionality of election outcomes.

Abstract

Proportional representation plays a crucial role in electoral systems. In ordinal elections, where voters rank candidates based on their preferences, the Single Transferable Vote (STV) is the most widely used proportional voting method. STV is considered proportional because it satisfies an axiom requiring that large enough solid coalitions of voters are adequately represented. Using real-world data from local Scottish elections, we observe that solid coalitions of the required size rarely occur in practice. This observation challenges the importance of proportionality axioms and raises the question of how the proportionality of voting methods can be assessed beyond their axiomatic performance. We address these concerns by developing quantitative measures of proportionality. We apply these measures to evaluate the proportionality of voting rules on real-world election data. Besides STV, we consider SNTV, the Expanding Approvals Rule, and Sequential Ranked-Choice Voting. We also study the effects of ballot truncation by artificially completing truncated ballots and comparing the proportionality of outcomes under complete and truncated ballots.