Proportionality in Practice: Quantifying Proportionality in Ordinal Elections

TL;DR

The paper tackles the discrepancy between proportionality axioms for multiwinner voting with ordinal ballots and real-world practice, using the Scottish local elections dataset to critique PSC's empirical force. It introduces alpha-PSC and related quantitative measures to assess proportionality when large solid coalitions are rare, and develops an ILP-based method to compute these values for committees and instances. Through extensive experiments across 1070 elections, it shows that real outcomes often render PSC weak, that semi-proportional SNTV can align closely with proportional measures, and that stronger axioms like EJR+ and priceability correlate with STV variants, while seq-RCV can perform poorly under these metrics. The findings highlight the value of empirical, quantitative evaluation of proportionality, inform rule choice in ordinal elections, and motivate further axioms or measures tailored to real-world ballot data.

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.

Figures (14)

• Figure 1: Histograms of PSC values and EJR+ values achievable in our elections, rounded to one decimal place.
• Figure 2: The PSC values achieved by voting rules, together with optimal PSC values (shown in the leftmost column).
• Figure 3: Left: PSC values achieved by Scottish STV and seq-RCV for the 485 elections where the rules disagree. Right: PSC values achieved by Scottish STV and EAR over the 262 elections where the rules disagree. Elections are ordered by increasing optimal PSC value.
• Figure 4: Histograms of LS and priceability values achievable in our elections, rounded to one decimal place.
• Figure 5: The EJR+ values achieved by voting rules

Theorems & Definitions (15)

• Definition 1: PSC
• Definition 2: LS
• Definition 3: PJR+ BrPe23a
• Definition 4: EJR+
• Definition 5: Priceability
• Example 1
• Definition 6
• Definition 7: $\alpha$-PSC
• Example 2
• Theorem 1