Reallocating Wasted Votes in Proportional Parliamentary Elections with Thresholds

TL;DR

This work tackles wasted votes produced by electoral thresholds in proportional parliamentary elections by introducing a replacement-vote framework where voters can rank parties. It formalizes a model with a threshold and analyzes three party-selection rules—Direct Winners Only ($DO$), Single Transferable Vote ($STV$), and Greedy Plurality ($GP$)—via axiomatic criteria and an empirical study based on the 2024 French European Parliament election data. Theoretical results show distinct tradeoffs: $GP$ and $STV$ generally improve representativity and satisfy different efficiency/clone-independence properties, while $DO$ offers stronger monotonicity and a reinforcement characterization. Empirically, ranking-based ballots substantially reduce wasted votes and increase representativity, with $STV$ and $GP$ delivering consistent outcomes across samples and demonstrating robustness to threshold variation and noise; the findings suggest viable paths to mitigate wasted votes while preserving governability in thresholded proportional systems.

Abstract

In many proportional parliamentary elections, electoral thresholds (typically 3-5%) are used to promote stability and governability by preventing the election of parties with very small representation. However, these thresholds often result in a significant number of "wasted votes" cast for parties that fail to meet the threshold, which reduces representativeness. One proposal is to allow voters to specify replacement votes, by either indicating a second choice party or by ranking a subset of the parties, but there are several ways of deciding on the scores of the parties (and thus the composition of the parliament) given those votes. We introduce a formal model of party voting with thresholds, and compare a variety of party selection rules axiomatically, and experimentally using a dataset we collected during the 2024 European election in France. We identify three particularly attractive rules, called Direct Winners Only (DO), Single Transferable Vote (STV) and Greedy Plurality (GP).

Figures (13)

• Figure 1: Number of unrepresented voters and of selected parties in our datasets when applying different rules.
• Figure 2: Distribution of the ranks of the representatives in the rankings of the voters for the STV and GP rules with a 5% threshold, in the ranking datasets.
• Figure 3: Percentage of unrepresented voters after truncating their rankings to a particular length, indicated on the horizontal axis.
• Figure 4: Distribution of the ranks of the representatives in the rankings of the voters with different thresholds. The threshold used is indicated on the horizontal axis. For each threshold, the vertical slice above it shows how the voters are divided into unrepresented voters (black area) and represented voters (colored areas, colored according to the rank of the party assigned to the voter).
• Figure 5: Median percentage of unrepresented voters (over 100 random profiles) for different thresholds (horizontal axis).

Theorems & Definitions (56)

• definition 1
• proposition 1
• proof
• definition 2
• definition 3
• proposition 2
• proof
• definition 4
• proposition 3
• proof