A Formal Model of the Relationship between the Number of Parties and the District Magnitude

TL;DR

This paper introduces a formal probabilistic model linking district magnitude to the number of relevant parties in elections, validated with data from multiple countries and Polish local elections, and explores electoral engineering applications.

Contribution

It presents a novel model based on the Jefferson-D'Hondt system and Pólya's urn, connecting district size with party dynamics, validated across diverse electoral data.

Findings

Model accurately predicts party number based on district magnitude.

Empirical validation across multiple countries and Polish elections.

Demonstrates potential for electoral engineering analysis.

Abstract

On the basis of a formula for calculating seat shares and natural thresholds in multidistrict elections under the Jefferson-D'Hondt system and a probabilistic model of electoral behavior based on P\'{o}lya's urn model, we propose a new model of the relationship between the district magnitude and the number / effective number of relevant parties. We test that model on both electoral results from multiple countries employing the D'Hondt method (relatively small number of elections, but wide diversity of political configurations) and data based on hundreds of Polish local elections (large number of elections, but much higher degree of parameter uniformity). We also explore some applications of the proposed model, demonstrating how it can be used to estimate the potential effects of electoral engineering.