Seat Allocation and Seat Bias under the Jefferson--D'Hondt Method

TL;DR

This paper analytically characterizes how seat shares under the Jefferson--D'Hondt method depend linearly on vote shares and district features, revealing insights into seat bias and thresholds in proportional representation.

Contribution

It provides a novel affine model linking seat shares to vote shares and district parameters under specific assumptions, extending understanding of seat bias in apportionment.

Findings

Seat share is an affine function of vote share and district parameters.

Derived estimates for natural thresholds and their properties.

Identified conditions under which seat bias formulas generalize to multiple districts.

Abstract

We prove that under the Jefferson--D'Hondt method of apportionment, given certain distributional assumptions regarding mean rounding residuals, as well as absence of correlations between party vote shares, district sizes (in votes), and multipliers, the seat share of each relevant party is an affine function of the aggregate vote share, the number of relevant parties, and the mean district magnitude. We further show that the first of those assumptions follows approximately from more general ones regarding smoothness, vanishing at the extremes, and total variation of the density of the distribution of vote shares. We also discuss how our main result differs from the simple generalization of the single-district asymptotic seat bias formulae, and how it can be used to derive an estimate of the natural threshold and certain properties thereof.