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

TL;DR

This paper analyzes seat allocation under the Jefferson–D'Hondt method across multi-district elections. It proves that, under mild distributional assumptions on district-level residuals and absence of cross-district correlations, the seat share $q_i$ of each relevant party is an affine function of the renormalized vote share $\hat{p}_i$, the number of relevant parties $\hat{n}$, and the mean district magnitude $m$, specifically $q_i = \hat{p}_i + \hat{p}_i\frac{\hat{n}}{2m} - \frac{1}{2m}$. A key component is showing the mean district multiplier satisfies $\langle\hat{\lambda}_k\rangle = m + \hat{n}/2$, and that the rounding residuals contribute an average of $1/2$, enabling a deterministic, affine seat-share expression. The authors justify Assumption A1 (average residuals and lack of interdistrict correlations) via a probabilistic model with i.i.d. district magnitudes and district vote shares, proving A1a holds approximately as the number of districts grows, and discuss the natural threshold and its implications for party relevance. The results provide a tractable framework for estimating the natural threshold and analyzing how seat bias scales with district magnitude and party participation, offering a bridge between single-district asymptotics and real-world multi-district apportionment.

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.

Figures (1)

• Figure 1: Disrepancy values on the unit simplex $\Delta_{n}$ for $n=3$, $m=8$, and ${ \if@compatibility \mathchar"0116 {} \mathchar"0116 }=m+n/2=9.5$. Blue, green, and red regions represent, respectively, ${ \if@compatibility \mathchar"0115 {} \mathchar"0115 }_{{ \if@compatibility \mathchar"0116 {} \mathchar"0116 }}=0$, ${ \if@compatibility \mathchar"0115 {} \mathchar"0115 }_{{ \if@compatibility \mathchar"0116 {} \mathchar"0116 }}=1$, and ${ \if@compatibility \mathchar"0115 {} \mathchar"0115 }_{{ \if@compatibility \mathchar"0116 {} \mathchar"0116 }}=-1$.

Theorems & Definitions (45)

• Definition 1: Apportionment method
• Definition 2: Rounding function
• Definition 3: Divisor method of apportionment
• Remark 1
• Proposition 1
• Corollary 2
• Definition 4: Rounding thresholds
• Definition 5: Electoral quotients
• Proposition 3
• Remark 2: Electoral ties