  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.
