Asymptotic bias of some election methods

Abstract

Abstract. Consider an election where N seats are distributed among parties with proportions p1, . . . , pm of the votes. We study, for the common divisor and quota methods, the asymptotic distribution, and in particular the mean, of the seat excess of a party, i.e. the difference between the number of seats given to the party and the (real) number N pi that yields exact proportionality. Our approach is to keep p1, . . . , pm fixed and let N → ∞, with N random in a suitable way.In particular, we give formulas showing the bias favouring large or small parties for the different election methods.

A power-weighted variant of the EU27 Cambridge Compromise

“…See 3 Table 1. As pointed out to the authors by a referee and also by Svante Janson, there are alternative ways to calculate sensible initial exponents E init , including the asymptotic bias formula of [7]. However, when implementing the procedure into the open source Augsburg software BAZI 4 we found that the naïve initialization E = 1 works very well, for all practical purposes.…”

The variance of the discrepancy distribution of rounding procedures, and sums of uniform random variables

“…] prove that the limiting distribution of the rounding residual becomes uniform if the underlying distribution admits a Riemann integrable density. Janson [6] shows that the conclusion remains valid if the underlying distribution is absolutely continuous. Only recently did we spot the prior reference Tukey [12] who establishes the same result.…”

European apportionment via the Cambridge Compromise

“…There is an explicit trade-off between base and rounding method (see [6,7,14] The Cambridge Compromise may be reformulated as a system of any of these three types, and we illustrate this with the case of D'Hondt's method. Allocate to every State the minimum m seats (currently m = 6).…”