Asymptotic bias of some election methods

TL;DR

This paper analyzes the asymptotic bias in seat allocation methods in elections, revealing how different methods favor large or small parties as the total number of seats grows large.

Contribution

It provides formulas for the bias of common divisor and quota methods, detailing their asymptotic behavior as the total seats increase.

Findings

Bias formulas favoring large or small parties depending on the method

Asymptotic distribution of seat excess for different election methods

Analysis under fixed party proportions with increasing total seats

Abstract

Consider an election where N seats are distributed among parties with proportions p_1,...,p_m 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 Np_i that yields exact proportionality. Our approach is to keep p_1,...,p_m fixed and let N tend to infinity, 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.