New apportionment methods and their quota property

Abstract

In this paper, we discuss the apportionment problem of distributing seats in a legislature, based proportionally on the population of electoral districts or on the vote totals of political parties. If an apportionment method can be defined via discrete optimization, then its continuous relaxation should have an ideally proportional solution (i.e., the quota) at optimality. First, we propose a new class of reasonable methods of apportionment satisfying such a property. Then we study symmetries of five apportionment methods in the new class. Finally, we estimate how often the five methods stay within the quota.