Apportionment: Uni- and Bi-Dimensional

Abstract

This paper characterizes divisor methods for vector and matrix apportion problems with very simple properties. For the vector problem—a vector gives the votes of parties or the populations of states, a single number the size of the house—they are shown to be the only methods that are coherent with the definition of the corresponding divisor method when applied to only two states or parties. For the matrix problem—rows correspond to districts, columns to parties, entries to votes for party-lists, and the number of seats due to each row (or district) and each column (or party) is known—one extra property is necessary. The method must be proportional: it must give identical answers to a problem obtained by re-scaling any rows and/or any columns of the matrix of votes.