Electoral cells of largest remainders method

Abstract

In an election process, p parties compete for S seats in a parliament. After votes are cast, the electoral result may be thought of as an element x in Rp. Given x, the so-called largest remainders method determines the number ai of seats party i gets in the parliament. The electoral cell determined by (a1,...,ap) is the closure of the set of all results x that determine ai seats for party i, 1<= i<= p. The electoral cells are convex polytopes and tile a hyperplane of Rp.

In this paper we give a description of the electoral cells. For a single cell we identify and classify the cell's faces, completely describe its face lattice, and determine its group of automorphisms. It turns out that each face of dimension d arises from a d-unit-cube by a co pression along a diagonal.