Elections with Three Candidates Four Candidates and Beyond: Counting Ties in the Borda Count with Permutahedra and Ehrhart Quasi-Polynomials

In voting theory, the Borda count’s tendency to produce a tie in an election varies as a function of n, the number of voters, and m, the number of candidates. To better understand this tendency, we embed all possible rankings of candidates in a hyperplane sitting in m-dimensional space, to form an (m - 1)-dimensional polytope: the m-permutahedron. The number of possible ties may then be determined computationally using a special class of polynomials with modular coefficients. However, due to the growing complexity of the system, this method has not yet been extended past the case of m = 3. We examine the properties of certain voting situations for m ≥ 4 to better understand an election’s tendency to produce a Borda tie between all candidates

A Finite Calculus Approach To Ehrhart Polynomials

A rational polytope is the convex hull of a finite set of points in R-d with rational coordinates. Given a rational polytope P subset of R-d, Ehrhart proved that, for t is an element of Z(\u3e= 0), the function #(tP boolean AND Z(d)) agrees with a quasi-polynomial L-P(t), called the Ehrhart quasi-polynomial. The Ehrhart quasi-polynomial can be regarded as a discrete version of the volume of a polytope. We use that analogy to derive a new proof of Ehrhart\u27s theorem. This proof also allows us to quickly prove two other facts about Ehrhart quasi-polynomials: McMullen\u27s theorem about the periodicity of the individual coefficients of the quasi-polynomial and the Ehrhart Macdonald theorem on reciprocity