Physical Sciences and Mathematics Commons^™

Elections With Three Candidates Four Candidates And Beyond: Counting Ties In The Borda Count With Permutahedra And Ehrhart Quasi-Polynomials, Adam Margulies

Honors Theses

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. …