On Ehrhart polynomials and probability calculations in voting theory

In voting theory, analyzing the frequency of an event (e.g. a voting paradox), under some specific but widely used assumptions, is equivalent to computing the exact number of integer solutions in a system of linear constraints. Recently, some algorithms for computing this number have been proposed in social choice literature by Huang and Chua (Soc Choice Welfare 17:143–155 2000) and by Gehrlein (Soc Choice Welfare 19:503–512 2002; Rev Econ Des 9:317–336 2006). The purpose of this paper is threefold. Firstly, we want to do justice to Eugène Ehrhart, who, more than forty years ago, discovered the theoretical foundations of the above mentioned algorithms. Secondly, we present some efficient algorithms that have been recently developed by computer scientists, independently from voting theorists. Thirdly, we illustrate the use of these algorithms by providing some original results in voting theory.