  A large amount of literature in social choice theory deals with quantifying
the probability of certain election outcomes. One way of computing the
probability of a specific voting situation under the Impartial Anonymous
Culture assumption is via counting integral points in polyhedra. Here, Ehrhart
theory can help, but unfortunately the dimension and complexity of the involved
polyhedra grows rapidly with the number of candidates. However, if we exploit
available polyhedral symmetries, some computations become possible that
previously were infeasible. We show this in three well known examples:
Condorcet's paradox, Condorcet efficiency of plurality voting and in Plurality
voting vs Plurality Runoff.
