Symmetry extensions of “neutrality”

For n≧3 candidates, a system voting vector Wn specifies the positional voting method assigned to each of the 2n−(n+1) subsets of two or more candidates. While most system voting vectors need not admit any relationships among the election rankings; the ones that do are characterized here. The characterization is based on a particular geometric structure (an algebraic variety) that is described in detail and then used to define a partial ordering “⇚” among system voting vectors. The impact of the partial ordering is that if Wn1 ⇚ Wn2, then Wn2 admits more kinds of (single profile) voting paradoxes than Wn1. Therefore the partial ordering provides a powerful, computationally feasible way to compare system voting vectors. The basic ideas are illustrated with examples that completely describe the partial ordering for n=3 and n=4 candidates.