How similar are two elections?

Abstract

We introduce and study isomorphic distances between ordinal elections (with the same numbers of candidates and voters). The main feature of these distances is that they are invariant to renaming the candidates and voters, and two elections are at distance zero if and only if they are isomorphic. Specifically, we consider isomorphic extensions of distances between preference orders: Given such a distance d, we extend it to distance $d\text{-}\mathrm{ID}$ between elections by unifying candidate names and finding a matching between the votes, so that the sum of the d-distances between the matched votes is as small as possible. We show that testing isomorphism of two elections can be done in polynomial time so, in principle, such distances can be tractable. Yet, we show that two very natural isomorphic distances are NP-complete and hard to approximate. We attempt to rectify the situation by showing FPT algorithms for several natural parameterizations.