Distances between PreferenceProfiles

Earthmover Distance

The Earthmover distance is a measure of how far apart two distributions are over a given metric space. In our case, the metric space is the BallotGraph endowed with the shortes path metric. We then consider a PreferenceProfile to be a distribution that assigns the number of times a ballot was cast to a node of the BallotGraph. Informally, the Earthmover distance is the minimum cost of moving the "dirt" piled on the nodes by the first profile to the second profile given the distance it must travel.

\(L_p\) Distance

The \(L_p\) distance is a metric parameterized by \(p\in (0,\infty]\). It is computed as \(d(P_1,P_2) = \left(\sum |P_1(b)-P_2(b)|^p\right)^{1/p}\), where the sum is indexed over all possible ballots, and \(P_i(b)\) denotes the number of times that ballot was cast.