Understanding Distance Measures Among Elections

Abstract

Motivated by putting empirical work based on (synthetic) election data on a more solid mathematical basis, we analyze six distances among elections, including, e.g., the challenging-to-compute but very precise swap distance and the distance used to form the so-called map of elections. Among the six, the latter seems to strike the best balance between its computational complexity and expressiveness.

Properties of Position Matrices and Their Elections

“…Intuitively, the isomorphic swap distance between two elections is the summed swap distance of their votes, provided we first rename the candidates and reorder the votes to minimize this value. Maps of elections could be generated using the isomorphic swap distance instead of the positionwise one, and they would be more accurate than those based on the positionwise distance [Boehmer et al, 2022b], but the isomorphic swap distance is NP-hard to compute and challenging to compute in practice [Faliszewski et al, 2019]; indeed, we use a brute-force implementation. Boehmer et al [2022b] have shown that the largest isomorphic swap distance between two elections with m candidates and n voters is 1 4 n(m 2 −m) (up to minor rounding errors; for this result, see their technical report).…”

“…Maps of elections could be generated using the isomorphic swap distance instead of the positionwise one, and they would be more accurate than those based on the positionwise distance [Boehmer et al, 2022b], but the isomorphic swap distance is NP-hard to compute and challenging to compute in practice [Faliszewski et al, 2019]; indeed, we use a brute-force implementation. Boehmer et al [2022b] have shown that the largest isomorphic swap distance between two elections with m candidates and n voters is 1 4 n(m 2 −m) (up to minor rounding errors; for this result, see their technical report). Whenever we give an isomorphic swap distance between two elections (with the same numbers of candidates and voters), we report it as a fraction of this value.…”

“…For our experiments, we use an 8x80 dataset that resembles those of Szufa et al [2020], Boehmer et al [2021b], and Boehmer et al [2022b]. It contains 480 elections with 8 candidates and 80 votes generated using the same statistical models, with the same parameters, as the map of Boehmer et al [2022b].…”

“…For our experiments, we use an 8x80 dataset that resembles those of Szufa et al [2020], Boehmer et al [2021b], and Boehmer et al [2022b]. It contains 480 elections with 8 candidates and 80 votes generated using the same statistical models, with the same parameters, as the map of Boehmer et al [2022b]. In particular, we used (i) impartial culture (IC), where each vote is equally likely, (ii) the Mallows and urn distributions, whose votes are more or less correlated, depending on a parameter, (iii) various Euclidean models, where candidates and voters are points in Euclidean spaces and the voters rank the candidates with respect to their geometric distance, and (iv) uniform distributions over balanced group-separable, caterpillar group-separable, and single-peaked elections (we refer to the uniform distribution of single-peaked elections as the Walsh model; we also use the model of Conitzer [2009] to generate single-peaked elections).…”

“…UN and ID, as well as two other special points, were introduced by Boehmer et al [2021b]. For each two elections, their positionwise distance is at most as large as the distance between UN and ID [Boehmer et al, 2022b].…”