Multidimensional Manhattan Preferences

Abstract

A preference profile (i.e., a collection of linear preference orders of the voters over a set of alternatives) with m alternatives and n voters is d-Manhattan (resp. d-Euclidean) if both the alternatives and the voters can be placed into a d-dimensional space such that between each pair of alternatives, every voter prefers the one which has a shorter Manhattan (resp. Euclidean) distance to the voter. We initiate the study of how d-Manhattan preference profiles depend on the values m and n. First, we provide explicit constructions to show that each preference profile with m alternatives and n voters is d-Manhattan whenever d ≥ min(n, m − 1). Second, for d = 2, we show that the smallest non d-Manhattan preference profile has either 3 voters and 6 alternatives, or 4 voters and 5 alternatives, or 5 voters and 4 alternatives. This is more complex than the case with d-Euclidean preferences (see [Bogomolnaia and Laslier, 2007] and [Bulteau and Chen, 2022]).