Multidimensional Manhattan Preferences
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A preference profile with m alternatives and n voters is d-Manhattan (resp. d-Euclidean) if both the alternatives and the voters can be placed into the 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. Following Bogomolnaia and Laslier [Journal of Mathematical Economics, 2007] and Chen and Grottke [Social Choice and Welfare, 2021] who look at d-Euclidean preference profiles, we study which preference profiles are d-Manhattan depending on the values m and n. First, we 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 three voters and six alternatives, or four voters and five alternatives, or five voters and four alternatives. This is more complex than the case with d-Euclidean preferences (see [Bogomolnaia and Laslier, 2007] and [Bulteau and Chen, 2020].
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