Multidimensional Manhattan Preferences

TL;DR

This paper investigates the conditions under which preference profiles are $d$-Manhattan, establishing bounds based on the number of voters and alternatives, and identifying minimal non-$d$-Manhattan profiles for $d=2$.

Contribution

It characterizes when preference profiles are $d$-Manhattan and identifies the smallest non-$d$-Manhattan profiles for two-dimensional cases.

Findings

Any profile with $m$ alternatives and $n$ voters is $d$-Manhattan if $d \\geq \\min(n, m-1)$.

For $d=2$, the smallest non-$d$-Manhattan profiles have specific voter and alternative counts.

The complexity of $d$-Manhattan profiles exceeds that of $d$-Euclidean profiles.

Abstract

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 $d$ $\geq$ 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].