On the Number of Single-Peaked Narcissistic or Single-Crossing Narcissistic Preference Profiles

TL;DR

This paper quantifies the number of preference profiles that are both narcissistic and either single-peaked or single-crossing, revealing exact counts for these structured voter preference profiles.

Contribution

It provides exact formulas for counting single-peaked and single-crossing narcissistic preference profiles with n voters and n alternatives.

Findings

Number of single-peaked narcissistic profiles: _{i=2}^{n-1} inom{n-1}{i-1}

Number of single-crossing narcissistic profiles: 2^{inom{n-1}{2}}

Explicit enumeration formulas for structured preference profiles.

Abstract

We investigate preference profiles for a set $\mathcal{V}$ of voters, where each voter $i$ has a preference order $\succ_i$ on a finite set $A$ of alternatives (that is, a linear order on $A$) such that for each two alternatives $a,b\in A$, voter $i$ prefers $a$ to $b$ if $a\succ_i b$. Such a profile is narcissistic if each alternative $a$ is preferred the most by at least one voter. It is single-peaked if there is a linear order $\triangleright^{\text{sp}}$ on the alternatives such that each voter's preferences on the alternatives along the order $\triangleright^{\text{sp}}$ are either strictly increasing, or strictly decreasing, or first strictly increasing and then strictly decreasing. It is single-crossing if there is a linear order $\triangleright^{\text{sc}}$ on the voters such that each pair of alternatives divides the order $\triangleright^{\text{sc}}$ into at most two suborders, where in each suborder, all voters have the same linear order on this pair. We show that for $n$ voters and $n$ alternatives,the number of single-peaked narcissistic profiles is $\prod_{i=2}^{n-1} \binom{n-1}{i-1}$ while the number of single-crossing narcissistic profiles is $2^{\binom{n-1}{2}}$.