TY - JOUR

T1 - On the number of single-peaked narcissistic or single-crossing narcissistic preference profiles

AU - Chen, Jiehua

AU - Finnendahl, Ugo P.

N1 - Funding Information:
We thank the anonymous reviewers from Discrete Mathematics for pointing out a shorter proof of Theorem 1 and for improving the presentation of the paper. We thank Robert Bredereck (TU Berlin, Germany) for initial discussion on this project and Laurent Bulteau (Laboratoire d’Informatique Gaspard Monge in Marne-la-Vallée, France) for some important references while Jiehua Chen was visiting him in March 2016; the visit was funded by Laboratoire d’Informatique Gaspard Monge in Marne-la-Vallée, France . The main work was done while Jiehua Chen was with TU Berlin, Germany. While with Ben-Gurion University of the Negev, she was supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007–2013)under REA grant agreement number 631163.11 , and by the Israel Science Foundation (Grant Number 551145/14 ).
Publisher Copyright:
© 2018 Elsevier B.V.

PY - 2018/5/1

Y1 - 2018/5/1

N2 - We investigate preference profiles for a set V of voters, where each voter i has a preference order ≻i on a finite set A of alternatives (that is, a linear order on A) such that for each two alternatives a,b∈A, voter i prefers a to b if a≻ib. 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 ▹sp on the alternatives such that each voter's preferences on the alternatives along the order ▹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 ▹sc on the voters such that each pair of alternatives divides the order ▹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 ∏i=2n−1[Formula presented] while the number of single-crossing narcissistic profiles is 2[Formula presented].

AB - We investigate preference profiles for a set V of voters, where each voter i has a preference order ≻i on a finite set A of alternatives (that is, a linear order on A) such that for each two alternatives a,b∈A, voter i prefers a to b if a≻ib. 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 ▹sp on the alternatives such that each voter's preferences on the alternatives along the order ▹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 ▹sc on the voters such that each pair of alternatives divides the order ▹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 ∏i=2n−1[Formula presented] while the number of single-crossing narcissistic profiles is 2[Formula presented].

KW - Narcissistic preferences

KW - Semi-standard Young tableaux

KW - Single-crossing preferences

KW - Single-peaked preferences

UR - http://www.scopus.com/inward/record.url?scp=85042329603&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2018.01.008

DO - 10.1016/j.disc.2018.01.008

M3 - Article

AN - SCOPUS:85042329603

VL - 341

SP - 1225

EP - 1236

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 5

ER -