Euclidean preferences in the plane under $$\varvec{\ell _1},$$ ℓ 1, $$\varvec{\ell _2}$$ ℓ 2 and $$\varvec{\ell _\infty }$$ ℓ ∞ norms

Bruno Escoffier (), Olivier Spanjaard () and Magdaléna Tydrichová ()
Additional contact information
Bruno Escoffier: Sorbonne Université, CNRS, LIP6
Olivier Spanjaard: Sorbonne Université, CNRS, LIP6
Magdaléna Tydrichová: MICS, CentraleSupélec, Universitè Paris-Saclay

Social Choice and Welfare, 2024, vol. 63, issue 1, No 6, 125-169

Abstract: Abstract We present various results about Euclidean preferences in the plane under $$\ell _1,$$ ℓ 1 , $$\ell _2$$ ℓ 2 and $$\ell _{\infty }$$ ℓ ∞ norms. When there are four candidates, we show that the maximum size (in terms of the number of pairwise distinct preferences) of Euclidean preference profiles in $${\mathbb {R}}^2$$ R 2 under norm $$\ell _1$$ ℓ 1 or $$\ell _{\infty }$$ ℓ ∞ is 19. Whatever the number of candidates, we prove that at most four distinct candidates can be ranked in the last position of a two-dimensional Euclidean preference profile under norm $$\ell _1$$ ℓ 1 or $$\ell _\infty ,$$ ℓ ∞ , which generalizes the case of one-dimensional Euclidean preferences (for which it is well known that at most two candidates can be ranked last). We generalize this result to $$2^d$$ 2 d (resp. 2d) for $$\ell _1$$ ℓ 1 (resp. $$\ell _\infty $$ ℓ ∞ ) for d-dimensional Euclidean preferences. We also establish that the maximum size of a two-dimensional Euclidean preference profile on m candidates under norm $$\ell _1$$ ℓ 1 is in $$\varTheta (m^4),$$ Θ ( m 4 ) , which is the same order of magnitude as the known maximum size under norm $$\ell _2.$$ ℓ 2 . Finally, we provide a new proof that two-dimensional Euclidean preference profiles under norm $$\ell _2$$ ℓ 2 for four candidates can be characterized by three inclusion-maximal two-dimensional Euclidean profiles. This proof is a simpler alternative to that proposed by Kamiya et al. (Adv Appl Math 47(2):379–400, 2011).

Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s00355-024-01525-2 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sochwe:v:63:y:2024:i:1:d:10.1007_s00355-024-01525-2

Ordering information: This journal article can be ordered from
http://www.springer. ... c+theory/journal/355

DOI: 10.1007/s00355-024-01525-2

Access Statistics for this article

Social Choice and Welfare is currently edited by Bhaskar Dutta, Marc Fleurbaey, Elizabeth Maggie Penn and Clemens Puppe

More articles in Social Choice and Welfare from Springer, The Society for Social Choice and Welfare Contact information at EDIRC.
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().