eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-08
7:1
7:15
10.4230/LIPIcs.SoCG.2020.7
article
On β-Plurality Points in Spatial Voting Games
Aronov, Boris
1
https://orcid.org/0000-0003-3110-4702
de Berg, Mark
2
https://orcid.org/0000-0001-5770-3784
Gudmundsson, Joachim
3
https://orcid.org/0000-0002-6778-7990
Horton, Michael
4
https://orcid.org/0000-0001-6388-9634
Tandon School of Engineering, New York University, Brooklyn, NY 11201, USA
Department of Computing Science, TU Eindhoven, 5600 MB Eindhoven, The Netherlands
School of Computer Science, University of Sydney, Sydney, NSW 2006, Australia
Sportlogiq, Inc., Montreal, Quebec H2T 3B3, Canada
Let V be a set of n points in ℝ^d, called voters. A point p ∈ ℝ^d is a plurality point for V when the following holds: for every q ∈ ℝ^d the number of voters closer to p than to q is at least the number of voters closer to q than to p. Thus, in a vote where each v ∈ V votes for the nearest proposal (and voters for which the proposals are at equal distance abstain), proposal p will not lose against any alternative proposal q. For most voter sets a plurality point does not exist. We therefore introduce the concept of β-plurality points, which are defined similarly to regular plurality points except that the distance of each voter to p (but not to q) is scaled by a factor β, for some constant 0<β⩽1. We investigate the existence and computation of β-plurality points, and obtain the following results.
- Define β^*_d := sup{β : any finite multiset V in ℝ^d admits a β-plurality point}. We prove that β^*₂ = √3/2, and that 1/√d ⩽ β^*_d ⩽ √3/2 for all d⩾3.
- Define β(V) := sup {β : V admits a β-plurality point}. We present an algorithm that, given a voter set V in {ℝ}^d, computes an (1-ε)⋅ β(V) plurality point in time O(n²/ε^(3d-2) ⋅ log(n/ε^(d-1)) ⋅ log²(1/ε)).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol164-socg2020/LIPIcs.SoCG.2020.7/LIPIcs.SoCG.2020.7.pdf
Computational geometry
Spatial voting theory
Plurality point
Computational social choice