 License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2020.7
URN: urn:nbn:de:0030-drops-121651
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12165/
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### On β-Plurality Points in Spatial Voting Games

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### Abstract

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)).

### BibTeX - Entry

```@InProceedings{aronov_et_al:LIPIcs:2020:12165,
author =	{Boris Aronov and Mark de Berg and Joachim Gudmundsson and Michael Horton},
title =	{{On β-Plurality Points in Spatial Voting Games}},
booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
pages =	{7:1--7:15},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-143-6},
ISSN =	{1868-8969},
year =	{2020},
volume =	{164},
editor =	{Sergio Cabello and Danny Z. Chen},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
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