On β-Plurality Points in Spatial Voting Games

Authors Boris Aronov , Mark de Berg , Joachim Gudmundsson , Michael Horton

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Author Details

Boris Aronov
  • Tandon School of Engineering, New York University, Brooklyn, NY 11201, USA
Mark de Berg
  • Department of Computing Science, TU Eindhoven, 5600 MB Eindhoven, The Netherlands
Joachim Gudmundsson
  • School of Computer Science, University of Sydney, Sydney, NSW 2006, Australia
Michael Horton
  • Sportlogiq, Inc., Montreal, Quebec H2T 3B3, Canada


The authors would like to thank Sampson Wong for improving an earlier version of Lemma 2.6.

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Boris Aronov, Mark de Berg, Joachim Gudmundsson, and Michael Horton. On β-Plurality Points in Spatial Voting Games. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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)).

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Computational geometry
  • Spatial voting theory
  • Plurality point
  • Computational social choice


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