On β-Plurality Points in Spatial Voting Games
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)).
Computational geometry
Spatial voting theory
Plurality point
Computational social choice
Theory of computation~Design and analysis of algorithms
7:1-7:15
Regular Paper
A full version of the paper is available at [Boris Aronov et al., 2020] https://arxiv.org/abs/2003.07513.
The authors would like to thank Sampson Wong for improving an earlier version of Lemma 2.6.
Boris
Aronov
Boris Aronov
Tandon School of Engineering, New York University, Brooklyn, NY 11201, USA
https://orcid.org/0000-0003-3110-4702
Partially supported by NSF grant CCF-15-40656 and by grant 2014/170 from the US-Israel Binational Science Foundation.
Mark
de Berg
Mark de Berg
Department of Computing Science, TU Eindhoven, 5600 MB Eindhoven, The Netherlands
https://orcid.org/0000-0001-5770-3784
Supported by the Netherlands' Organisation for Scientific Research (NWO) under project no. 024.002.003.
Joachim
Gudmundsson
Joachim Gudmundsson
School of Computer Science, University of Sydney, Sydney, NSW 2006, Australia
https://orcid.org/0000-0002-6778-7990
Supported under the Australian Research Council Discovery Projects funding scheme (project numbers DP150101134 and DP180102870).
Michael
Horton
Michael Horton
Sportlogiq, Inc., Montreal, Quebec H2T 3B3, Canada
https://orcid.org/0000-0001-6388-9634
Part of the work performed on this paper was done while visiting NYU, supported by NSF grant CCF 12-18791.
10.4230/LIPIcs.SoCG.2020.7
Boris Aronov, Mark de Berg, Joachim Gudmundsson, and Michael Horton. On β-plurality points in spatial voting games, 2020. URL: http://arxiv.org/abs/2003.07513.
http://arxiv.org/abs/2003.07513
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Boris Aronov, Mark de Berg, Joachim Gudmundsson, and Michael Horton
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