Abstract
Let V be a set of n points in Rd, called voters. A point p∈Rd is a plurality point for V when the following holds: for every q∈Rd 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 Rd 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 Rd, computes an (1−ε)⋅β(V) plurality point in time O(n2ε3d−2⋅lognεd−1⋅log21ε).
Original language | English |
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Article number | 2003.07513 |
Number of pages | 18 |
Journal | arXiv |
Volume | 2020 |
DOIs | |
Publication status | Published - 17 Mar 2020 |
Keywords
- Computer Science
- Computational Geometry