On β-Plurality Points in Spatial Voting Games

Abstract

Let V be a set of n points in mathcal R d , called voters . A point p ∈ mathcal R d is a plurality point for V when the following holds: For every q ∈ mathcal 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 := {β : any finite multiset V in mathcal R d admits a β-plurality point. We prove that β * d = √3/2, and that 1/√ d ⩽ β * d ⩽ √ 3/2 for all d ⩾ 3. • Define β ( p, V ) := sup {β : p is a β -plurality point for V }. Given a voter set V in mathcal R 2 , we provide an algorithm that runs in O ( n log n ) time and computes a point p such that β ( p , V ) ⩾ β * b . Moreover, for d ⩾ 2, we can compute a point p with β ( p , V ) ⩾ 1/√ d in O ( n ) time. • Define β ( V ) := sup { β : V admits a β -plurality point}. We present an algorithm that, given a voter set V in mathcal R d , computes an ((1-ɛ)ċ β ( V ))-plurality point in time O n 2 ɛ 3d-2 ċ log n ɛ d-1 ċ log 2 1ɛ).

Plurality in Spatial Voting Games with constant $β$

“…The difference between the definitions is that in Definition 1, a voter v that is "undecided", i.e., β•d(p, v) ≤ d(q, v), will choose p over q, while in the original definition, such voters remain "undecided". Definition 1 is equivalent to the original definition in [AdBGH20]. A proof of this equivalence appears in Appendix A.…”

“…Recently, Aronov, de Berg, Gudmundsson, and Horton [AdBGH20], introduced a relaxation, by defining a point p ∈ X to be a β-plurality point, for β ∈ (0, 1], if for every other point q ∈ X,…”

“…to find or estimate these parameters. We refer to [AdBGH20] for further motivation, and a review of the (considerable) related literature.…”

“…Euclidean SpaceIn this section we consider the case of the Euclidean metric space, and give a bound on β * (R d , • 2 ) which is independent of d and greater than 1 √ d for any d ≥ 4, thus improving the lower bound of[AdBGH20] for d ≥ 4.…”

“…Acknowledgments. After sharing our proof of Theorem 1 with the authors of [AdBGH20], Mark de Berg proved a weaker version of Theorem 3, and generously allowed us to publish our proof which is based on his observation. Specifically, de Berg proved that β *…”