On β-Plurality Points in Spatial Voting Games

Let V be a set of n points in mathcal Rd, called voters. A point p ∈ mathcal Rd is a plurality point for V when the following holds: For every q ∈ mathcal 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 := {β : any finite multiset V in mathcal Rd 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 R2, 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 Rd, computes an ((1-ɛ)ċ β (V))-plurality point in time On2ɛ 3d-2 ċ log n ɛ d-1 ċ log 2 1ɛ).