On beta-Plurality Points in Spatial Voting Games

Abstract

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( n 2 ε 3d−2 • log n ε d−1 • log 2 1 ε ).