On beta-Plurality Points in Spatial Voting Games
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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. * 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(p, V) := sup{β : p is a β-plurality point for V}. Given a voter set V ∈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^*_2. 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 ℝ^d, computes an (1-ε)·β(V) plurality point in time O(n^2/ε^3d-2·logn/ε^d-1·log^2 1/ε).
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