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).

中文翻译：

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空间投票博弈中的β-复数点

让五成为一组n数学 R 中的点d, 称为选民. 一个点p∈ 数学 Rd是一个多点为了五当以下成立时：对于每个q∈ 数学 Rd, 选民人数更接近p比q至少是选民人数更接近q比p. 因此，在投票中，每个v∈五投票给最近的提案（以及提案距离相等的选民弃权），提案p不会输给任何替代提案q. 对于大多数选民集，不存在复数点。因此，我们引入了β-复数点，其定义类似于常规复数点，除了每个选民到p（但不是q) 由一个因子缩放β, 对于某个常数 0< β ⩽ 1。我们研究了β-多个点并获得以下结果。• 定义β* d:= {β : 任何有限多重集五在数学 Rd承认一个β-复数点。我们证明 β* d= √3/2，那 1/√d⩽ β* d⩽ √ 3/2 全部d⩾ 3. • 定义 β (p, V) := 支持 {β :p是一个 β -复数点五}。给定一个选民集五在数学 R2，我们提供了一个运行在○(n日志n) 时间并计算一个点p使得 β (p,五) ⩾ β* b. 此外，对于d⩾ 2，我们可以计算一个点p与 β (p,五) ⩾ 1/√d在○(n） 时间。• 定义 β (五) := 支持 { β :五承认一个β-复数点}。我们提出了一个算法，给定一个选民集五在数学 Rd, 计算 ((1-ɛ)ċ β (五))-多个时间点○ n 2ε3d-2ċ 日志nεd-1ċ 日志21ɛ)。