空间投票博弈中的β-多点问题
On beta-Plurality Points in Spatial Voting Games
摘要：
让$V$是$\mathbb{R}^d$中的一组$n$点，称为选民。点$p\in \mathbb{R}^d$是$V$的多点，当以下成立时：对于每个$q\in\mathbb{R}^d$，与$q$相比更接近$p$的投票者的数量至少是与$q$相比更接近$p$的投票者的数量。因此，在每个$v\in V$投票支持最近的提案（并且提案距离相等的投票者弃权）的投票中，提案$p$不会输给任何替代提案$q$。对于大多数选民集，不存在多数点。因此，我们引入了$β$ -复数点的概念，其定义类似于常规复数点，除了每个投票者到$p$（而不是到$q$）的距离由因子$β$缩放，对于某个常数$0<β\leq 1$。我们研究了$β$ -复数点的存在性和计算，并得到如下结果。* 定义$β^*_d := \sup \{ β: \text{any finite multiset $ V $ in $ \mathbb{R}^d $ admits a $ β $-plurality point} \}$。我们证明了$β^*_2 = \sqrt{3}/2$，和$1/\sqrt{d} \leq β^*_d \leq \sqrt{3}/2$的所有$d\geq 3$。* 定义$β(p, V) := \sup \{ β: \text{$ p $ is a $ β $-plurality point for $ V $}\}$。给定一个选民集$V \in \mathbb{R}^2$，我们提供了一个算法，在$O(n \log n)$时间运行，并计算一个点$p$，$β(p, V) \geq β^*_2$。此外，对于$d\geq 2$，我们可以在$O(n)$时间内用$β(p,V) \geq 1/\sqrt{d}$计算点$p$。* 定义$β(V) := \sup \{ β: \text{$ V $ admits a $ β $-plurality point}\}$。我们提出了一种算法，给定$\mathbb{R}^d$中的投票者集合$V$，计算$(1-\varepsilon)\cdot β(V)$多个时间点$O(\frac{n^2}{\varepsilon^{3d-2}} \cdot \log \frac{n}{\varepsilon^{d-1}} \cdot \log^2 \frac {1}{\varepsilon})$。
英文摘要：
Let $V$ be a set of $n$ points in $\mathbb{R}^d$, called voters. A point $p\in \mathbb{R}^d$ is a plurality point for $V$ when the following holds: for every $q\in\mathbb{R}^d$ 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 $v\in V$ 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 $0<β\leq 1$. We investigate the existence and computation of $β$-plurality points, and obtain the following. * Define $β^*_d := \sup \{ β: \text{any finite multiset $V$ in $\mathbb{R}^d$ admits a $β$-plurality point} \}$. We prove that $β^*_2 = \sqrt{3}/2$, and that $1/\sqrt{d} \leq β^*_d \leq \sqrt{3}/2$ for all $d\geq 3$. * Define $β(p, V) := \sup \{ β: \text{$p$ is a $β$-plurality point for $V$}\}$. Given a voter set $V \in \mathbb{R}^2$, we provide an algorithm that runs in $O(n \log n)$ time and computes a point $p$ such that $β(p, V) \geq β^*_2$. Moreover, for $d\geq 2$ we can compute a point $p$ with $β(p,V) \geq 1/\sqrt{d}$ in $O(n)$ time. * Define $β(V) := \sup \{ β: \text{$V$ admits a $β$-plurality point}\}$. We present an algorithm that, given a voter set $V$ in $\mathbb{R}^d$, computes an $(1-\varepsilon)\cdot β(V)$ plurality point in time $O(\frac{n^2}{\varepsilon^{3d-2}} \cdot \log \frac{n}{\varepsilon^{d-1}} \cdot \log^2 \frac {1}{\varepsilon})$.
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