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On beta-Plurality Points in Spatial Voting Games
arXiv - CS - Computational Geometry Pub Date : 2020-03-17 , DOI: arxiv-2003.07513
Boris Aronov and Mark de Berg and Joachim Gudmundsson and Michael Horton

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 $\beta$-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 $\beta$, for some constant $0<\beta\leq 1$. We investigate the existence and computation of $\beta$-plurality points, and obtain the following. * Define $\beta^*_d := \sup \{ \beta : \text{any finite multiset $V$ in $\mathbb{R}^d$ admits a $\beta$-plurality point} \}$. We prove that $\beta^*_2 = \sqrt{3}/2$, and that $1/\sqrt{d} \leq \beta^*_d \leq \sqrt{3}/2$ for all $d\geq 3$. * Define $\beta(p, V) := \sup \{ \beta : \text{$p$ is a $\beta$-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 $\beta(p, V) \geq \beta^*_2$. Moreover, for $d\geq 2$ we can compute a point $p$ with $\beta(p,V) \geq 1/\sqrt{d}$ in $O(n)$ time. * Define $\beta(V) := \sup \{ \beta : \text{$V$ admits a $\beta$-plurality point}\}$. We present an algorithm that, given a voter set $V$ in $\mathbb{R}^d$, computes an $(1-\varepsilon)\cdot \beta(V)$ plurality point in time $O(\frac{n^2}{\varepsilon^{3d-2}} \cdot \log \frac{n}{\varepsilon^{d-1}} \cdot \log^2 \frac {1}{\varepsilon})$.

中文翻译:

空间投票博弈中的beta-Plurality点

令 $V$ 是 $\mathbb{R}^d$ 中的一组 $n$ 点,称为选民。一个点 $p\in\mathbb{R}^d$ 是 $V$ 的复数点,当以下条件成立时:对于每个 $q\in\mathbb{R}^d$ 更接近 $p$ 的选民数量比 $q$ 至少是更接近 $q$ 而不是 $p$ 的选民数量。因此,在每个 $v\in V$ 投票给最近的提案(并且提案距离相等的选民弃权)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,不存在复数点。因此,我们引入了 $\beta$-plurality 点的概念,其定义与常规复数点类似,不同之处在于每个投票者到 $p$(但不是到 $q$)的距离由一个因子 $\beta$ 缩放,对于一些常数 $0<\beta\leq 1$。我们调查了 $\beta$-plurality 点的存在和计算,并获得以下内容。* 定义 $\beta^*_d := \sup \{ \beta : \text{$\mathbb{R}^d$ 中的任何有限多重集 $V$ 承认一个 $\beta$-复数点} \}$。我们证明 $\beta^*_2 = \sqrt{3}/2$,并且 $1/\sqrt{d} \leq \beta^*_d \leq \sqrt{3}/2$ 对于所有 $d\ geq 3 美元。* 定义 $\beta(p, V) := \sup \{ \beta : \text{$p$ 是 $V$}\}$ 的 $\beta$-复数点。给定选民集 $V \in \mathbb{R}^2$,我们提供了一个算法,该算法在 $O(n \log n)$ 时间内运行并计算点 $p$ 使得 $\beta(p, V ) \geq \beta^*_2$。此外,对于 $d\geq 2$,我们可以在 $O(n)$ 时间内用 $\beta(p,V) \geq 1/\sqrt{d}$ 计算点 $p$。* 定义 $\beta(V) := \sup \{ \beta : \text{$V$ 承认一个 $\beta$-复数点}\}$。我们提出了一个算法,给定 $\mathbb{R}^d$ 中的选民集 $V$,
更新日期:2020-05-19
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