常数β下空间投票博弈中的多数性
Plurality in Spatial Voting Games with constant $β$
AI总结：
该研究针对空间投票中孔多塞胜者极少存在的问题，拓展β-多数松弛准则，改进了d维欧氏空间及一般度量空间下β*的下界，并构造了β*上界为1/2的度量空间。
摘要：
考虑选民集合V，其由度量空间(X,d)中的一个多重集表示。选民需要达成一项决策——即X中的一个点。若对于任意其他备选点q∈X，都满足|{v∈V | β·d(p,v) ≤ d(q,v)}| ≥ |V|/2，则称点p∈X为V的β-多数点。换言之，当引入有利于p的额外因子β时，至少有一半选民“更偏好”p而非q。当β=1时，这一概念等价于孔多塞胜者，而孔多塞胜者极少存在。β-多数概念由Aronov、de Berg、Gudmundsson和Horton [TALG 2021]提出，作为孔多塞准则的一种松弛形式。\n令β*_{(X,d)} = sup{β | X中每个有限多重集V都存在一个β-多数点}。参数β*决定了为达成稳定决策所需的松弛程度。Aronov等人已证明，对于欧几里得平面，β*_{(ℝ²,‖·‖₂)} = √3/2；更一般地，对于d维欧几里得空间，有1/√d ≤ β*_{(ℝᵈ,‖·‖₂)} ≤ √3/2。本文证明，对任意维度d，都有0.557 ≤ β*_{(ℝᵈ,‖·‖₂)}（注意对于所有d≥4，1/√d < 0.557）。此外，我们证明对任意度量空间(X,d)，都有√2 - 1 ≤ β*_{(X,d)}，并证明存在一个度量空间满足β*_{(X,d)} ≤ 1/2。
英文摘要：
Consider a set $V$ of voters, represented by a multiset in a metric space $(X,d)$. The voters have to reach a decision -- a point in $X$. A choice $p\in X$ is called a $β$-plurality point for $V$, if for any other choice $q\in X$ it holds that $|\{v\in V\mid β\cdot d(p,v)\le d(q,v)\}|\ge\frac{|V|}{2}$. In other words, at least half of the voters ``prefer'' $p$ over $q$, when an extra factor of $β$ is taken in favor of $p$. For $β=1$, this is equivalent to Condorcet winner, which rarely exists. The concept of $β$-plurality was suggested by Aronov, de Berg, Gudmundsson, and Horton [TALG 2021] as a relaxation of the Condorcet criterion. Let $β^*_{(X,d)}=\sup\{β\mid \mbox{every finite multiset $V$ in $X$ admits a $β$-plurality point}\}$. The parameter $β^*$ determines the amount of relaxation required in order to reach a stable decision. Aronov et al. showed that for the Euclidean plane $β^*_{(\mathbb{R}^2,\|\cdot\|_2)}=\frac{\sqrt{3}}{2}$, and more generally, for $d$-dimensional Euclidean space, $\frac{1}{\sqrt{d}}\le β^*_{(\mathbb{R}^d,\|\cdot\|_2)}\le\frac{\sqrt{3}}{2}$. In this paper, we show that $0.557\le β^*_{(\mathbb{R}^d,\|\cdot\|_2)}$ for any dimension $d$ (notice that $\frac{1}{\sqrt{d}}<0.557$ for any $d\ge 4$). In addition, we prove that for every metric space $(X,d)$ it holds that $\sqrt{2}-1\leβ^*_{(X,d)}$, and show that there exists a metric space for which $β^*_{(X,d)}\le \frac12$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
