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 $\beta$-plurality point for $V$, if for any other choice $q\in X$ it holds that $|\{v\in V\mid \beta\cdot d(p,v)\le d(q,v)\}|\ge\frac{|V|}{2}$. In other words, at least half of the voters ``prefer'' $p$ over $q$, when an extra factor of $\beta$ is taken in favor of $p$. For $\beta=1$, this is equivalent to Condorcet winner, which rarely exists. The concept of $\beta$-plurality was suggested by Aronov, de Berg, Gudmundsson, and Horton [TALG 2021] as a relaxation of the Condorcet criterion. Let $\beta^*_{(X,d)}=\sup\{\beta\mid \mbox{every finite multiset $V$ in $X$ admits a $\beta$-plurality point}\}$. The parameter $\beta^*$ determines the amount of relaxation required in order to reach a stable decision. Aronov et al. showed that for the Euclidean plane $\beta^*_{(\mathbb{R}^2,\|\cdot\|_2)}=\frac{\sqrt{3}}{2}$, and more generally, for $d$-dimensional Euclidean space, $\frac{1}{\sqrt{d}}\le \beta^*_{(\mathbb{R}^d,\|\cdot\|_2)}\le\frac{\sqrt{3}}{2}$. In this paper, we show that $0.557\le \beta^*_{(\mathbb{R}^d,\|\cdot\|_2)}$ 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 $\sqrt{2}-1\le\beta^*_{(X,d)}$, and show that there exists a metric space for which $\beta^*_{(X,d)}\le \frac12$.

翻译：考虑度量空间$(X,d)$中由多重集表示的选民集合$V$。选民需要达成决策——即空间中的一点$p\in X$。若对任意其他选择$q\in X$，满足$|\{v\in V\mid \beta\cdot d(p,v)\le d(q,v)\}|\ge\frac{|V|}{2}$，则称$p$为$V$的$β$-多元点。换言之，当附加系数$β$偏向$p$时，至少半数选民“偏好”$p$胜于$q$。当$β=1$时，这等价于康多塞获胜者（此类获胜者极少存在）。$β$-多元概念由Aronov、de Berg、Gudmundsson与Horton [TALG 2021]提出，作为康多塞准则的松弛。令$\beta^*_{(X,d)}=\sup\{\beta\mid \mbox{空间$X$中任意有限多重集$V$均存在$β$-多元点}\}$。参数$\beta^*$决定了达成稳定决策所需的松弛程度。Aronov等人证明，对于欧几里得平面有$\beta^*_{(\mathbb{R}^2,\|\cdot\|_2)}=\frac{\sqrt{3}}{2}$，更一般地，在$d$维欧几里得空间中$\frac{1}{\sqrt{d}}\le \beta^*_{(\mathbb{R}^d,\|\cdot\|_2)}\le\frac{\sqrt{3}}{2}$。本文证明对任意维数$d$，$0.557\le \beta^*_{(\mathbb{R}^d,\|\cdot\|_2)}$（注意到当$d\ge 4$时$\frac{1}{\sqrt{d}}<0.557$）。此外，我们证明对所有度量空间$(X,d)$有$\sqrt{2}-1\le\beta^*_{(X,d)}$，并表明存在某度量空间使得$\beta^*_{(X,d)}\le \frac12$。