  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 $\beta^*_{(X,d)}\le \frac12$.
