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 \(\beta ^*_{(X,d)}=\sup \{\beta \mid \text{ every } \text{ finite } \text{ multiset } V{ in}X{ admitsa}\beta \text{-plurality } \text{ point }\}\) . 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 \(\beta ^*_{({\mathbb {R}}^2,\Vert \cdot \Vert _2)}=\frac{\sqrt{3}}{2}\) , and more generally, for d-dimensional Euclidean space, \(\frac{1}{\sqrt{d}}\le \beta ^*_{({\mathbb {R}}^d,\Vert \cdot \Vert _2)}\le \frac{\sqrt{3}}{2}\) . In this paper, we show that \(0.557\le \beta ^*_{({\mathbb {R}}^d,\Vert \cdot \Vert _2)}\) 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 \(\beta ^*_{(X,d)}\le \frac{1}{2}\) .
