Plurality in Spatial Voting Games with constant $\beta$

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 $p\in X$iscalleda$\beta$-pluralitypointfor$V$,ifforanyotherchoice$q\in X$itholdsthat$|{v\in V\mid \beta\cdot d(p,v)\le d(q,v)}|\ge\frac{|V|}{2}$. 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 multisetV$in$X$admitsa$\beta-plurality point}\}. The parameter ${\beta }^{\ast }$ determines the amount of relaxation required in order to reach a stable decision. Aronov et al. showed that for the Euclidean plane ${\beta }_{({\mathbb{R}}^2,\mathrm{\parallel }\cdot {\mathrm{\parallel }}_2)}^{\ast }=\frac{\sqrt{3}}{2}$, 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 $0.557\le {\beta }_{({\mathbb{R}}^d,\mathrm{\parallel }\cdot {\mathrm{\parallel }}_2)}^{\ast }$ 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 $\sqrt{2}-1\le {\beta }_{(X,d)}^{\ast }$, and show that there exists a metric space for which ${\beta }_{(X,d)}^{\ast }\le \frac{1}{2}$.