Consider a set of voters V, 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|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. Define \beta^*_{(X,d) = \sup\{\beta\mid \mbox{every finite multiset V in X admits a \beta-plurality point\}. }  Aronov, de Berg, Gudmundsson, and Horton [SoCG 2020] ,=\sup\{\beta\mid 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^*_{(R^2,\|\cdot\|_2)}=\sqrt{3}{2}, and more generally, for d-dimensional Euclidean space, 1{d}\le \beta^*_{(R^d,\|\cdot\|_2)}\le\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 exist a metric space for which \beta^*_{(X,d)}\le \frac12.
Consider a set of voters V, 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|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 [SoCG 2020] as a relaxation of the Condorcet criterion. Let \beta^*_{(X,d)=\sup\{\beta\mid 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^*_{(R^2,\|\cdot\|_2)}=\sqrt{3}{2}, and more generally, for d-dimensional Euclidean space, 1{d}\le \beta^*_{(R^d,\|\cdot\|_2)}\le\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.
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Define \beta^*_{(X,d)
For \beta
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\sup\{\beta\mid \mbox{every finite multiset V in X admits a \beta-plurality point
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1, this is equivalent to Condorcet winner, which rarely exists. The concept of \beta-plurality was suggested by
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as a relaxation of the Condorcet criterion. Let \beta^*_{(X,d)
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exist
exists