Plurality in Spatial Voting Games with constant β

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∈ X is called a β-plurality point for V, if for any other choice q∈ X it holds that |v∈ Vmid β· d(p,v)≤ d(q,v)|≥frac|V|2. In other words, at least half of the voters ``prefer'' p over q, when an extra factor of β is taken in favor of p. For β=1, this is equivalent to Condorcet winner, which rarely exists. The concept of β-plurality was suggested by Aronov, de Berg, Gudmundsson, and Horton [TALG 2021] as a relaxation of the Condorcet criterion. Let β^*(X,d)=supβmid mboxevery finite multiset V in X admits a β-plurality point. The parameter β^* determines the amount of relaxation required in order to reach a stable decision. Aronov et al. showed that for the Euclidean plane β^*(ℝ2,|·|2)=frac√32, and more generally, for d-dimensional Euclidean space, frac1√d≤ β^*(ℝd,|·|2)≤frac√32. In this paper, we show that 0.557≤ β^*(ℝd,|·|2) for any dimension d (notice that frac1√d<0.557 for any d≥ 4). In addition, we prove that for every metric space (X,d) it holds that √2-1≤β^*(X,d), and show that there exists a metric space for which β^*(X,d)≤ frac12.