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∈V∣β·d(p,v)≤d(q,v)}|≥|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)∗=sup{β∣everyfinitemultisetVinXadmitsaβ-pluralitypoint}. 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)∗=32, and more generally, for d-dimensional Euclidean space, 1d≤β(Rd,‖·‖2)∗≤32. In this paper, we show that 0.557≤β(Rd,‖·‖2)∗ for any dimension d (notice that 1d<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)∗≤12.