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| every finite multiset V inX admitsaβ-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 ^*_(ℝ^2,‖·‖ _2)=√(3)/2 , and more generally, for d -dimensional Euclidean space, 1/√(d)≤β ^*_(ℝ^d,‖·‖ _2)≤√(3)/2 . In this paper, we show that 0.557≤β ^*_(ℝ^d,‖·‖ _2) for any dimension d (notice that 1/√(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)≤1/2 .