# Condorcet Relaxation In Spatial Voting

## Keywords:

Social Choice / Voting## Abstract

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∈ 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'' 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 [SoCG 2020] as a relaxation of the Condorcet criterion. Denote by β*(X,d) the value sup{ β ∣ every 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)=√3/2 , and more generally, for d-dimensional Euclidean space, 1/√d ≤ β*(ℝd,\|⋅\|2)≤√3/2 . In this paper, we show that 0.557≤ β*(ℝd,\|⋅\|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 .## Downloads

## Published

2021-05-18

## How to Cite

*Proceedings of the AAAI Conference on Artificial Intelligence*,

*35*(6), 5407-5414. Retrieved from https://ojs.aaai.org/index.php/AAAI/article/view/16681

## Issue

## Section

AAAI Technical Track on Game Theory and Economic Paradigms