Computing the Yolk in Spatial Voting Games without Computing Median Lines

Abstract

The yolk is an important concept in spatial voting games: the yolk center generalises the equilibrium and the yolk radius bounds the uncovered set. We present near-linear time algorithms for computing the yolk in the plane. To the best of our knowledge our algorithm is the first that does not precompute median lines, and hence is able to break the best known upper bound of O(n 4/3 ) on the number of limiting median lines. We avoid this requirement by carefully applying Megiddo's parametric search technique, which is a powerful framework that could lead to faster algorithms for other spatial voting problems.

On β-Plurality Points in Spatial Voting Games

“…It is unsatisfactory for the model to be unable to provide a solution in most cases, and so we may want to find a point that is close to being a plurality point. One way to formalize this is to consider the center of the yolk (or plurality ball) of V , where the yolk [14,18,22,23] is the smallest ball intersecting every median hyperplane of V . We introduce β-plurality points as an alternative way to relax the requirements for a plurality point, and study several combinatorial and algorithmic questions regarding β-plurality points.…”

Condorcet Relaxation In Spatial Voting

“…2 . Furthermore, they showed that for the case where V consist of the three vertices of an equilateral triangle, it holds that 1986;Miller, Grofman, and Feld 1989;Feld, Grofman, and Miller 1988;Gudmundsson and Wong 2019;Miller 2015), which is the smallest ball intersecting every median hyperplane 3 of V . The center of the yolk is a good heuristic for a plurality point (see (Miller and Godfrey 2008) for a list of properties the yolk possesses).…”