Computing the Yolk in Spatial Voting Games without Computing Median Lines

The yolk is an important concept in spatial voting games as it generalises the equilibrium and provides bounds on the uncovered set. We present near-linear time algorithms for computing the yolk in the spatial voting model in the plane. To the best of our knowledge our algorithm is the first algorithm that does not require precomputing the median lines and hence able to break the existing $O(n^{4/3})$ bound which equals the known upper bound on the number of median lines. We avoid this requirement by using Megiddo's parametric search, which is a powerful framework that could lead to faster algorithms for many other spatial voting problems

The Spatial Analysis of Elections and Committees: Four Decades of Research

It has been more than thirty five years since the publication of Downs's (1957) seminal volume on elections and spatial theory and more than forty since Black and Newing (1951) offered their analysis of majority rule and committees. Thus, in response to the question "What have we accomplished since then?" it is not unreasonable to suppose that the appropriate answer would be "a great deal." Unfortunately, reality admits of only a more ambiguous response

The Spatial Analysis of Elections and Committees: Four Decades of Research

It has been more than thirty five years since the publication of Downs's (1957) seminal volume on elections and spatial theory and more than forty since Black and Newing (1951) offered their analysis of majority rule and committees. Thus, in response to the question "What have we accomplished since then?" it is not unreasonable to suppose that the appropriate answer would be "a great deal." Unfortunately, reality admits of only a more ambiguous response

The almost surely shrinking yolk

The yolk, defined by McKelvey as the smallest ball intersecting all median hyperplanes, is a key concept in the Euclidean spatial model of voting. Koehler conjectured that the yolk radius of a random sample from a uniform distribution on a square tends to zero. The following sharper and more general results are proved here: Let the population be a random sample from a probability measure [mu] on [real]m. Then the yolk of the sample does not necessarily converge to the yolk of [mu]. However, if [mu] is strictly centered, i.e. the yolk radius of [mu] is zero, then the radius of the sample yolk will converge to zero almost surely, and the center of the sample yolk will converge almost surely to the center of the yolk of [mu]. Moreover, if the yolk radius of [mu] is nonzero, the sample yolk radius will not converge to zero if [mu] contains three non-collinear mass points or if somewhere it has density bounded away from zero in some ball of positive volume. All results hold for both odd and even population sizes.Yolk Spatial model Convergence Euclidean model Voting Probability