The size of the yolk: Computations for odd and even-numbered committees
In the Euclidean spatial model of voting, voter ideal points are located in an m-dimensional issue space, where voters prefer policies closer to their ideal points under the Euclidean norm. The yolk has become an important concept for analysis of the instability of majority choice by committees. The yolk is commonly defined as the smallest ball intersecting all median hyperplanes. A median hyperplane partitions a set of n voters with ideal points located in the issue space such that at least n/2 of the ideal points are on or to one side of the hyperplane and at least n/2 are on or to the other side of the hyperplane. There is a close relationship between the yolk and other solution concepts such as the uncovered set, the Pareto set and the win set, the latter being defined with respect to some proposal and constituting the set of all options preferred to that proposal. The author shows by means of a two- dimensional geometric construction how the yolk determines outer and inner bounds on win sets. The paper starts with the analysis of an odd- numbered committee. A more general formulation of the definition of the yolk is needed in order to deal with even-numbered committees. So-called a-median lines and b-median lines are introduced that determine the outer extremes and the inner extreme of the win set respectively. According to \textit{S. L. Feld}, \textit{B. Grofman} and \textit{N. R. Miller} [Math. Comput. Modelling 12, No.4/5, 405-416 (1989; Zbl 0673.90008)] the location of the yolk indicates the generalized center of the voter distribution, and the size of the yolk indicates how much the voter distribution deviates from one that would generate a majority rule equilibrium. Several social choice theorists have conjectured that the yolk may tend to decrease as the voter population grows. The author supports this conjecture in the second half of the paper by means of simulation studies. Voter ideal points are randomly selected from a uniform distribution in a two-dimensional issue space. The simulations show both for odd-numbered and for even-numbered committees that the mean and variance of the yolk radii decrease with increases in committee size (the size ranging from 3 to 101 members). An application to members of the U.S. Senate completes the analysis.
scientific article; zbMATH DE number 4172709