A critique of distributional analysis in the spatial model

Distributional analysis is widely used to study social choice in Euclidean models [35, 36, 1, 5, 11, 19, 8, 2, e.g]. This method assumes a continuum of voters distributed according a probability measure. Since infinite populations do not exist, the goal of distributional analysis is to give insight into the behavior of large finite populations. However, properties of finite populations do not necessarily converge to the properties of infinite populations. Thus the method of distributional analysis is flawed. In some cases [1] it will predict that a point is in the core with probability 1, while the true probability converges to 0. In other cases it can be combined with probabilistic analysis to make accurate predictions about the asymptotic behavior of large populations, as in [2]. Uniform convergence of empirical measures [23] is employed here to yield a simpler, more general proof of α-majority convergence, a short proof of yolk shrinkage, and suggests a rule of thumb to determine the accuracy of distributionbased predictions. The results also help clarify the mathematical underpinnings of statistical analysis of empirical voting data.