A critique of distributional analysis in social choice

Abstract

Distributional analysis is widely used to study social choice in Euclidean models [28, 29, 1, 3, 8, 15, 5, 2, e.g.]. This method assumes a continuum of voters distributed according to a distribution function. 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 in general converge to the properties of infinite populations. Thus the method of distributional analysis is flawed. In some cases it will predict that a point is in the core with probability 1, while the true probability converges to 0. On the other hand, it is sometime possible to combine distributional analysis with probabilistic analysis to make correct predictions about the asymptotic behavior of large populations, as in [2, e.g.]. Results on the uniform convergence of empirical measures [18, e.g.] are employed to yield simpler proofs of min-max Simpson-Cramer majority [5,2] and yolk shrinkage [26]. The analysis suggests a rule of thumb as to whether or not a prediction based on distributional analysis will be valid for large finite populations. From the experimental point of view, the discussion helps clarify the mathematical underpinnings of statistical analysis of empirical voting data. A careful reading shows Tullock's original paper [28] to be consistent with the analysis given here.