Welfare ordering of voting weight allocations

TL;DR

This paper introduces a stochastic comparison-based partial order for voting weight allocations in committees with many groups, showing asymptotic equivalence to cosine proportionality and monotonicity of key welfare functions.

Contribution

It develops a new partial ordering framework for voting weights and proves its asymptotic equivalence to cosine proportionality in large committees.

Findings

Asymptotic equivalence of the ordering to cosine proportionality

Monotonicity of expected welfare and related functions in the new order

Applicability to large-group voting systems

Abstract

This paper studies the allocation of voting weights in a committee representing groups of different sizes. We introduce a partial ordering of weight allocations based on stochastic comparison of social welfare. We show that when the number of groups is sufficiently large, this ordering asymptotically coincides with the total ordering induced by the cosine proportionality between the weights and the group sizes. A corollary is that a class of expectation-form objective functions, including expected welfare, the mean majority deficit and the probability of inversions, are asymptotically monotone in the cosine proportionality.