Abstract

We investigate a class of weighted voting games for which weights are randomly distributed over the standard probability simplex. We provide close-formed formulae for the expectation and density of the distribution of weight of the k-th largest player under the uniform distribution. We analyze the average voting power of the k-th largest player and its dependence on the quota, obtaining analytical and numerical results for small values of n and a general theorem about the functional form of the relation between the average Penrose–Banzhaf power index and the quota for the uniform measure on the simplex. We also analyze the power of a collectivity to act (Coleman efficiency index) of random weighted voting games, obtaining analytical upper bounds therefor.

Highlights

  • An n–player weighted voting game G is described by a weight vector w := (w1, . . . , wn) ∈ ∆n, where ∆n is the standard (n − 1)–dimensional probability simplex, and a qualified majority quota q

  • By random weighted voting game we mean a weighted voting game in which the number of players n and the quota q are fixed, and the weight vector w is drawn from the standard probability simplex ∆n with some probability

  • We consider a random weighted voting game, where the weight vector W ∈ ∆n is a random variable with the uniform probability distribution, which will be thereafter denoted as Unif (∆n)

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Summary

Introduction

Let ∆n be the standard (n − 1)–dimensional probability simplex, which represents the set of normalized weight vectors. We consider a random weighted voting game, where the weight vector W ∈ ∆n is a random variable with the uniform probability distribution, which will be thereafter denoted as Unif (∆n). Note that the coordinates of W, i.e., the voting weights of the players, can almost surely be strictly ordered. This ordering provides a natural basis for distinguishing the players a posteriori. We start with the simplest question: what is the expected value and density of the distribution of voting weight of the k–th largest player in a random weighted voting game?

Related work
Expected value: barycenter of the asymmetric simplex
Densities
Definitions
The power of a collectivity to act
Splines
Methods
Full Text
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