# Average Weights and Power in Weighted Voting Games

## TL;DR

This paper derives formulas for the distribution and power indices of players in randomly weighted voting games, providing insights into how weights and quotas influence voting power.

## Contribution

It introduces analytical formulas for weight distributions and power indices in weighted voting games with weights uniformly distributed over the simplex.

## Key findings

- Closed-form expressions for weight expectations and densities.
- Analytical relations between power indices and quotas.
- Upper bounds for collective action power in random weighted games.

## 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.