Average Weights and Power in Weighted Voting Games.

We investigate a class of weighted voting games for which weights are randomly distributed over the unit 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 Banzhaf power index and the quota for the uniform measure on the simplex.