  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.
