Extremum-count conjecture for average normalized Penrose–Banzhaf power indices

Let $\Delta_n$ be the probability simplex, equipped with the uniform distribution, and let ${\Greekmath 010C}_k^{\downarrow}$ denote the normalized Penrose–Banzhaf power index of the $k$-th largest player in a weighted voting game with quota $q$. Write $\mathbb{E}({\Greekmath 010C}_k^{\downarrow})$ for its average over the simplex.

Extremum-count conjecture. For every $k=1,\ldots,n$, the function $q\mapsto\mathbb{E}({\Greekmath 010C}_k^{\downarrow})$ has exactly $k-1$ local extrema on $(1/2,1)$.

This conjecture describes the regularity suggested by the paper’s Monte Carlo experiments: the largest player’s average power is monotone, while successive players exhibit increasing numbers of extrema. The statement is supported there only by numerical observation, and no proof or resolution is supplied.