6 problems

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 $(1/2,1)$.

Let $n\ge 2$, let $\mathfrak{W}$ denote the class of weighted games, and let $c_{\mathcal{P}}(n,\mathfrak{W})$ be the cost of local monotonicity for the power-index collection…

Let an Alon–Edelman type result mean a bound showing that a simple game's power vector is close to the power vector of a game with only the concentrated $k$ voters and the remainin…

Density conjecture. Achievable power vectors should be dense in the regions of the space of power distributions that are of practical importance, and the poor approximability of…

Shapley–Shubik analogue conjecture. There should be a result analogous to the Alon–Edelman theorem for the Shapley–Shubik index.

Inverse power index lower-bound conjecture. Every such simple game satisfies