9 problems

Let $w_1,w_2,w_3, \ldots, w_m$ be positive weights, let $w$ be their maximum, and set … For a voting game with these weights, let $\beta_i$ denote the relative Banzhaf index of pla…

A weighted majority game is a simple voting game representable by a quota and nonnegative weights, and a complete simple game is a simple game with a complete desirability relation…

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

A complete simple game is a simple voting game whose desirability relation on voters is a total preorder. Voters are partitioned into types according to this preorder, and a shift-…

Let $c_\mathcal{C}(n)$ denote the maximum critical threshold value of a complete simple game on $n$ voters. Complete-game growth conjecture. … This is the asymptotic form of the pa…

Let $c_\mathcal{S}(n)$ denote the maximum critical threshold value among simple games on $n$ voters. Simple-game threshold conjecture. For every $n\ge 4$, … The paper proves the di…

Let $c_\mathcal{C}(n)$ denote the maximum critical threshold value of a complete simple game on $n$ voters. Complete-game threshold conjecture. There are positive constants $c_1$ a…