13 problems

For each $n\ge6$, define … A simple game is a monotone voting game on $n$ voters, and $\operatorname{Bz}(v)$ denotes its normalized Banzhaf power vector. Exact two-level Banzhaf co…

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…

Classification conjecture. Each type composition $\widehat{n}$ is either weighted or there exists a type composition $\widehat{m}$ in this list with $\widehat{n}\ge\widehat{m}$. Th…

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

Exact attainment conjecture. For every $n\geq 19$, there exists a simple game $v^*\in\mathcal{S}$ whose PBI attains $\beta^n$ exactly.

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…

Let $[n]=\{1,\ldots,n\}$ and let $\Delta\subseteq2^{[n]}$ be a strongly acyclic simplicial complex. Define the Winder desirability relation $\leq_W$ on $2^{[n]}$ by $A\leq_W B$ if…

Let $[n]=\{1,\ldots,n\}$ and let $\Delta\subseteq 2^{[n]}$ be a simplicial complex. Define the Winder desirability relation $\leq_W$ on $2^{[n]}$ by $A\leq_W B$ if and only if, for…

A system of $n$ weights is a collection of $n$ weights, and satisfying the $n'$th Fishburn condition means meeting the combinatorial condition defined earlier in the paper. Fishbur…