13 problems

For $n$ voters, the normalized Banzhaf power vectors of weighted majority games are points in the unit simplex. The interior of the unit simplex consists of normalized nonnegative…

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…

For each $n\ge2$, define … A weighted majority game is a game represented by a quota and nonnegative voter weights. Two-level Banzhaf lower-bound conjecture. There is a constant…

Let $n\ge3$, let $\sigma\in\mathbb{R}_{\ge0}^n$ satisfy $\Vert\sigma\Vert_1=1$, and write $\sigma^n$ for the desired power distribution in dimension $n$. Shapley–Shubik approximati…

For each $n\in\mathbb{N}$, let $\sigma^n=(0.75,0.25,0,\dots,0)\in\mathbb{R}_{\ge0}^n$, and let $v$ be a simple game on $n$ voters. Uniform approximation lower-bound conjecture. … T…

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…

Let $[q;w_1,\dots,w_n]$ be a weighted majority game with $q\in(0,1)$ and $w\in\mathbb{R}_{\ge0}^n$ normalized by $\Vert w\Vert_1=1$. Let $\Delta=\max_{1\le i\le n}w_i$, and let…

Let $P$ be a power index, let $[q;w_1,\dots,w_n]$ denote a weighted majority game with $q\in(0,1)$, normalized nonnegative weights $w\in\mathbb{R}_{\ge0}^n$ satisfying…

Let $W\subseteq\mathbb{R}_{\ge 0}$ be a set of weights, let $N^{(0)}\subsetneq N^{(1)}\subsetneq\dots$ be an infinite increasing chain of finite non-empty sets, and put…

Let $Bz_1$ denote the relative Banzhaf power index for players of type $1$, and let Corollary denote the source's ordering result for the Shapley--Shubik index. Generalized power-i…

Let $n_1,n_2$ be the numbers of players of the two types, and let $Bz_2(a,b,n_1,n_2)$ denote the relative Banzhaf power index of a player of type $2$ in the corresponding complete…

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

Normal approximation conjecture. The greater difference between the $q^*$- and $\bar q$-heuristics for $n>15$ is conjectured to result from the normal density approximation underly…