12 problems

Let $m=3$ candidates have the six possible rankings, with ranking probabilities $p_1,p_2,p_3,p_4,p_5,p_6$ satisfying $p_i\geq 0$ and $\sum_{i=1}^6p_i=1$. For $n$ voters, write…

Existence conjecture. Every preference profile has a $\frac{1}{3}$-winning committee of $2$ alternatives.

Simultaneous non-monotonicity likelihood conjecture. The fraction of election outcomes satisfying all conditions simultaneously lies between $0.003$ and $0.004$:

Lower-bound optimality conjecture. If $\alpha \geq \frac{2}{k+1}$, then an $\alpha$-undominated set of size $k$ always exists.

Let $(X,f)$ be a connectivity system, and let $\mathcal{U}$ be a non-principal ultrafilter of order $k+1$ on $(X,f)$. A majority system of order $k+1$ on $(X,f)$ is a family satisf…

Consider voting tables of type c, parametrized by $(\rho,\alpha)$, and suppose they are added to the set of positive tables as the competence level decreases. For each table, defin…

Exponential deviation conjecture. The exponential deviation holds right off the threshold … The conjecture is supported by numerical experiments, but the supplied text gives no pro…

Let $m^{(n)}$ be a sequence of weight distributions with corresponding probability distributions $(P^{(n)})_{n \in \mathbb{N}}$, where $P^{(n)}=(p_i^{(n)})_{i \in \mathbb{N}}$ is a…

Prime-multiple power-index conjecture. The divisors of the divisor voting systems of $pn$ and $mn$ have the same Shapley–Shubik power indices and the same Banzhaf power indices.

ADO optimality conjecture. For any fixed set of an odd number of distinct abilities for a jury, the ordering that produces the highest reliability is the ADO.

Universal high-temperature Quantitative Arrow's Theorem. For every $q\ge 3$, there exists $\beta'=\beta'(q)>0$ such that, for every $\epsilon>0$, there is…

The Condorcet generating-function conjecture. (i) The series $B(x,P)$ converges for $0\leq x<\frac12$ and $x\leq P\leq\frac12$; (ii) $B(x,P)\geq 0$ on the same region; and (iii)