29 problems

Fishburn's conjecture. For $n=6,7$, a Condorcet domain is maximum if and only if it is isomorphic to a domain constructed by Fishburn's alternating scheme.

For $d\geq1$, let $C_n$ be the collection of odd, transitive-symmetric functions $f:\{-1,1\}^n\to\{-1,1\}$. Let $\widehat f(S)$ denote Fourier coefficients, and let…

Let $0\leq\rho<1$ and $\epsilon>0$. Let $f:\{-1,1\}^n\to\{-1,1\}$ satisfy ${\bf E}[f]=0$, and write $\mathrm{Inf}_i(f)$ for the influence of coordinate $i$. Let $V\subseteq[n]$ inc…

Balanced-subset family conjecture. For each $k$ and all sufficiently large $n$, there is a family $\mathcal{F}$ of $n$-subsets of $[2n]$ such that:

Unbounded-index conjecture. There is no uniform bound on the index of incoherent social decision frames: for every $n$ there is an incoherent frame $\mathscr{M}$ of index at least…

Let $P$ be the set of players and let $\mathcal{K}$ be a coalition structure, with core stability meaning that no subset $S \subseteq P$ can deviate to form a new coalition and mak…

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

An election consists of a set of candidates and voters with preferences over the candidates, and its Condorcet dimension is the minimum number of dimensions needed for a Condorcet…

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

Non-normality conjecture. Asymptotically almost surely, MUCDs are not normal.

Puppe and Slinko's conjecture. A MUCD is connected if and only if it is a peak-pit domain.

Disconnectedness conjecture. Asymptotically almost surely, MUCDs are not connected.

Irreducibility conjecture. MUCDs are asymptotically almost surely irreducible.

Abello and Johnson's conjecture. The maximum size satisfies

Johnson's conjecture. The maximum possible size of a Condorcet domain is $2^{n-1}$.

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…

Boland's conjecture. The lowest collective accuracy is reached when the number of voters per group equals the number of groups, that is, when $k=l$.

Boland's conjecture. The collective performance of a group of independent voters is larger for a large number of small groups than for a small number of large groups.

Let $p_{\min}:=\min_{i\ne j}p_{i,j}$ denote the minimum probability of an edge between distinct alternatives in the generalized random tournament model. A 5-kings threshold conject…

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 conjectured bounds. The number $\mathcal{N}(m)$ satisfies

Let a voter distribution have a minority voter population, let $\mathsf{ClusP}$ denote its clustering measure, and let $\mathbb{E}(\mathsf{Rep}_\Delta)$ denote the expected represe…

Galambos–Reiner's conjecture. The quantity $\mathsf{fb}(m)$ is a tight upper bound on the cardinality of acyclic sets described in terms of higher Bruhat orders.

Quadratic voting (QV) is used for binary collective decision problems in the symmetric private-information environment considered here. Quadratic voting efficiency conjecture. QV s…