3 problems

Plurality is Stablest conjecture. If $\rho\geq0$ and $\mathbb{E}f=\frac1k\sum_{j=1}^{k}e_j$, then

Let $0\leq\rho\leq1$ and let $q\geq2$ be an integer. For $f:[q]^n\to[0,1]$, let $\mathbb{S}_\rho(f)$ denote its noise stability and let $\mathrm{Inf}_i(f)$ denote the influence of…

Let $n$ be large, let $k\geq 3$ be the number of candidates, and let voters cast independent uniformly random votes. A voting method is balanced when each candidate has equal proba…