On Shapley Values and Threshold Intervals

Let $f\text{\hspace{0.17em}⁣}:\{0,1{\}}^n\to \{0,1\}$ be a monotone Boolean functions, let ${\psi }_k(f)$ denote the Shapley value of the $k$th variable and $b_k(f)$ denote the Banzhaf value (influence) of the $k$th variable. We prove that if we have ${\psi }_k(f)\le t$ for all $k$, then the threshold interval of $f$ has length ${\displaystyle O\left(\frac{1}{\log (1\mathrm{/}t)}\right)}$. We also prove that if $f$ is balanced and $b_k(f)\le t$ for every $k$, then ${\displaystyle \underset{k}{\max }{\psi }_k(f)\le O\left(\frac{\log \log (1\mathrm{/}t)}{\log (1\mathrm{/}t)}\right)}$.