On Shapley Values and Threshold Intervals

TL;DR

This paper establishes bounds on the threshold interval length of monotone Boolean functions based on upper limits of Shapley and Banzhaf influence values, linking influence measures to function thresholds.

Contribution

It proves new bounds relating influence measures (Shapley and Banzhaf values) to the threshold intervals of monotone Boolean functions.

Findings

Threshold interval length is $O(1/\log(1/t))$ when all Shapley values are at most $t$.

For balanced functions with Banzhaf influence at most $t$, the maximum Shapley value is $O(rac{\log \log(1/t)}{\log(1/t)})$.

Provides theoretical bounds connecting influence measures to function thresholds.

Abstract

Let $f\colon \{0,1\}^n\to \{0,1\}$ 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 $\displaystyle O \left(\frac {1}{\log (1/t)}\right)$. We also prove that if $f$ is balanced and $b_k(f) \le t$ for every $k$, then $\displaystyle \max_{k} \psi_k(f) \le O\left(\frac {\log \log (1/t)}{\log(1/t)}\right) $.