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arXiv · 2502.05990

On Shapley Values and Threshold Intervals

Gil Kalai·Noam Lifshitz
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Abstract

Let $f\colon \{0,1\}^n\to \{0,1\}$ be a monotone Boolean functions, let $ψ_k(f)$ 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 $ψ_k(f) \le t$ 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} ψ_k(f) \le O\left(\frac {\log \log (1/t)}{\log(1/t)}\right) $.

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Gil Kalai, Noam Lifshitz. 2025-02-09. On Shapley Values and Threshold Intervals. https://arxiv.org/abs/2502.05990

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