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On Shapley Values and Threshold Intervals

Gil Kalai, Noam Lifshitz

TL;DR

Motivated by the interplay between power indices in cooperative games and sharp threshold phenomena, the paper studies monotone Boolean functions under the $p$-biased cube and connects Shapley values to threshold sizes. It introduces a sharp KKL-type argument using one-sided noise and Fourier-analytic tools to bound the influence structure when Shapley or Banzhaf values are small. The main results show that small Shapley values bound the threshold width as $O\left(\frac{\log(1/\epsilon)}{\log(1/t)}\right)$, and, under balance and Banzhaf constraints, shrink Shapley values further to $O\left(\frac{\log\log(1/t)}{\log(1/t)}\right)$, with near-tightness illustrated by explicit examples. These findings quantify when voting rules exhibit sharp transitions and have implications for Condorcet/McGarvey-type results.

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) $.

On Shapley Values and Threshold Intervals

TL;DR

Motivated by the interplay between power indices in cooperative games and sharp threshold phenomena, the paper studies monotone Boolean functions under the -biased cube and connects Shapley values to threshold sizes. It introduces a sharp KKL-type argument using one-sided noise and Fourier-analytic tools to bound the influence structure when Shapley or Banzhaf values are small. The main results show that small Shapley values bound the threshold width as , and, under balance and Banzhaf constraints, shrink Shapley values further to , with near-tightness illustrated by explicit examples. These findings quantify when voting rules exhibit sharp transitions and have implications for Condorcet/McGarvey-type results.

Abstract

Let be a monotone Boolean functions, let denote the Shapley value of the th variable and denote the Banzhaf value (influence) of the th variable. We prove that if we have for all , then the threshold interval of has length . We also prove that if is balanced and for every , then .

Paper Structure

This paper contains 13 sections, 10 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Boolean functions and threshold intervals
  3. Cooperative games and the Shapley value
  4. Influence and the Banzhaf value
  5. The KKL theorem, sharp threshold, and our previos work
  6. Our main results
  7. Two examples
  8. Voting paradoxes, Condorcet and McGarvey
  9. Preliminaries
  10. $p$-biased characters
  11. One sided noise operator
  12. A KKL-type argument
  13. Proof of results

Key Result

Theorem 1.1

There exists an absolute constant $C>0$, such that the following holds. Let $f$ be a monotone Boolean function, $\epsilon,t>0$. Suppose that $p_{1-\epsilon}(f)-p_\epsilon (f) >t$. Then there exists a set $T\subset [n]$ of size $\le \epsilon^{-Ct}$ such that for every $p$ with $p_\epsilon \le p \le p

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: Lif20
  • Theorem 2.2: Odo14
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 6 more