Influential coalitions for Boolean Functions

TL;DR

This paper advances understanding of influential coalitions in Boolean functions by improving bounds and providing counterexamples to longstanding conjectures, highlighting significant gaps for future research.

Contribution

It refines bounds on influential coalitions and presents counterexamples to previous conjectures from the 1980s, revealing unexpected behaviors.

Findings

Improved bounds on influence of large coalitions

Counterexamples to longstanding conjectures

Large gaps between bounds suggest open problems

Abstract

We improve results of Kahn, Kalai, and Linial from the late 80s on the existence of influential large coalitions for Boolean functions, and we give counterexamples to conjectures (of Benny Chor and others) also from the late 80s, by exhibiting functions for which the influences of large coalitions are unexpectedly small relative to the expectations of the functions. The large gaps between the new upper and lower bounds leave a lot of room for further study.