Functions without influential coalitions

TL;DR

This paper provides counterexamples to longstanding conjectures by showing functions with unexpectedly small influences from large coalitions, challenging previous assumptions in the field.

Contribution

It introduces specific functions that disprove earlier conjectures about influence measures in Boolean functions.

Findings

Counterexamples to two conjectures from the 1980s.

Functions with small influences from large coalitions.

Challenges previous beliefs about influence in Boolean functions.

Abstract

We give counterexamples to a conjecture of Benny Chor and another of the second author, both from the late 80s, by exhibiting functions for which the influences of large coalitions are unexpectedly small relative to the expectations of the functions.