# Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions

## TL;DR

This paper extends key results on coalition biasing of Boolean functions from the uniform measure to arbitrary product measures, advancing towards Friedgut's conjecture and analyzing multi-round protocols.

## Contribution

It generalizes the biasing results of Kahn, Kalai, Linial, and Russell et al. to arbitrary product distributions, including multi-round protocols and functions with finite ranges.

## Key findings

- Coalitions of o(n) players can bias Boolean functions under arbitrary product measures.
- The results include multi-round protocols with o(log* n) rounds.
- Introduces a novel boosting argument for general product distributions.

## Abstract

The seminal result of Kahn, Kalai and Linial shows that a coalition of $O(\frac{n}{\log n})$ players can bias the outcome of any Boolean function $\{0,1\}^n \to \{0,1\}$ with respect to the uniform measure. We extend their result to arbitrary product measures on $\{0,1\}^n$, by combining their argument with a completely different argument that handles very biased coordinates. We view this result as a step towards proving a conjecture of Friedgut, which states that Boolean functions on the continuous cube $[0,1]^n$ (or, equivalently, on $\{1,\dots,n\}^n$) can be biased using coalitions of $o(n)$ players. This is the first step taken in this direction since Friedgut proposed the conjecture in 2004. Russell, Saks and Zuckerman extended the result of Kahn, Kalai and Linial to multi-round protocols, showing that when the number of rounds is $o(\log^* n)$, a coalition of $o(n)$ players can bias the outcome with respect to the uniform measure. We extend this result as well to arbitrary product measures on $\{0,1\}^n$. The argument of Russell et al. relies on the fact that a coalition of $o(n)$ players can boost the expectation of any Boolean function from $\epsilon$ to $1-\epsilon$ with respect to the uniform measure. This fails for general product distributions, as the example of the AND function with respect to $\mu_{1-1/n}$ shows. Instead, we use a novel boosting argument alongside a generalization of our first result to arbitrary finite ranges.