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

• Published in: arXiv.org 2019-02
• Main Authors: Filmus, Yuval, Hambardzumyan, Lianna, Hatami, Hamed, Hatami, Pooya, Zuckerman, David
• Format: Article
• Language: English
• Subjects: Bias; Boolean algebra; Boolean functions; Continuity (mathematics); Players
• ISSN: 2331-8422

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.