  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.
