  We consider the problem of stochastic monotone submodular function
maximization, subject to constraints. We give results on adaptivity gaps, and
on the gap between the optimal offline and online solutions. We present a
procedure that transforms a decision tree (adaptive algorithm) into a
non-adaptive chain. We prove that this chain achieves at least ${\tau}$ times
the utility of the decision tree, over a product distribution and binary state
space, where ${\tau} = \min_{i,j} \Pr[x_i=j]$. This proves an adaptivity gap of
$1/{\tau}$ (which is $2$ in the case of a uniform distribution) for the problem
of stochastic monotone submodular maximization subject to state-independent
constraints. For a cardinality constraint, we prove that a simple adaptive
greedy algorithm achieves an approximation factor of $(1-1/e^{\tau})$ with
respect to the optimal offline solution; previously, it has been proven that
the algorithm achieves an approximation factor of $(1-1/e)$ with respect to the
optimal adaptive online solution. Finally, we show that there exists a
non-adaptive solution for the stochastic max coverage problem that is within a
factor $(1-1/e)$ of the optimal adaptive solution and within a factor of
${\tau}(1-1/e)$ of the optimal offline solution.
