  We study the problem of maximizing a stochastic monotone submodular function
with respect to a matroid constraint. Due to the presence of diminishing
marginal values in real-world problems, our model can capture the effect of
stochasticity in a wide range of applications. We show that the adaptivity gap
-- the ratio between the values of optimal adaptive and optimal non-adaptive
policies -- is bounded and is equal to e/(e-1). We propose a polynomial-time
non-adaptive policy that achieves this bound. We also present an adaptive
myopic policy that obtains at least half of the optimal value. Furthermore,
when the matroid is uniform, the myopic policy achieves the optimal
approximation ratio of 1-1/e.
