  In this work we study the problem of {\em Stochastic Budgeted Multi-round
Submodular Maximization} (SBMSm), in which we would like to adaptively maximize
the sum over multiple rounds of the value of a monotone and submodular
objective function defined on a subset of items, subject to the fact that the
values of this function depend on the realization of stochastic events and the
number of items that we can select over all rounds is limited by a given
budget. This problem extends, and generalizes to multiple round settings,
well-studied problems such as (adaptive) influence maximization and stochastic
probing.
  We first show that, if the number of items and stochastic events is somehow
bounded, there is a polynomial time dynamic programming algorithm for SBMSm.
Then, we provide a simple greedy approximation algorithm for SBMSm, that first
non-adaptively allocates the budget to be spent at each round, and then
greedily and adaptively maximizes the objective function by using the budget
assigned at each round. Such algorithm guarantees a
$(1-1/e-\epsilon)$-approximation to the optimal adaptive value.
  Finally, by introducing a metric called {\em budget-adaptivity gap}, we
measure how much an optimal policy for SBMSm, that is adaptive in both the
budget allocation and item selection, is better than an optimal partially
adaptive policy that, as in our greedy algorithm, determined the budget
allocation in advance. We show a tight bound of $e/(e-1)$ on the
budget-adaptivity gap, and this result implies that our greedy algorithm
guarantees the best approximation among all partially adaptive policies.
