  We generalize the problem of online submodular welfare maximization to
incorporate a variety of new elements arising from reusability, stochastic
rewards, combinatorial actions and similar features that have received
significant attention in recent years. For our general formulation, we show
that a non-adaptive Greedy algorithm achieves the highest possible competitive
ratio against an adaptive offline benchmark in the adversarial arrival model
and in the unknown IID stochastic arrival model. In addition to generalizing
several previous results, this shows that, in general, adaptivity to stochastic
rewards (and similar features) offers no theoretical (worst-case) benefits.
