We consider the problem of revenue-optimal dynamic mechanism design in
settings where agents' types evolve over time as a function of their (both
public and private) experience with items that are auctioned repeatedly over an
infinite horizon. A central question here is understanding what natural
restrictions on the environment permit the design of optimal mechanisms (note
that even in the simpler static setting, optimal mechanisms are characterized
only under certain restrictions).
We study the problem of maximizing a stochastic monotone submodular function
with respect to a matroid constraint. We study the adaptivity gap - the ratio
between the values of optimal adaptive and non-adaptive policies - and show
that it is equal to e/(e-1). This result implies that the benefit of adaptivity
is bounded. We also study the myopic policy and show that it is a
1/2-approximation. Furthermore, when the matroid is uniform, approximation
ratio of the myopic policy becomes 1-1/e which is optimum.