Revenue maximization in the dynamic knapsack problem

We characterize the revenue maximizing policy in the dynamic and stochastic knapsack problem where a given capacity needs to be allocated by a given deadline to sequentially arriving agents. Each agent is described by a twodimensional type that re‡ects his capacity requirement and his willingness to pay per unit of capacity. Types are private information. We …rst characterize implementable policies. Then we solve the revenue maximization problem for the special case where there is private information about per-unit values, but weights are observable. After that we derive two sets of additional conditions on the joint distribution of values and weights under which the revenue maximizing policy for the case with observable weights is implementable, and thus optimal also for the case with two-dimensional private information. Finally, we analyze a simple policy for which per-unit prices vary with requested weight but do not vary with time. Its implementation requirements are similar to those of the optimal policy and it turns out to be asymptotically revenue maximizing when available capacity/ time to the deadline both go to in…nity.