On revenue maximization with sharp multi-unit demands

We consider markets consisting of a set of indivisible items, and buyers that have sharp multi-unit demand. This means that each buyer i wants a specific number di of items; a bundle of size less than di has no value. We consider the objective of setting prices and allocations in order to maximize the total revenue of the market maker. The pricing problem with sharp multi-unit demand buyers has a number of properties that the unit-demand model does not possess, and is an important question in algorithmic pricing. We consider the problem of computing a revenue maximizing solution for two solution concepts: competitive equilibrium and envy-free pricing. For unrestricted valuations, these problems are NP-complete; we focus on a realistic special case of "correlated values" where each buyer i has a valuation viqj for item j, where vi and qj are positive quantities associated with buyer i and item j respectively. We present a polynomial time algorithm to solve the revenue-maximizing competitive equilibrium problem. For envy-free pricing, if the demand of each buyer is bounded by a constant, a revenue maximizing solution can be found efficiently; the general demand case is shown to be NP-hard.