Optimal Multi-Unit Mechanisms with Private Demands

In the multi-unit pricing problem, multiple units of a single item are for sale. A buyer's valuation for n units of the item is v min n, d , where the per unit valuation v and the capacity d are private information of the buyer. We consider this problem in the Bayesian setting, where the pair (v,d) is drawn jointly from a given probability distribution. In the unlimited supply setting, the optimal (revenue maximizing) mechanism is a pricing problem, i.e., it is a menu of lotteries. In this paper we show that under a natural regularity condition on the probability distributions, which we call decreasing marginal revenue, the optimal pricing is in fact deterministic. It is a price curve, offering i units of the item for a price of pi, for every integer i. Further, we show that the revenue as a function of the prices pi is a concave function, which implies that the optimum price curve can be found in polynomial time. This gives a rare example of a natural multi-parameter setting where we can show such a clean characterization of the optimal mechanism. We also give a more detailed characterization of the optimal prices for the case where there are only two possible demands.