Optimal Multi-Unit Mechanisms with Private Demands

Abstract

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 p i , for every integer i. Further, we show that the revenue as a function of the prices p i 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.

Selling Two Identical Objects

“…The sale of homogenous objects is analyzed by Malakhov and Vohra (2009) and by Devanur et al (2020). These models differ from ours in that buyers have the same privately known value for all units, but the number of units desired is privately known.…”

“…Apart from Malakhov and Vohra (2009) and Devanur et al (2020), two other papers obtain sufficient conditions for the existence of optimal mechanisms that are deterministic. Manelli and Vincent (2006) obtain sufficient conditions in a model with two heterogenous goods with independent, additive values.…”

The Value of Information Concealment

“…The driving horse for our main result is the duality framework which was recently promoted by Cai et al [2] and subsequently applied to various settings [e.g. 11,8,7]. The basic approach is to write the optimal revenue of a target mechanism (in our case, the revenue with private signals, or the optimal BIC revenue) as the objective of a linear program and then Lagrangify the IC and IR constraints; the value of the resulting Lagrangian with any dual variables serves as an upper bound on the optimal revenue and can be used as a benchmark for approximation.…”