We compare the expected efficiency of revenue maximizing (or optimal ) mechanisms with that of efficiency maximizing ones. We show that the efficiency of the revenue maximizing mechanism for selling a single item with k + log_ e (e-1) k + 1 bidders is at least as much as the efficiency of the efficiency maximizing mechanism with k bidders, when bidder valuations are drawn i.i.d. from a Monotone Hazard Rate distribution. Surprisingly, we also show that this bound is tight within a small additive constant of 5.7. In other words, Theta(log k) extra bidders suffice for the revenue maximizing mechanism to match the efficiency of the efficiency maximizing mechanism, while o(log k) do not. This is in contrast to the result of Bulow and Klemperer comparing the revenue of the two mechanisms, where only one extra bidder suffices. More precisely, they show that the revenue of the efficiency maximizing mechanism with k+1 bidders is no less than the revenue of the revenue maximizing mechanism with k bidders. We extend our result for the case of selling t identical items and show that 2.2 log k + t Theta(log log k) extra bidders suffice for the revenue maximizing mechanism to match the efficiency of the efficiency maximizing mechanism. In order to prove our results, we do a classification of Monotone Hazard Rate (MHR) distributions and identify a family of MHR distributions, such that for each class in our classification, there is a member of this family that is pointwise lower than every distribution in that class. This lets us prove interesting structural theorems about distributions with Monotone Hazard Rate.
We study selling one indivisible object to multiple potential buyers. Depending on the objective of the seller, different selling mechanisms are desirable. The Vickrey auction with a truthful reserve price is optimal when the objective is efficiency (i.e., allocating the object to the party who values it the most). The Myerson auction is optimal when the objective is the seller's expected utility. These two objectives are generally in conflict, and cannot be maximized with one mechanism. In many real-world settings---such as privatization and competing electronic marketplaces---it is not clear that the objective should be either efficiency or seller's expected utility. Typically, one of these objectives should weigh more than the other, but both are important. We account for importance of both objectives by designing a new deterministic auction mechanism that maximizes expected social welfare subject to a minimum constraint on the seller's expected utility. This way the seller can expect to do well enough for himself, while maintaining the attractive properties of the mechanism.