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.
Which is the more profitable way to sell a company: an auction with no reserve price or an optimally structured negotiation with one less bidder? The authors show, under reasonable assumptions, that the auction is always preferable when bidders' signals are independent. For affiliated signals, the result holds under certain restrictions on the seller's choice of negotiating mechanism. The result suggests that the value of negotiating skill is small relative to the value of additional competition. The paper also shows how the analogies between monopoly theory and auction theory can help derive new results in auction theory. Copyright 1996 by American Economic Association.