Second best efficiency and the English auction

We characterize the incentive compatible allocation that maximizes the expected social surplus in a single-unit sale when the efficient allocation is not implementable. We then show that allowing for the possibility that the good remains unsold may increase the expected social surplus even when allocating the good to no bidder generates less social surplus than allocating to any of the bidders.
Aside from this option, the English auction implements the second best allocation when there are only two bidders but not always with more than two bidders.

JEL Classification Numbers: D44, D82. Keywords: efficiency, auctions, mechanism design.