Optimal Two-stage Auctions with Costly Information Acquisition

We consider an auction environment with costly entry wherein the cost mainly stems from information acquisition. Bidders are endowed with original estimates (“types”) about their private values and can further learn their true values of the object for sale by incurring an entry cost. We first derive an integral form of the envelope formula as required by incentive compatible two-stage mechanisms, based on which we demonstrate that optimality of the generalized Myerson allocation rule is robust to our setting with costly information acquisition. Optimal entry is thus to admit the set of bidders that maximizes expected virtual surplus adjusted by both the second-stage signal and entry cost. We show that our optimal entry and allocation rules are both IR and IC implementable, and furthermore, in an important class of environments, they can be implemented via a two-stage auction that is essentially a handicap auction augmented with an entry mechanism.