Optimal equity auctions with two-dimensional types

We analyze the design and performance of equity auctions when bidder's valuations and opportunity costs are private information, distributed according to an arbitrary joint density that can differ across bidders. We identify, for any incentive compatible mechanism, an equivalent single-dimensional representation for uncertainty. We then characterize the revenue-maximizing and surplus-maximizing equity mechanisms, and compare revenues in optimal equity and cash auctions. Unlike in cash auctions, the adverse selection arising from bidders' two-dimensional types in equity auctions can lead to a global violation of the regularity condition, which represents a maximal mismatch between incentive compatibility and maximization of revenue or surplus. Such mismatch can lead a seller to exclude bidders and demand a bidder-specific stake from a non-excluded bidder, providing insights into when a firm should employ an auction and when it should just negotiate with a single bidder.