Optimal equity auctions with heterogeneous bidders

I analyze the effects of heterogeneity in terms of bidders' valuation distributions and standalone values in equity auctions-auctions in which bidders offer equities rather than cash. Heterogeneity misaligns equity bids' face values, monetary values, and bidder types, making the seller's revenues sensitive to the auction design. Given heterogeneity, I identify the mechanism that maximizes the expected revenue among all incentive-compatible mechanisms of equity auctions. I show how different sources of heterogeneity alter the optimal design, revenues, and allocational efficiency. Unlike optimal cash auctions (Myerson, 1981), where bidders' standalone values are irrelevant, the allocation of optimal equity auctions favors bidders with lower standalone values, because the seller can extract a larger proportion of their rents. By contrast, when bidders differ only in valuation distributions, optimal equity auctions feature more efficient allocations than optimal cash auctions.