Relative Proot Auctions

We study independent private value auctions where bidders have preferences over relative payoos. We derive symmetric equilibria for four important auction forms: rst-price, second-price, English, and Dutch. We nd that equilibrium bidding is more aggressive than standard predictions. Indeed, in second-price auctions it is optimal to bid above one's valuation when bidders care about relative payoos. Further, strategic equivalence between second-price and English auctions no longer holds; for all but the two highest-valued bidders, bidding is more aggressive in the second-price auction. Despite the fact that all the auctions we study are ex post eecient, revenue equivalence also no longer holds|English auctions always outperform second-price auctions. Moreover, when two bidders are competing, or when valuations are uniformly distributed, second-price auctions have higher expected revenues than rst-price auctions. Finally, we show that bidding in rst-price relative proot auctions is behaviorally equivalent to bidders having preferences exhibiting constant relative risk aversion.