We show that first-price auctions can maximize a wide variety of objectives, includ- ing revenue, welfare, bidder surplus, and equality, while second-price (or ascending)

auctions do not maximize revenue except in Myerson’s regular case. This stark contrast between canonical auction pricing rules qualifies the celebrated revenue equivalence. Furthermore, the optimality of first-price auctions does not hinge on any distributional

assumptions, which enables us to analyze problems that are beyond the scope of My- ersonian mechanism design. The resulting optimal auctions might employ not only

reserve prices but also bid caps and other exclusions in the space of allowed bids. We also provide tools for the optimal design of first-price auctions and establish conditions for equilibrium existence and uniqueness. (joint with Byeong-hyeon Jeong)