We characterize single-item auction formats that are shill-proof in the sense that a profit-maximizing seller has no incentive to submit shill bids. We distinguish between strong shill-proofness, in which a seller with full knowledge of bidders' valuations can never profit from shilling, and weak shill-proofness, which requires only that the expected equilibrium profit from shilling is non-positive. The Dutch auction (with a suitable reserve) is the unique (revenue-)optimal and strongly shill-proof auction. Any deterministic auction can satisfy only two properties in the set {static, strategy-proof, weakly shill-proof}. Our main results extend to settings with affiliated and interdependent values.

Allocating goods efficiently can take a long time. That time spent is costly to participants. We study mechanism design in a symmetric, IPV, heterogeneous item allocation environment where the designer's objective is a flexible function of the allocative efficiency and the time taken to complete a mechanism. Assuming that the size of action sets is sufficiently small, a mechanism offering participants a sequence of potential clearing prices is optimal. We show how activity rules, such as the Milgrom-Wilson activity rule in the FCC spectrum auctions, help speed up auctions: They prevent the formation of sub-optimal equilibria. When the number of bidders is sufficiently large and the value distribution for any bundle is sufficiently thin-tailed, a descending auction is the optimal way to allocate goods. An equivalence result between the designer's speed objective and a complexity objective based on the number of possible actions is also discussed.