We consider large double auctions with private values. Values need be neither symmetric nor independent. Multiple units may be owned or desired. Participation may be stochastic. We introduce a very mild notion of "a little independence." We prove that all nontrivial equilibria of auctions that satisfy this notion are asymptotically efficient. For any alpha>0, inefficiency disappears at rate (1/n)^(2 alpha). Copyright The Econometric Society 2006.