1. Introduction. Consider the problem faced by someone who has an object to sell, and who does not know how much his prospective buyers might be willing to pay for the object. This seller would like to find some auction procedure which can give him the highest expected revenue or utility among all the different kinds of auctions known (progressive auctions, Dutch auctions, sealed bid auctions, discriminatory auctions, etc.). In this paper, we will construct such optimal auctions for a wide class of sellers' auction design problems. Although these auctions generally sell the object at a discount below what the highest bidder is willing to pay, and sometimes they do not even sell to highest bidder, we shall prove that no other auction mechanism can give higher expected utility to the seller. To analyze the potential performance of different kinds of auctions, we follow