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AN AUCTION IS A MECHANISM for allocating a single indivisible object to one of several competing bidders. The winner is the bidder who is awarded the object. The rules of the auction specify two functions. The first is the probability with which a bidder wins, as a function of everyone's bids. The second is the payment each bidder makes to the seller, as a function of all the bids and whether or not he wins. For instance, a first-price auction awards the object to the highest bidder with probability one (providing there are no tie bids), the winner pays his bid, and the losers pay nothing. The bidders in an auction differ significantly. These differences are captured by the bidder's type. A type may be the bidder's personal valuation of the object for sale, his degree of risk aversion, or perhaps his information about the object. (Maskin and Riley (1984) discuss a number of different economically meaningful examples of bidder types.) From the viewpoint of the seller and the other bidders, each bidder's type is a random variable. In this analysis we confine attention to auctions in which the types are independently and identically distributed according to a known probability distribution. The Revelation Principle asserts that every auction is strategically equivalent to an auction in which bidders bid by announcing their type and no bidder has any incentive to lie. Such an auction is called an incentive compatible direct auction. We will confine our attention to the probability functions for direct auctions, and let the incentive compatibility conditions restrict the payment functions. Each bidder can compute the probability that he wins, conditional on his own type, by averaging over the types of the other bidders. The function relating a bidder's type to his probability of winning is the reduced form of the auction. The literature on optimal auctions usually addresses the problem of maximizing expected revenue for the seller. For this purpose, all the relevant information about the probability function of an auction is contained in its reduced form. It is the reduced form that determines each bidder's behavior and hence the seller's expected revenue. In a symmetric auction each bidder's reduced form is identical, so that expected revenue is a functional defined on reduced forms, which are functions of one variable, namely, types. This makes the seller's problem somewhat tractable. To design an auction, a seller must be able to recognize a reduced form and recover the underlying auction. Reduced forms satisfy an intuitive feasibility condition. Given a set of types, the