# Optimal Parametric Auctions

We study the problem of an auctioneer who wants to maximize her profits. In our model, there are n buyers with private valuations drawn from independent distributions F_1,...,F_n. When these distributions are known to the seller, Myerson's optimal auction is a well known mechanism that maximizes revenue. However, in many cases it is too strong to assume that the seller knows these distributions. We propose an alternative model where the seller only knows the mean mu_i and variance sigma_i^2 of each distribution F_i. We call mechanisms that only use this information parametric auctions. We construct such auctions for all single-dimensional downward closed environments. For a very large class of distributions, including (but not limited to) distributions with a monotone hazard rate, our auctions achieve a constant fraction of the revenue of Myerson's auction. When the seller has absolutely no knowledge about the distributions, it is well known that no auction can achieve a constant fraction of the optimal revenue when the players are not identically distributed. Our parametric model gives the seller a small amount of extra information, allowing her to construct auctions for which (1) she does not know the full distribution of valuations, (2) no two bidders need to be drawn from identical distributions and (3) the revenue obtained is a constant fraction of the revenue in Myerson's optimal auction. For digital goods environments we present a different parametric auction that not only gives a better approximation to the optimal auction, but that is also optimal in a new sense, which we call maximin optimality. Informally, an auction is maximin optimal if it maximizes revenue in the worst case over an adversary's choice of the distribution. We show that our digital parametric is maximin optimal among the class of posted price mechanisms.