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Revenue maximization with a single sample

We design and analyze approximately revenue-maximizing auctions in general single-parameter settings. Bidders have publicly observable attributes, and we assume that the valuations of indistinguishable bidders are independent draws from a common distribution. Crucially, we assume all valuation distributions are a priori unknown to the seller. Despite this handicap, we show how to obtain approximately optimal expected revenue - nearly as large as what could be obtained if the distributions were known in advance - under quite general conditions.
 Our most general result concerns arbitrary downward-closed single-parameter environments and valuation distributions that satisfy a standard hazard rate condition. We also assume that no bidder has a unique attribute value, which is obviously necessary with unknown and attribute-dependent valuation distributions. Here, we give an auction that, for every such environment and unknown valuation distributions, has expected revenue at least a constant fraction of the expected optimal welfare (and hence revenue). A key idea in our auction is to associate each bidder with another that has the same attribute, with the second bidder's valuation acting as a random reserve price for the first. Conceptually, our analysis shows that even a single sample from a distribution - the second bidder's valuation - is sufficient information to obtain near-optimal expected revenue, even in quite general settings.