Improved Approximations for Free-Order Prophets and Second-Price Auctions
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We study the fundamental problem of selling a single indivisible item to one of n buyers with independent and potentially nonidentical value distributions. We focus on two simple and widely used selling mechanisms: the second price auction with eager personalized reserve prices and the sequential posted price mechanism. Using a new approach, we improve the best-known performance guarantees for these mechanisms. We show that for every value of the number of buyers n, the eager second price (ESP) auction and sequential posted price mechanisms respectively earn at least 0.6620 and 0.6543 fractions of the optimal revenue. We also provide improved performance guarantees for these mechanisms when the number of buyers is small, which is the more relevant regime for many applications of interest. This in particular implies an improved bound of 0.6543 for free-order prophet inequalities. Motivated by our improved revenue bounds, we further study the problem of optimizing reserve prices in the ESP auctions when the sorted order of personalized reserve prices among bidders is exogenous. We show that this problem can be solved polynomially. In addition, by analyzing a real auction dataset from Google's advertising exchange, we demonstrate the effectiveness of order-based pricing.
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