Optimal Competitive Auctions

We study the design of truthful auctions for selling identical items in unlimited supply (e.g., digital goods) to n unit demand buyers. This classic problem stands out from profit-maximizing auction design literature as it requires no probabilistic assumptions on buyers' valuations and employs the framework of competitive analysis. Our objective is to optimize the worst-case performance of an auction, measured by the ratio between a given benchmark and revenue generated by the auction. We establish a sufficient and necessary condition that characterizes competitive ratios for all monotone benchmarks. The characterization identifies the worst-case distribution of instances and reveals intrinsic relations between competitive ratios and benchmarks in the competitive analysis. With the characterization at hand, we show optimal competitive auctions for two natural benchmarks. The most well-studied benchmark $\mathcal{F}^{(2)}(\cdot)$ measures the envy-free optimal revenue where at least two buyers win. Goldberg et al. [13] showed a sequence of lower bounds on the competitive ratio for each number of buyers n. They conjectured that all these bounds are tight. We show that optimal competitive auctions match these bounds. Thus, we confirm the conjecture and settle a central open problem in the design of digital goods auctions. As one more application we examine another economically meaningful benchmark, which measures the optimal revenue across all limited-supply Vickrey auctions. We identify the optimal competitive ratios to be $(\frac{n}{n-1})^{n-1}-1$ for each number of buyers n, that is $e-1$ as $n$ approaches infinity

Optimal Parametric Auctions

We study the problem of profit maximization in auctions of one good where the buyers' valuations are drawn from independent distributions. When these distributions are known to the seller, Myerson's optimal auction is a well-known mechanism for maximizing revenue. In many cases, however, the seller may not know the buyers' distributions. We propose an alternative model where the seller only knows the mean and the variance of each distribution. We call parametric an auction whose mechanism only uses these parameters. We construct parametric auctions both when the seller only has one copy of the good to sell, and when she has an infinite number of identical copies (i.e., when the good is digital). 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) no two bidders need to be drawn from identical distributions and (2) the revenue obtained is a constant fraction of the revenue in Myerson's optimal auction

Limited and online supply and the bayesian foundations of prior-free mechanism design

We study auctions for selling a limited supply of a single commodity in the case where the supply is known in advance and the case it is unknown and must be instead allocated in an online fashion. The latter variant was proposed by Mahdian and Saberi [12] as a model of an important phe-nomena in auctions for selling Internet advertising: adver-tising impressions must be allocated as they arrive and the total quantity available is unknown in advance. We describe the Bayesian optimal mechanism for these variants and ex-tend the random sampling auction of Goldberg et al. [8] to address the prior-free case