This paper provides an elementary, non-technical, survey of auction theory, by introducing and describing some of the critical papers in the subject. (The most important of these are reproduced in a companion book, The Economic Theory of Auctions, Paul Klemperer (ed.), Edward Elgar (pub.), forthcoming.) We begin with the most fundamental concepts, and then introduce the basic analysis of optimal auctions, the revenue equivalence theorem, and marginal revenues. Subsequent sections address risk-aversion, affiliation, asymmetries, entry, collusion, multi-unit auctions, double auctions, royalties, incentive contracts, and other topics. Appendices contain technical details, some simple worked examples, and a bibliography for each section.

I review the uses of economic theory in the initial design and later improvement of the ‘‘simultaneous ascending auction,’ ’ which was developed initially for the sale of radio spectrum licenses in the United States. I analyze some capabilities and limitations of the auction, the roles of various detailed rules, the possibilities for introducing combinatorial bidding, and some considerations in adapting the auction for sales in which revenue, rather than efficiency, is the primary goal. I.

A benchmark "package auction" is introduced in which bidders may determine their own packages on which to bid. If all bidders bid straightforwardly, then the outcome is a point in the core of the exchange economy that minimizes the seller's revenue. When goods are substitutes, straightforward bidding strategies comprise an ex post Nash equilibrium. Compared to the Vickrey auction, the benchmark ascending package auction has cheaper information processing, better handling of budget constraints, and less vulnerability to joint bidding strategies among bidders who would otherwise be losers. Improvements are suggested that speed the auction and limit opportunities for collusion.

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We study the recognized open problem of designing revenuemaximizing combinatorial auctions. It is unsolved even for two bidders and two items for sale. Rather than pursuing the pure economic approach of attempting to characterize the optimal auction, we explore techniques for automatically modifying existing mechanisms in a way that increase expected revenue. We introduce a general family of auctions, based on bidder weighting and allocation boosting, which we call virtual valuations combinatorial auctions (VVCA). All auctions in the family are based on the Vickrey-Clarke-Groves (VCG) mechanism, executed on virtual valuations that are linear transformations of the bidders' real valuations. The restriction to linear transformations is motivated by incentive compatibility. The auction family is parameterized by the coefficients in the linear transformations. The problem

A key trend in the world—especially in electronic commerce—is a demand for higher levels of expressiveness in the mechanisms that mediate interactions, such as the allocation of resources, matching of peers, and elicitation of opinions from large and diverse communities. Intuitively, one would think that this increase in expressiveness would lead to more efficient mechanisms (e.g., due to better matching of supply and demand). However, until now we have lacked a general way of characterizing the expressiveness of these mechanisms, analyzing how it impacts the actions taken by rational agents—and ultimately the outcome of the mechanism. In this technical report we introduce a general model of expressiveness for mechanisms. Our model is based on a new measure which we refer to as the maximum impact dimension. The measure captures the number of different ways that an agent can impact the outcome of a mechanism. We proceed to uncover a fundamental connection between this measure and the concept of shattering from computational learning theory. We also provide a way to determine an upper bound on the expected efficiency of any mechanism under its most efficient Nash equilibrium which, remarkably, depends only on the mechanism’s expressiveness. We show that for any setting and any prior over agent preferences, the

We examine a formal model of sponsored search in which advertisers can bid not only on search terms, but on search terms under specific contexts. A context is any auxiliary information that might accompany a search, and might include information that is factual, estimated or inferred. Natural examples of contexts include the zip code, gender, or abstract “intentions ” (such as researching a vacation) of the searcher. After introducing a natural probabilistic model for context-based auctions, we prove several theoretical results, including the fact that under rather general circumstances, the overall social welfare of the advertisers and auctioneer together can only increase when moving from standard to context-based mechanisms. In contrast, we also provide and discuss specific examples in which only one party (advertisers or auctioneer) benefit at the expense of the other in moving to context-based search, and we give extensive simulations contrasting standard and context-based mechanisms in light of these observations.

Two auctions are held in sequence with the possibility of learning between the second and the first. The buyer optimally chooses to discriminate against the winner of the first auction in the second. The optimal mechanism has a discontinuity in the winner’s second bid; this strictly dominates a sequence of independent sealed bid auctions. 1.

A context in sponsored search is additional information about a query, such as the user’s age, gender or location, that can change an advertisement’s relevance or an advertiser’s value for that query. Given a set of contexts, advertiser welfare is maximized if the search engine runs a separate auction for each context; however, due to lack of competition within contexts, this can lead to a significant loss in revenue. In general, neither separate auctions nor pure bundling need maximize revenue. With this motivation, we study the algorithmic question of computing the revenue-maximizing partition of a set of items under a second-price mechanism and additive valuations for bundles. We show that the problem is strongly NP-hard, and present an algorithm that yields a 1-approximation of the 2 revenue from the optimal partition. The algorithm simultaneously yields a 1-approximation of the opti-2 mal welfare, thus ensuring that the gain in revenue is not at the cost of welfare. Finally we show that our algorithm can be applied to the sponsored search setting with multiple slots, to obtain a constant factor approximation of the revenue from the optimal partition. 1