We consider the "Offline Ad Slot Scheduling" problem, where advertisers must be scheduled to "sponsored search" slots during a given period of time. Advertisers specify a budget constraint, as well as a maximum cost per click, and may not be assigned to more than one slot for a particular search. We give a truthful mechanism under the utility model where bidders try to maximize their clicks, subject to their personal constraints. In addition, we show that the revenue-maximizing mechanism is not truthful, but has a Nash equilibrium whose outcome is identical to our mechanism. As far as we can tell, this is the first treatment of sponsored search that directly incorporates both multiple slots and budget constraints into an analysis of incentives. Our mechanism employs a descending-price auction that maintains a solution to a certain machine scheduling problem whose job lengths depend on the price, and hence is variable over the auction. The price stops when the set of bidders that can afford that price pack exactly into a block of ad slots, at which point the mechanism allocates that block and continues on the remaining slots. To prove our result on the equilibrium of the revenue-maximizing mechanism, we first show that a greedy algorithm suffices to solve the revenue-maximizing linear program; we then use this insight to prove that bidders allocated in the same block of our mechanism have no incentive to deviate from bidding the fixed price of that block.
@cite_10: 
@cite_17: Billions of dollars are spent each year on sponsored search, a form of advertising where merchants pay for placement alongside web search results. Slots for ad listings are allocated via an auction-style mechanism where the higher a merchant bids, the more likely his ad is to appear above other ads on the page. In this paper we analyze the incentive, efficiency, and revenue properties of two slot auction designs: "rank by bid" (RBB) and "rank by revenue" (RBR), which correspond to stylized versions of the mechanisms currently used by Yahoo! and Google, respectively. We also consider first- and second-price payment rules together with each of these allocation rules, as both have been used historically. We consider both the "short-run" incomplete information setting and the "long-run" complete information setting. With incomplete information, neither RBB nor RBR are truthful with either first or second pricing. We find that the informational requirements of RBB are much weaker than those of RBR, but that RBR is efficient whereas RBB is not. We also show that no revenue ranking of RBB and RBR is possible given an arbitrary distribution over bidder values and relevance. With complete information, we find that no equilibrium exists with first pricing using either RBB or RBR. We show that there typically exists a multitude of equilibria with second pricing, and we bound the divergence of (economic) value in such equilibria from the value obtained assuming all merchants bid truthfully.
@cite_20: We present a truthful auction for pricing advertising slots on a web-page assuming that advertisements for different merchants must be ranked in decreasing order of their (weighted) bids. This captures both the Overture model where bidders are ranked in order of the submitted bids, and the Google model where bidders are ranked in order of the expected revenue (or utility) that their advertisement generates. Assuming separable click-through rates, we prove revenue-equivalence between our auction and the non-truthful next-price auctions currently in use.
@cite_8: We investigate the "generalized second price" auction (GSP), a new mechanism which is used by search engines to sell online advertising that most Internet users encounter daily. GSP is tailored to its unique environment, and neither the mechanism nor the environment have previously been studied in the mechanism design literature. Although GSP looks similar to the Vickrey-Clarke-Groves (VCG) mechanism, its properties are very different. In particular, unlike the VCG mechanism, GSP generally does not have an equilibrium in dominant strategies, and truth-telling is not an equilibrium of GSP. To analyze the properties of GSP in a dynamic environment, we describe the generalized English auction that corresponds to the GSP and show that it has a unique equilibrium. This is an ex post equilibrium that results in the same payoffs to all players as the dominant strategy equilibrium of VCG.
There are some papers on sponsored search that analyze the generalized second-price (GSP) auction, which is the auction currently in use at Google and Yahoo. The equilibria of this auction are characterized and compared with VCG @cite_8 @cite_17 @cite_20 @cite_10 . Here the utility function is the profit-maximizing utility where each bidder attempts to maximize her clicks @math profit per click, and budget constraints are generally not treated.