We consider the budget-constrained bidding optimization problem for sponsored search auctions, and model it as an online (multiple-choice) knapsack problem. We design both deterministic and randomized algorithms for the online (multiple-choice) knapsack problems achieving a provably optimal competitive ratio. This translates back to fully automatic bidding strategies maximizing either profit or revenue for the budget-constrained advertiser. To maximize revenue from sponsored search advertising, our bidding strategy can be oblivious (i.e., without knowledge) of other bidders ’ prices and/or clickthrough-rates for those positions. We evaluate our bidding algorithms using both synthetic data and real bidding data gathered manually, and also discuss a sniping heuristic that strictly improves bidding performance. With sniping and parameter tuning enabled, our bidding algorithms can achieve a performance ratio above 90 % against the optimum by the omniscient bidder. 1.

We present the design of a banner advertising auction which is considerably more expressive than current designs. We describe a general model of expressive ad contracts/bidding and an allocation model that can be executed in real time through the assignment of fractions of relevant ad channels to specific advertiser contracts. The uncertainty in channel supply and demand is addressed by the formulation of a stochastic combinatorial optimization problem for channel allocation that is rerun periodically. We solve this in two different ways: fast deterministic optimization with respect to expectations; and a novel online sample-based stochastic optimization method— that can be applied to continuous decision spaces—which exploits the deterministic optimization as a black box. Experiments demonstrate the importance of expressive bidding and the value of stochastic optimization. 1

Abstract. 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. ⋆ This work was done while the author was visiting Google, Inc., New York, NY. 1

Position auctions were widely used by search engines to sell keyword advertising before being well understood (and, indeed, studied) theoretically. To date, theorists have made significant progress, for example showing that a given auction is efficient or revenue-dominates a benchmark auction such as VCG. This paper augments that line of work, relying on computational equilibrium analysis. By computing Nash equilibria and calculating their expected revenue and social welfare, we can quantitatively answer questions that theoretical methods have not. Broadly, the questions we answer are: (1) How often do the theoretically predicted “good” (i.e., efficient, high-revenue) equilibria of GSP occur? (2) In models where GSP is known to be inefficient, how much welfare does it waste? We also use our data to examine the larger question of whether GSP is a good choice, compared with the alternatives.

We model budget-constrained keyword bidding in sponsored search auctions as a stochastic multiple-choice knapsack problem (S-MCKP) and design an algorithm to solve S-MCKP and the corresponding bidding optimization problem. Our algorithm selects items online based on a threshold function which can be built/updated using historical data. Our algorithm achieved about 99 % performance compared to the offline optimum when applied to a real bidding dataset. With synthetic dataset and iid item-sets, its performance ratio against the offline optimum converges to one empirically with increasing number of periods.

Position auctions such as the Generalized Second Price (GSP) are commonly used for sponsored search, e.g., by Yahoo! and Google. We now have an understanding of the equilibria of these auctions, via game-theoretic concepts like Generalized English Auctions and the “locally envy-free ” property, as well as through a relationship to the well-known, truthful Vickrey-Clarke-Groves (VCG) mechanism. In practice, however, position auctions are implemented with additional constraints, in particular, bidder-specific minimum prices. Such minimum prices are used to control the quality of the ads that appear on the page. We study the effect of bidder-specific minimum prices in position auctions. Naïvely enforcing minimum prices in the VCG mechanism breaks the truthfulness of the auction; we describe two variants of VCG for which revealing the truth is a dominant strategy. The implications of bidder-specific minimum prices are more intricate for the GSP auction. Some properties proved for standard GSP no longer hold in this setting. For example, we show that the GSP allocation is now not always efficient (in terms of advertiser value). Also, the property of“envy-locality”enjoyed by GSP—which is essential in the prior analysis of strategies and equilibria— no longer holds. Our main result is to show that despite losing envy locality, GSP with bidder-specific minimum prices still has an envy-free equilibrium. 1.

Due to the economic importance of the position auctions used by search engines to sell advertising, these auctions have received considerable recent study. However, most of this study has been analytic, and these analyses have relied on strong assumptions about the structure of the setting. In this paper, we show that it is feasible to perform computational equilibrium analyses of complex, realistic auction problems like advertising auctions. In particular, we show for the first time that the Nash equilibria of position auctions can be computed exactly, and we do so without relying on any of the assumptions that are necessary for closed-form analysis. We achieve these results by deriving a polynomialsized action graph game representation of the position auction (discretizing bid amounts) and then finding a Nash equilibrium of that game. Our formulation makes it possible to show how equilibrium behavior, revenue and efficiency vary across auction types (Generalized First Price or Generalized Second Price), payment structures (pay-per-click or pay-perimpression), and click-through bias (position does/does not also depend on advertiser’s click-through rate).

We present algorithms for a class of resource allocation problems both in the online setting with stochastic input and in the offline setting. This class of problems contains many interesting special cases such as the Adwords problem. In the online setting we introduce a new distributional model called the adversarial stochastic input model, which is a generalization of the i.i.d model with unknown distributions, where the distributions can change over time. In this model we give a 1 − O(ǫ) approximation algorithm for the resource allocation problem, with almost the weakest possible assumption: the ratio of the maximum amount of resource consumed by any single request to the total capacity of the resource, and the ratio of the profit contributed by any single request to the optimal profit is at most ǫ 2 /log(1/ǫ) 2 where n is the number of resources log n+log(1/ǫ) available. There are instances where this ratio is ǫ 2 /log n such that no randomized algorithm can have a competitive ratio of 1 − o(ǫ) even in the i.i.d model. The upper bound on ratio that we require improves on the previous upper-bound for the i.i.d case by a factor of n. Our proof technique also gives a very simple proof that the greedy algorithm has a competitive ratio of 1 −1/e for the Adwords problem in the i.i.d model with unknown distributions, and more generally in the adversarial stochastic input model, when there is no bound on the bid to budget ratio. All the previous proofs assume A full version of this paper, with all the proofs, is available at

Abstract. We consider a crucial aspect of displaying advertisements on the internet: the individual user. In particular, we consider ad fatigue, where a user tires of an advertisement as it is seen more often. We would like to show advertisements such that, given the impact of ad fatigue, the overall efficiency of the system is optimized. We design an approximation algorithm, for the case that we study, that approaches the optimum as the number of unique ads shown, if there is only one available position, increases. 1

The use of auction mechanisms like the GSP in online advertising can lead to loss of both efficiency and revenue when advertisers have rich preferences: even simple forms of expressiveness like budget constraints can lead to suboptimal outcomes. This has led to the recognition of the value of (sequential and/or stochastic) optimization in ad allocation. Unfortunately, natural formulations of such optimization problems fall prey to channel explosion. Specifically, available ad inventory must be partitioned into subsets, or channels, of indistinguishable supply, each channel containing inventory that is interchangeable from the perspective of each active advertiser. The number of such channels grows exponentially in the number of features of interest. We propose a means for automatically abstracting these channels, grouping together channels so that irrelevant distinctions are ignored. Our approach, based on LP/MIP column and constraint generation, dramatically reduces the number of distinct channels over which ads are allocated, thus rendering optimization computationally feasible at practical scales. Our algorithms also allow revenue/efficiency to be sacrificed in a principled fashion by ignoring potentially relevant distinctions, but retaining the most important distinctions, ignoring only those that have low impact on solution quality. This allows tradeoffs to be made between tractability and solution quality. Numerical experiments demonstrate the computational practicality of our approach as well as the quality of the abstractions generated. 1.