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55
Online budgeted matching in random input models with applications to adwords
- In SODA 2008
"... We study an online assignment problem, motivated by Adwords Allocation, in which queries are to be assigned to bidders with budget constraints. We analyze the performance of the Greedy algorithm (which assigns each query to the highest bidder) in a randomized input model with queries arriving in a r ..."
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Cited by 72 (10 self)
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We study an online assignment problem, motivated by Adwords Allocation, in which queries are to be assigned to bidders with budget constraints. We analyze the performance of the Greedy algorithm (which assigns each query to the highest bidder) in a randomized input model with queries arriving in a random permutation. Our main result is a tight analysis of Greedy in this model showing that it has a competitive ratio of 1 − 1/e for maximizing the value of the assignment. We also consider the more standard i.i.d. model of input, and show that our analysis holds there as well. This is to be contrasted with the worst case analysis of [MSVV05] which shows that Greedy has a ratio of 1/2, and that the optimal algorithm presented there has a ratio of 1 − 1/e. The analysis of Greedy is important in the Adwords setting because it is the natural allocation algorithm for an auction-style process. From a theoretical perspective, our result simplifies and generalizes the classic algorithm of Karp, Vazirani and Vazirani for online bipartite matching. Our results include a new proof to show that the Ranking algorithm of [KVV90] has a ratio of 1 − 1/e in the worst case. It has been recently discovered [KV07] (independent of our results) that one of the crucial lemmas in [KVV90], related to a certain reduction, is incorrect. Our proof is direct, in that it does not go via such a reduction, which also enables us to generalize the analysis to our online assignment problem. 1
The adwords problem: Online keyword matching with budgeted bidders under random permutations
- In Proc. 10th Annual ACM Conference on Electronic Commerge (EC
, 2009
"... We consider the problem of a search engine trying to assign a sequence of search keywords to a set of competing bidders, each with a daily spending limit. The goal is to maximize the revenue generated by these keyword sales, bearing in mind that, as some bidders may eventually exceed their budget, n ..."
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Cited by 70 (6 self)
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We consider the problem of a search engine trying to assign a sequence of search keywords to a set of competing bidders, each with a daily spending limit. The goal is to maximize the revenue generated by these keyword sales, bearing in mind that, as some bidders may eventually exceed their budget, not all keywords should be sold to the highest bidder. We assume that the sequence of keywords (or equivalently, of bids) is revealed on-line. Our concern will be the competitive ratio for this problem versus the off-line optimum. We extend the current literature on this problem by considering the setting where the keywords arrive in a random order. In this setting we are able to achieve a competitive ratio of 1 − ɛ under some mild, but necessary, assumptions.
Sponsored Search Auctions with Markovian Users
"... Abstract. Sponsored search involves running an auction among advertisers who bid in order to have their ad shown next to search results for specific keywords. The most popular auction for sponsored search is the “Generalized Second Price ” (GSP) auction where advertisers are assigned to slots in the ..."
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Cited by 62 (3 self)
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Abstract. Sponsored search involves running an auction among advertisers who bid in order to have their ad shown next to search results for specific keywords. The most popular auction for sponsored search is the “Generalized Second Price ” (GSP) auction where advertisers are assigned to slots in the decreasing order of their score, which is defined as the product of their bid and click-through rate. One of the main advantages of this simple ranking is that bidding strategy is intuitive: to move up to a more prominent slot on the results page, bid more. This makes it simple for advertisers to strategize. However this ranking only maximizes efficiency under the assumption that the probability of a user clicking on an ad is independent of the other ads shown on the page. We study a Markovian user model that does not make this assumption. Under this model, the most efficient assignment is no longer a simple ranking function as in GSP. We show that the optimal assignment can be found efficiently (even in near-linear time). As a result of the more sophisticated structure of the optimal assignment, bidding dynamics become more complex: indeed it is no longer clear that bidding more moves one higher on the page. Our main technical result is that despite the added complexity of the bidding dynamics, the optimal assignment has the property that ad position is still monotone in bid. Thus even in this richer user model, our mechanism retains the core bidding dynamics of the GSP auction that make it useful for advertisers. 1
A cascade model for externalities in sponsored search
- In ACM EC-08 Workshop on Ad Auctions
, 2008
"... Abstract. One of the most important yet insufficiently studied issues in online advertising is the externality effect among ads: the value of an ad impression on a page is affected not just by the location that the ad is placed in, but also by the set of other ads displayed on the page. For instance ..."
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Cited by 56 (1 self)
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Abstract. One of the most important yet insufficiently studied issues in online advertising is the externality effect among ads: the value of an ad impression on a page is affected not just by the location that the ad is placed in, but also by the set of other ads displayed on the page. For instance, a high quality competing ad can detract users from another ad, while a low quality ad could cause the viewer to abandon the page altogether. In this paper, we propose and analyze a model for externalities in sponsored search ads. Our model is based on the assumption that users will visually scan the list of ads from the top to the bottom. After each ad, they make independent random decisions with ad-specific probabilities on whether to continue scanning. We then generalize the model in two ways: allowing for multiple separate blocks of ads, and allowing click probabilities to explicitly depend on ad positions as well. For the most basic model, we present a polynomial-time incentive-compatible auction mechanism for allocating and pricing ad slots. For the generalizations, we give approximation algorithms for the allocation of ads. 1
Online stochastic packing applied to display ad allocation
- In ESA
, 2010
"... Inspired by online ad allocation, we study online stochastic packing linear programs from theoretical and practical standpoints. We first present a near-optimal online algorithm for a general class of packing linear programs which model various online resource allocation problems including online va ..."
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Cited by 40 (4 self)
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Inspired by online ad allocation, we study online stochastic packing linear programs from theoretical and practical standpoints. We first present a near-optimal online algorithm for a general class of packing linear programs which model various online resource allocation problems including online variants of routing, ad allocations, generalized assignment, and combinatorial auctions. As our main theoretical result, we prove that a simple primal-dual training-based algorithm achieves a (1 − o(1))-approximation guarantee in the random order stochastic model. This is a significant improvement over logarithmic or constant-factor approximations for the adversarial variants of the same problems (e.g. factor 1 − 1e for online ad allocation, and log(m) for online routing). We then focus on the online display ad allocation problem and study the efficiency and fairness of various training-based and online allocation algorithms on data sets collected from real-life display ad allocation system. Our experimental evaluation confirms the effectiveness of training-based primal-dual algorithms on real data sets, and also indicate an intrinsic trade-off between fairness and efficiency. 1
Online bipartite matching with random arrivals: an approach based on strongly factor-revealing lps
- In Proceedings of the 43rd annual ACM symposium on Theory of computing, STOC ’11
, 2011
"... In a seminal paper, Karp, Vazirani, and Vazirani [9] show that a simple ranking algorithm achieves a competitive ratio of 1 − 1/e for the online bipartite matching problem in the standard adversarial model, where the ratio of 1−1/e is also shown to be optimal. Their result also implies that in the r ..."
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Cited by 40 (0 self)
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In a seminal paper, Karp, Vazirani, and Vazirani [9] show that a simple ranking algorithm achieves a competitive ratio of 1 − 1/e for the online bipartite matching problem in the standard adversarial model, where the ratio of 1−1/e is also shown to be optimal. Their result also implies that in the random arrivals model defined by Goel and Mehta [6], where the online nodes arrive in a random order, a simple greedy algorithm achieves a competitive ratio of 1−1/e. In this paper, we study the ranking algorithm in the random arrivals model, and show that it has a competitive ratio of at least 0.696, beating the 1 − 1/e ≈ 0.632 barrier in the adversarial model. Our result also extends to the i.i.d. distribution model of Feldman et al. [5], removing the assumption that the distribution is known. Our analysis has two main steps. First, we exploit certain dominance and monotonicity properties of the ranking algorithm to derive a family of factor-revealing linear programs (LPs). In particular, by symmetry of the ranking algorithm in the random arrivals model, we have the monotonicity property on both sides of the bipartite graph, giving good “strength ” to the LPs. Second, to obtain a good lower bound on the optimal values of all these LPs and hence on the competitive ratio of the algorithm, we introduce the technique of strongly factor-revealing LPs. In particular, we derive a family of modified LPs with similar strength such that the optimal value of any single one of these new LPs is a lower bound on the competitive ratio of the algorithm. This enables us to leverage the power of computer LP solvers to solve for large instances of the new LPs to establish bounds that would otherwise be difficult to attain by human analysis.
Online bipartite matching with unknown distributions
- In STOC
, 2011
"... We consider the online bipartite matching problem in the unknown distribution input model. We show that the Ranking algorithm of [KVV90] achieves a competitive ratio of at least 0.653. This is the first analysis to show an algorithm which breaks the natural 1 − 1/e ‘barrier ’ in the unknown distribu ..."
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Cited by 35 (2 self)
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We consider the online bipartite matching problem in the unknown distribution input model. We show that the Ranking algorithm of [KVV90] achieves a competitive ratio of at least 0.653. This is the first analysis to show an algorithm which breaks the natural 1 − 1/e ‘barrier ’ in the unknown distribution model (our analysis in fact works in the stricter, random order model) and answers an open question in [GM08]. We also describe a family of graphs on which Ranking does no better than 0.727 in the random order model. Finally, we show that for graphs which have k> 1 disjoint perfect matchings, Ranking achieves a competitive ratio of at least 1 −
Optimal delivery of sponsored search advertisements subject to budget constraints
- In ACM EC
, 2007
"... We discuss an auction framework in which sponsored search advertisements are delivered in response to queries. In practice, the presence of bidder budgets can have a significant impact on the ad delivery process. We propose an approach based on linear programming which takes bidder budgets into acco ..."
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Cited by 33 (3 self)
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We discuss an auction framework in which sponsored search advertisements are delivered in response to queries. In practice, the presence of bidder budgets can have a significant impact on the ad delivery process. We propose an approach based on linear programming which takes bidder budgets into account, and uses them in conjunction with forecasting of query frequencies, and pricing and ranking schemes, to optimize ad delivery. Simulations show significant improvements in revenue and efficiency. Categories and Subject Descriptors G.4 [Mathematics of Computing]: Mathematical Software—Algorithm
Stochastic models for budget optimization in search-based advertising
- In Proc. Workshop on Internet and Network Economics (WINE
"... Internet search companies sell advertisement slots based on users ’ search queries via an auction. Advertisers have to solve a complex optimization problem of how to place bids on the keywords of their interest so that they can maximize their return (the number of user clicks on their ads) for a giv ..."
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Cited by 23 (5 self)
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Internet search companies sell advertisement slots based on users ’ search queries via an auction. Advertisers have to solve a complex optimization problem of how to place bids on the keywords of their interest so that they can maximize their return (the number of user clicks on their ads) for a given budget. This is the budget optimization problem. In this paper, we model budget optimization as it arises in Internet search companies and formulate stochastic versions of the problem. The premise is that Internet search companies can predict probability distributions associated with queries in the future. We identify three natural stochastic models. In the spirit of other stochastic optimization problems, two questions arise. • (Evaluation Problem) Given a bid solution, can we evaluate the expected value of the objective function under different stochastic models? • (Optimization Problem) Can we determine a bid solution that maximizes the objective function in expectation under different stochastic models? Our main results are algorithmic and complexity results for both these problems for our three stochastic models. In particular, our algorithmic results show that simple prefix strategies that bid on all cheap keywords up to some level are either optimal or good approximations for many cases; we show other cases to be NP-hard. 1
