A new era of theoretical computer science addresses fundamental problems about auctions, networks, and human behavior.

http://www.maths.lse.ac.uk/Personal/stengel/TEXTE/agt-stengel.pdf

Algorithmic Game Theory

The popularity of the Web and Internet commerce provides many extremely large datasets from which information can be gleaned by data mining. This book focuses on practical algorithms that have been used to solve key problems in data mining and which can be used on even the largest datasets. It begins with a discussion of the map-reduce framework, an important tool for parallelizing algorithms automatically. The authors explain the tricks of locality-sensitive hashing and stream processing algorithms for mining data that arrives too fast for exhaustive processing. The PageRank idea and related tricks for organizing the Web are covered next. Other chapters cover the problems of finding frequent itemsets and clustering. The final chapters cover two applications: recommendation systems and Web advertising, each vital in e-commerce. Written by two authorities in database and Web technologies, this book is essential reading for students and practitioners alike.

/pdf/mining-of-massive-datasets-2wssu726p1.pdf

Mining of Massive Datasets

We investigate the "generalized second-price" (GSP) auction, a new mechanism used by search engines to sell online advertising. Although GSP looks similar to the Vickrey-Clarke-Groves (VCG) mechanism, its properties are very different. 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, we describe the generalized English auction that corresponds to GSP and show that it has a unique equilibrium. This is an ex post equilibrium, with the same payoffs to all players as the dominant strategy equilibrium of VCG.

/pdf/internet-advertising-and-the-generalized-second-price-1uh5jxi66t.pdf

Internet Advertising and the Generalized Second-Price Auction: Selling Billions of Dollars Worth of Keywords

Journal of the ACM

Let $f:2^X \rightarrow \cal R_+$ be a monotone submodular set function, and let $(X,\cal I)$ be a matroid. We consider the problem ${\rm max}_{S \in \cal I} f(S)$. It is known that the greedy algorithm yields a $1/2$-approximation [M. L. Fisher, G. L. Nemhauser, and L. A. Wolsey, Math. Programming Stud., no. 8 (1978), pp. 73-87] for this problem. For certain special cases, e.g., ${\rm max}_{|S| \leq k} f(S)$, the greedy algorithm yields a $(1-1/e)$-approximation. It is known that this is optimal both in the value oracle model (where the only access to $f$ is through a black box returning $f(S)$ for a given set $S$) [G. L. Nemhauser and L. A. Wolsey, Math. Oper. Res., 3 (1978), pp. 177-188] and for explicitly posed instances assuming $P \neq NP$ [U. Feige, J. ACM, 45 (1998), pp. 634-652]. In this paper, we provide a randomized $(1-1/e)$-approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [J. Combin. Optim., 8 (2004), pp. 307-328], and a continuous greedy process that may be of independent interest. As a special case, our algorithm implies an optimal approximation for the submodular welfare problem in the value oracle model [J. Vondrak, Proceedings of the $38$th ACM Symposium on Theory of Computing, 2008, pp. 67-74]. As a second application, we show that the generalized assignment problem (GAP) is also a special case; although the reduction requires $|X|$ to be exponential in the original problem size, we are able to achieve a $(1-1/e-o(1))$-approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.

/pdf/maximizing-a-monotone-submodular-function-subject-to-a-1ag1ags3m8.pdf

Maximizing a Monotone Submodular Function Subject to a Matroid Constraint

How does a search engine company decide what ads to display with each query so as to maximize its revenueq This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a trade-off revealing LP and use it to derive an optimal algorithm achieving a competitive ratio of 1−1/e for this problem.

/pdf/adwords-and-generalized-online-matching-13dglrv5l6.pdf

AdWords and generalized online matching

We prove the existence of e-Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payoffs to all players in any (exact) Nash equilibrium can be e-approximated by the payoffs to the players in some such logarithmic support e-Nash equilibrium. These strategies are also uniform on a multiset of logarithmic size and therefore this leads to a quasi-polynomial algorithm for computing an e-Nash equilibrium. To our knowledge this is the first subexponential algorithm for finding an e-Nash equilibrium. Our results hold for any multiple-player game as long as the number of players is a constant (i.e., it is independent of the number of pure strategies). A similar argument also proves that for a fixed number of players m, the payoffs to all players in any m-tuple of mixed strategies can be e-approximated by the payoffs in some m-tuple of constant support strategies.We also prove that if the payoff matrices of a two person game have low rank then the game has an exact Nash equilibrium with small support. This implies that if the payoff matrices can be well approximated by low rank matrices, the game has an e-equilibrium with small support. It also implies that if the payoff matrices have constant rank we can compute an exact Nash equilibrium in polynomial time.

/pdf/playing-large-games-using-simple-strategies-1121pwnx0k.pdf

Playing large games using simple strategies

Matching is a classic problem with a rich history and a significant impact, both on the theory of algorithms and in practice. Recently there has been a surge of interest in the online version of matching and its generalizations, due to the important new application domain of Internet advertising. The theory of online matching and allocation has played a critical role in designing algorithms for ad allocation. This monograph surveys the key problems, models and algorithms from online matchings, as well as their implication in the practice of ad allocation. The goal is to provide a classification of the problems in this area, an introduction into the techniques used, a glimpse into the practical impact, and to provide direction in terms of open questions. Matching continues to find core applications in diverse domains, and the advent of massive online and streaming data emphasizes the future applicability of the algorithms and techniques surveyed here.

/pdf/online-matching-and-ad-allocation-2bagnfnhyu.pdf

Online Matching and Ad Allocation

How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competitive ratios of 1-1/e for this problem.

AdWords and generalized on-line matching

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 alforithm 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.

Aranyak Mehta

Papers

AdWords and generalized online matching

Playing large games using simple strategies

Online Matching and Ad Allocation

AdWords and generalized on-line matching

Online budgeted matching in random input models with applications to Adwords