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

We present an auction-based algorithm for computing market equilibrium prices in a production model, in which producers have a single linear production constraint, and consumers have linear utility functions. We provide algorithms for both the Fisher and Arrow-Debreu versions of the problem.

An auction-based market equilibrium algorithm for a production model

A classic theorem by Vizing proves that if the maximum degree of a graph is Δ then it is possible to color its edges, in polynomial time, using at most Δ+1 colors. However, this algorithm is offline, i.e., it assumes the whole graph is known in advance. A natural question then is how well we can do in the online setting, where the edges of the graph are revealed one by one, and we need to color each edge as soon as it is added to the graph. Online edge coloring has an important application in fast switch scheduling. Here, a natural model is that edges arrive online, but in a random permutation. Even in the random permutations model, the best analysis for any algorithm is factor 2, which comes from the simple greedy algorithm (which is factor 2 even in the worst case online model). The algorithm of Aggarwal et al. [1] provides a 1+o(1) factor algorithm, but for the case of multigraphs, when Δ = ω(n2), where n is the number of vertices. In this paper, we show that for graphs with Δ = ω(log n), it is possible to color the graph with 1.43Delta; + o(Δ) colors in the online random order model. Our algorithm is inspired by a 1.6 factor distributed offline algorithm of Panconesi and Srinivasan [9], which we extend by reusing colors online in multiple rounds.

http://static.googleusercontent.com/media/research.google.com/en/us/pubs/archive/36741.pdf

A 1.43-competitive online graph edge coloring algorithm in the random order arrival model

We consider the fundamental problem of selecting $k$ out of $n$ random variables in a way that the expected highest or second-highest value is maximized. This question captures several applications where we have uncertainty about the quality of candidates (e.g. auction bids, search results) and have the capacity to explore only a small subset due to an exogenous constraint. For example, consider a second price auction where system constraints (e.g., costly retrieval or model computation) allow the participation of only $k$ out of $n$ bidders, and the goal is to optimize the expected efficiency (highest bid) or expected revenue (second highest bid). 
We study the case where we are given an explicit description of each random variable. We give a PTAS for the problem of maximizing the expected highest value. For the second-highest value, we prove a hardness result: assuming the Planted Clique Hypothesis, there is no constant factor approximation algorithm that runs in polynomial time. Surprisingly, under the assumption that each random variable has monotone hazard rate (MHR), a simple score-based algorithm, namely picking the $k$ random variables with the largest $1/\sqrt{k}$ top quantile value, is a constant approximation to the expected highest and second highest value, \emph{simultaneously}.

/pdf/hitting-the-high-notes-subset-selection-for-maximizing-owb2t90tn4.pdf

Hitting the High Notes: Subset Selection for Maximizing Expected Order Statistics

How should a seller price his goods in a market where each buyer prefers a single good among his desired goods, and will buy the cheapest such good, as long as it is within his budget? We provide efficient algorithms that compute near-optimal prices for this problem, focusing on a commodity market, where the range of buyer budgets is small. We also show that our technique (which is based on LP-rounding) easily extends to a different scenario, in which the buyers want to buy all the desired goods, as long as they are within budget.

/pdf/pricing-commodities-or-how-to-sell-when-buyers-have-164661fe19.pdf

Pricing commodities, or how to sell when buyers have restricted valuations

The connection between games and no-regret algorithms has been widely studied in the literature. A fundamental result is that when all players play no-regret strategies, this produces a sequence of actions whose time-average is a coarse-correlated equilibrium of the game. However, much less is known about equilibrium selection in the case that multiple equilibria exist. 
In this work, we study the convergence of no-regret bidding algorithms in auctions. Besides being of theoretical interest, bidding dynamics in auctions is an important question from a practical viewpoint as well. We study repeated game between bidders in which a single item is sold at each time step and the bidder's value is drawn from an unknown distribution. We show that if the bidders use any mean-based learning rule then the bidders converge with high probability to the truthful pure Nash Equilibrium in a second price auction, in VCG auction in the multi-slot setting and to the Bayesian Nash equilibrium in a first price auction. We note mean-based algorithms cover a wide variety of known no-regret algorithms such as Exp3, UCB, $\epsilon$-Greedy etc. Also, we analyze the convergence of the individual iterates produced by such learning algorithms, as opposed to the time-average of the sequence. Our experiments corroborate our theoretical findings and also find a similar convergence when we use other strategies such as Deep Q-Learning.

Aranyak Mehta

Papers

An auction-based market equilibrium algorithm for a production model

A 1.43-competitive online graph edge coloring algorithm in the random order arrival model

Hitting the High Notes: Subset Selection for Maximizing Expected Order Statistics

Pricing commodities, or how to sell when buyers have restricted valuations

Convergence Analysis of No-Regret Bidding Algorithms in Repeated Auctions