We present approximation and online algorithms for a number of problems of pricing items for sale so as to maximize seller's revenue in an unlimited supply setting. Our first result is an O(k)-approximation algorithm for pricing items to single-minded bidders who each want at most k items. This improves over recent independent work of Briest and Krysta [6] who achieve an O(k ) bound. For the case k = 2, where we obtain a 4-approximation, this can be viewed as the following graph vertex pricing problem: given a (multi) graph G with valuations w e on the edges, find prices p i 0 for the vertices to maximize (p i + p j ) .

In the multi-armed bandit problem, an online algorithm must choose from a set of strategies in a sequence of n trials so as to minimize the total cost of the chosen strategies. While nearly tight upper and lower bounds are known in the case when the strategy set is finite, much less is known when there is an infinite strategy set. Here we consider the case when the set of strategies is a subset of R d, and the cost functions are continuous. In the d = 1 case, we improve on the best-known upper and lower bounds, closing the gap to a sublogarithmic factor. We also consider the case where d> 1 and the cost functions are convex, adapting a recent online convex optimization algorithm of Zinkevich to the sparser feedback model of the multi-armed bandit problem. 1

Abstract We consider the online auction problem proposed byBar-Yossef, Hildrum, and Wu [4] in which an auctioneer is selling identical items to bidders arriving one at atime. We give an auction that achieves a constant factor of the optimal profit less an O(h) additive loss term,where h is the value of the highest bid. Furthermore,this auction does not require foreknowledge of the range of bidders ' valuations. On both counts, this answersopen questions from [4, 5]. We further improve on the results from [5] for the online posted-price problem by re-ducing their additive loss term from O(h log h log log h)to O(h log log h). Finally, we define the notion of an(offline) attribute auction for modeling the problem of auctioning items to consumers who are not a-priori in-distinguishable. We apply our online auction solution to achieve good bounds for the attribute auction problemwith 1-dimensional attributes.

Algorithms in varied fields use the idea of maintaining a distribution over a certain set and use the multiplicative update rule to iteratively change these weights. Their analysis are usually very similar and rely on an exponential potential function. We present a simple meta algorithm that unifies these disparate algorithms and drives them as simple instantiations of the meta algorithm. 1

Abstract There has been substantial work developing simple, efficient no-regret algorithms for a wideclass of repeated decision-making problems including online routing. These are adaptive strategies an individual can use that give strong guarantees on performance even in adversarially-changing environments. There has also been substantial work on analyzing properties of Nash equilibria in routing games. In this paper, we consider the question: if each player in a rout-ing game uses a no-regret strategy, will behavior converge to a Nash equilibrium? In general games the answer to this question is known to be no in a strong sense, but routing games havesubstantially more structure. In this paper we show that in the Wardrop setting of multicommodity flow and infinitesimalagents, behavior will approach Nash equilibrium (formally, on most days, the cost of the flow will be close to the cost of the cheapest paths possible given that flow) at a rate that dependspolynomially on the players ' regret bounds and the maximum slope of any latency function. We also show that price-of-anarchy results may be applied to these approximate equilibria, and alsoconsider the finite-size (non-infinitesimal) load-balancing model of Azar [2].

Abstract. A classical problem in query optimization is to find the optimal ordering of a set of possibly correlated selections. We provide an abstraction of this problem as a generalization of set cover called pipelined set cover, where the sets are applied sequentially to the elements to be covered and the elements covered at each stage are discarded. We show that several natural heuristics for this NP-hard problem, such as the greedy set-cover heuristic and a local-search heuristic, can be analyzed using a linear-programming framework. These heuristics lead to efficient algorithms for pipelined set cover that can be applied to order possibly correlated selections in conventional database systems as well as datastream processing systems. We use our linear-programming framework to show that the greedy and local-search algorithms are 4-approximations for pipelined set cover. We extend our analysis to minimize the lp-norm of the costs paid by the sets, where p ≥ 2 is an integer, to examine the improvement in performance when the total cost has increasing contribution from initial sets in the pipeline. Finally, we consider the online version of pipelined set cover and present a competitive algorithm with a logarithmic performance guarantee. Our analysis framework may be applicable to other problems in query optimization where it is important to account for correlations. 1

Abstract. We give an algorithm for learning a permutation on-line. The algorithm maintains its uncertainty about the target permutation as a doubly stochastic matrix. This matrix is updated by multiplying the current matrix entries by exponential factors. These factors destroy the doubly stochastic property of the matrix and an iterative procedure is needed to re-normalize the rows and columns. Even though the result of the normalization procedure does not have a closed form, we can still bound the additional loss of our algorithm over the loss of the best permutation chosen in hindsight. 1

In an online decision problem, an algorithm performs a sequence of trials, each of which involves selecting one element from a fixed set of alternatives (the “strategy set”) whose costs vary over time. After T trials, the combined cost of the algorithm’s choices is compared with that of the single strategy whose combined cost is minimum. Their difference is called regret, and one seeks algorithms which are efficient in that their regret is sublinear in T and polynomial in the problem size. We study an important class of online decision problems called generalized multiarmed bandit problems. In the past such problems have found applications in areas as diverse as statistics, computer science, economic theory, and medical decision-making. Most existing algorithms were efficient only in the case of a small (i.e. polynomialsized) strategy set. We extend the theory by supplying non-trivial algorithms and lower bounds for cases in which the strategy set is much larger (exponential or infinite) and

Abstract. In an online linear optimization problem, on each period t, an online algorithm chooses st ∈ S from a fixed (possibly infinite) set S of feasible decisions. Nature (who may be adversarial) chooses a weight vector wt ∈ R n, and the algorithm incurs cost c(st, wt), where c is a fixed cost function that is linear in the weight vector. In the full-information setting, the vector wt is then revealed to the algorithm, and in the bandit setting, only the cost experienced, c(st, wt), is revealed. The goal of the online algorithm is to perform nearly as well as the best fixed s ∈ S in hindsight. Many repeated decision-making problems with weights fit naturally into this framework, such as online shortest-path, online TSP, online clustering, and online weighted set cover. Previously, it was shown how to convert any efficient exact offline optimization algorithm for such a problem into an efficient online algorithm in both the full-information and the bandit settings, with average cost nearly as good as that of the best fixed s ∈ S in hindsight. However, in the case where the offline algorithm is an approximation algorithm with ratio α> 1, the previous approach only worked for special types of approximation algorithms. We show how to convert any offline approximation algorithm for a linear optimization problem into a corresponding online approximation algorithm, with a polynomial blowup in runtime. If the offline algorithm has an α-approximation guarantee, then the expected cost of the online algorithm on any sequence is not much larger than α times that of the best s ∈ S, where the best is chosen with the benefit of hindsight. Our main innovation is combining Zinkevich’s algorithm for convex optimization with a geometric transformation that can be applied to any approximation algorithm. Standard techniques generalize the above result to the bandit setting, except that a “Barycentric Spanner ” for the problem is also (provably) necessary as input. Our algorithm can also be viewed as a method for playing large repeated games, where one can only compute approximate best-responses, rather than best-responses. 1. Introduction. In the 1950’s

We explore the striking mathematical connections that exist between market scoring rules, cost function based prediction markets, and no-regret learning. We first show that any cost function based prediction market can be interpreted as an algorithm for the commonly studied problem of learning from expert advice by equating the set of outcomes on which bets are placed in the market with the set of experts in the learning setting, and equating trades made in the market with losses observed by the learning algorithm. If the loss of the market organizer is bounded, this bound can be used to derive an O ( √ T) regret bound for the corresponding learning algorithm. We then show that the class of markets with convex cost functions exactly corresponds to the class of Follow the Regularized Leader learning algorithms, with the choice of a cost function in the market corresponding to the choice of a regularizer in the learning problem. Finally, we show an equivalence between market scoring rules and prediction markets with convex cost functions. This implies both that any market scoring rule can be implemented as a cost function based market maker, and that market scoring rules can be interpreted naturally as Follow the Regularized Leader algorithms. These connections provide new insight into how it is that commonly studied markets, such as the Logarithmic Market Scoring Rule, can aggregate opinions into accurate estimates of the likelihood of future events.