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76
Algorithmic pricing via virtual valuations
 In Proc. of 8th EC
, 2007
"... Algorithmic pricing is the computational problem that sellers (e.g., in supermarkets) face when trying to set prices for their items to maximize their profit in the presence of a known demand. Guruswami et al. [9] propose this problem and give logarithmic approximations (in the number of consumers) ..."
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Cited by 56 (5 self)
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Algorithmic pricing is the computational problem that sellers (e.g., in supermarkets) face when trying to set prices for their items to maximize their profit in the presence of a known demand. Guruswami et al. [9] propose this problem and give logarithmic approximations (in the number of consumers) for the unitdemand and singleparameter cases where there is a specific set of consumers and their valuations for bundles are known precisely. Subsequently several versions of the problem have been shown to have polylogarithmic inapproximability. This problem has direct ties to the important open question of better understanding the Bayesian optimal mechanism in multiparameter agent settings; however, for this purpose approximation factors logarithmic in the number of agents are inadequate. It is therefore of vital interest to consider special cases where constant approximations are possible. We consider the unitdemand variant of this pricing problem. Here a consumer has a valuation for each different item and their value for a set of items is simply the maximum value they have for any item in the set. Instead of considering a set of consumers with precisely known preferences, like the prior algorithmic pricing literature, we assume that the preferences of the consumers are drawn from a distribution. This is the standard assumption in economics; furthermore, the setting of a specific set of customers with specific preferences, which is employed in all of the prior work in algorithmic pricing, is a special case of this general Bayesian pricing problem, where there is a discrete Bayesian distribution for preferences specified by picking one consumer uniformly from the given set of consumers. Notice that the distribution over the valuations for the individual items that this generates is obviously correlated. Our work complements these existing works by considering the case where the consumer’s valuations for the different items are independent random variables. Our main
Item Pricing for Revenue Maximization
"... We consider the problem of pricing n items to maximize revenue when faced with a series of unknown buyers with complex preferences, and show that a simple pricing scheme achieves surprisingly strong guarantees. We show that in the unlimited supply setting, a random single price achieves expected rev ..."
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Cited by 41 (4 self)
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We consider the problem of pricing n items to maximize revenue when faced with a series of unknown buyers with complex preferences, and show that a simple pricing scheme achieves surprisingly strong guarantees. We show that in the unlimited supply setting, a random single price achieves expected revenue within a logarithmic factor of the total social welfare for customers with general valuation functions, which may not even necessarily be monotone. This generalizes work of Guruswami et. al [18], who show a logarithmic factor for only the special cases of singleminded and unitdemand customers. In the limited supply setting, we show that for subadditive valuations, a random single price achieves revenue within a factor of 2 O( √ log n log log n) of the total social welfare, i.e., the optimal revenue the seller could hope to extract even if the seller could price each bundle differently for every buyer. This is the best approximation known for any item pricing scheme for subadditive (or even submodular) valuations, even using multiple prices. We complement this result with a lower bound showing a sequence of subadditive (in fact, XOS) buyers for which any single price has approximation ratio 2 Ω(log1/4 n), thus showing that single price schemes cannot achieve a polylogarithmic ratio. This lower bound demonstrates a clear distinction between revenue maximization and social welfare maximization in this setting, for which [12, 10] show that a fixed price achieves a logarithmic approximation in the case of XOS [12], and more generally subadditive [10], customers.
Uniform budgets and the envyfree pricing problem
 In Proceedings of the 35th International Colloquium on Automata, Languages and Programming
, 2008
"... We consider the unitdemand minbuying pricing problem, in which we want to compute revenue maximizing prices for a set of products P assuming that each consumer from a set of consumer samples C will purchase her cheapest affordable product once prices are fixed. We focus on the special uniformbudg ..."
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Cited by 31 (5 self)
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We consider the unitdemand minbuying pricing problem, in which we want to compute revenue maximizing prices for a set of products P assuming that each consumer from a set of consumer samples C will purchase her cheapest affordable product once prices are fixed. We focus on the special uniformbudget case, in which every consumer has only a single nonzero budget for some set of products. This constitutes a special case also of the unitdemand envyfree pricing problem. We show that, assuming specific hardness of the balanced bipartite independent set problem in constant degree graphs or hardness of refuting random 3CNF formulas, the unitdemand minbuying pricing problem with uniform budgets cannot be approximated in polynomial time within O(log ε C) for some ε> 0. This is the first result giving evidence that unitdemand envyfree pricing, as well, might be hard to approximate essentially better than within the known logarithmic ratio. We then introduce a slightly more general problem definition in which consumers are given as an explicit probability distribution and show that in this case the envyfree pricing problem can be shown to be inapproximable within O(P  ε) assuming NP � T δ>0 BPTIME(2O(nδ)). Finally, we briefly argue that all the results apply to the important setting of pricing with singleminded consumers as well. 1
A quasiPTAS for profitmaximizing pricing on line graphs
 In Proc. 15th Ann. European Symp. on Algorithms (ESA’07), Lecture Notes in Computer Science
, 2007
"... Abstract. We consider the problem of pricing items so as to maximize the profit made from selling these items. An instance is given by a set E of n items and a set of m clients, where each client is specified by one subset of E (the bundle of items she/he wants to buy), and a budget (valuation), whi ..."
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Cited by 30 (6 self)
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Abstract. We consider the problem of pricing items so as to maximize the profit made from selling these items. An instance is given by a set E of n items and a set of m clients, where each client is specified by one subset of E (the bundle of items she/he wants to buy), and a budget (valuation), which is the maximum price she/he is willing to pay for that subset. We restrict our attention to the model where the subsets can be arranged such that they form intervals of a line graph. Assuming an unlimited supply of any item, this problem is known as the highway problem and so far only an O(log n)approximation algorithm is known. We show that a PTAS is likely to exist by presenting a quasipolynomial time approximation scheme. We also combine our ideas with a recently developed quasiPTAS for the unsplittable flow problem on line graphs to extend this approximation scheme to the limited supply version of the pricing problem. 1
Playing games with approximation algorithms
 In Proceedings of the 39 th annual ACM Symposium on Theory of Computing
, 2007
"... 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 fu ..."
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Cited by 27 (2 self)
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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 fullinformation 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 decisionmaking problems with weights fit naturally into this framework, such as online shortestpath, 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 fullinformation 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 αapproximation 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 bestresponses, rather than bestresponses. 1. Introduction. In the 1950’s
Sequential posted pricing and multiparameter mechanism design
 Proc. of 42 nd ACM STOC
"... We consider the classical mathematical economics problem of Bayesian optimal mechanism design where a principal aims to optimize a given objective when allocating resources to selfinterested agents. In singleparameter settings (where each agent preference is given by a private value for being allo ..."
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Cited by 22 (6 self)
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We consider the classical mathematical economics problem of Bayesian optimal mechanism design where a principal aims to optimize a given objective when allocating resources to selfinterested agents. In singleparameter settings (where each agent preference is given by a private value for being allocated the resource and zero for not being allocated) this problem is solved [19]. While this economic solution is tractable whenever the noneconomic optimization problem is tractable, it is complicated enough that it is rarely employed. Moreover, the techniques do not seem to generalize to multiparameter settings. Our first result is that for general product distributions on agent preferences and resource allocation problems that satisfy matroid properties (e.g., multiunit auctions, matchings, spanning trees), sequential posted price mechanisms, where agents are approached inturn and offered a precomputed takeitorleaveit offer, are at most a 4approximation to the optimal singleround mechanism. Furthermore, a suitable sequence of prices can be effectively computed by sampling the agents ’ distributional preferences. Notably, the analysis of this sequential posted price mechanism can be extended to give approximation mechanisms for the unsolved multiparameter setting. In stark contrast to the singleparameter setting, in multiparameter settings there is no general description or tractable implementation of optimal mechanisms. For decades, this unanswered issue has been widely considered one of the most important in the economic theory on mechanism design. We focus on
Buying Cheap is Expensive: Hardness of NonParametric MultiProduct Pricing
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 68
, 2006
"... We investigate nonparametric unitdemand pricing problems, in which the goal is to find revenue maximizing prices for products P based on a set of consumer profiles C obtained, e.g., from an eCommerce website. A consumer profile consists of a number of nonzero budgets and a ranking of all the pro ..."
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Cited by 21 (6 self)
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We investigate nonparametric unitdemand pricing problems, in which the goal is to find revenue maximizing prices for products P based on a set of consumer profiles C obtained, e.g., from an eCommerce website. A consumer profile consists of a number of nonzero budgets and a ranking of all the products the consumer is interested in. Once prices are fixed, each consumer chooses to buy one of the products she can afford based on some predefined selection rule. We distinguish between the minbuying, maxbuying, and rankbuying models. For the minbuying and general rankbuying models the best known approximation ratio is O(log C) and, previously, the problem was only known to be APXhard. We obtain the first (near) tight lower bound showing that the problem is not approximable within O(log ε C) for some ε> 0, unless NP ⊆ DTIME(n loglog n). Going to slightly stronger (still reasonable) complexity theoretic assumptions we prove inapproximability within O(ℓ ε) (ℓ being an upper bound on the number of nonzero budgets per consumer) and O(P  ε) and provide matching upper bounds. Surprisingly, these hardness results hold even if a price ladder constraint, i.e., a predefined total order on the prices of all products, is given. This changes if we require that in the rankbuying model consumers’ budgets are consistent with their
Approximation algorithms for singleminded envyfree profitmaximization problems with limited supply
 FOCS
"... We present the first polynomialtime approximation algorithms for singleminded envyfree profitmaximization problems [13] with limited supply. Our algorithms return a pricing scheme and a subset of customers that are designated the winners, which satisfy the envyfreeness constraint, whereas in ou ..."
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Cited by 20 (0 self)
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We present the first polynomialtime approximation algorithms for singleminded envyfree profitmaximization problems [13] with limited supply. Our algorithms return a pricing scheme and a subset of customers that are designated the winners, which satisfy the envyfreeness constraint, whereas in our analyses, we compare the profit of our solution against the optimal value of the corresponding socialwelfaremaximization (SWM) problem of finding a winnerset with maximum total value. Our algorithms take any LPbased αapproximation algorithm for the corresponding SWM problem as input and return a solution that achieves profit at least OPT /O(α · log umax), where OPT is the optimal value of the SWM problem, and umax is the maximum supply of an item. This immediately yields approximation guarantees of O ( √ m log umax) for the general singleminded envyfree problem; and O(log umax) for the tollbooth and highway problems [13], and the graphvertex pricing problem [3] (α = O(1) for all the corresponding SWM problems). Since OPT is an upper bound on the maximum profit achievable by any solution (i.e., irrespective of whether the solution satisfies the envyfreeness constraint), our results directly carry over to the nonenvyfree versions of these problems too. Our result also thus (constructively) establishes an upper bound of O(α · log umax) on the ratio of (i) the optimum value of the profitmaximization problem and OPT; and (ii) the optimum profit achievable with and without the constraint of envyfreeness. 1.
Reducing Mechanism Design to Algorithm Design via Machine Learning
 IN THE PROCEEDINGS OF THE 46TH ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS) 2005, UNDER THE TITLE “MECHANISM DESIGN VIA MACHINE LEARNING
, 2007
"... We use techniques from samplecomplexity in machine learning to reduce problems of incentivecompatible mechanism design to standard algorithmic questions, for a broad class of revenuemaximizing pricing problems. Our reductions imply that for these problems, given an optimal (or βapproximation) al ..."
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Cited by 19 (4 self)
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We use techniques from samplecomplexity in machine learning to reduce problems of incentivecompatible mechanism design to standard algorithmic questions, for a broad class of revenuemaximizing pricing problems. Our reductions imply that for these problems, given an optimal (or βapproximation) algorithm for an algorithmic pricing problem, we can convert it into a (1 + ɛ)approximation (or β(1 + ɛ)approximation) for the incentivecompatible mechanism design problem, so long as the number of bidders is sufficiently large as a function of an appropriate measure of complexity of the class of allowable pricings. We apply these results to the problem of auctioning a digital good, to the attribute auction problem which includes a wide variety of discriminatory pricing problems, and to the problem of itempricing in unlimitedsupply combinatorial auctions. From a machine learning perspective, these settings present several challenges: in particular, the “loss function” is discontinuous, is asymmetric, and has a large range. We address these issues in part by introducing a new form of coveringnumber bound that is especially wellsuited to these problems and may be of independent interest.
Optimal envyfree pricing with metric substitutability
 In ACM Conference on Electronic Commerce
"... We study the envyfree pricing problem faced by a profit maximizing seller when there is metric substitutability among the items — consumer i’s value for item j is vi − ci,j, and the substitution costs, {ci,j}, form a metric. Our model is motivated from the observation that sellers often sell the sa ..."
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Cited by 18 (2 self)
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We study the envyfree pricing problem faced by a profit maximizing seller when there is metric substitutability among the items — consumer i’s value for item j is vi − ci,j, and the substitution costs, {ci,j}, form a metric. Our model is motivated from the observation that sellers often sell the same product at different prices in different locations, and rational consumers optimize the tradeoff between prices and substitution costs. While the general envyfree pricing problem is hard to approximate, the addition of metric substitutability constraints allows us to solve the problem exactly in polynomial time by reducing it to an instance of weighted independent set on a perfect graph. When the substitution costs do not form a metric, even in cases when a (1 + ǫ)approximate triangle inequality holds, the problem becomes NPhard. Our results show that triangle inequality is the exact sharp threshold for the problem of going from “tractable ” to “hard”. We then turn our attention to the multiunit demand case, where consumers request multiple copies of the item. This problem has an interesting paradoxical nonmonotonicity: The optimal revenue the seller can extract can actually decrease when consumers ’ demands increase. We show that in this case the revenue maximization problem becomes APXhard and give an O(log D) approximation algorithm, where D is the ratio of the largest to smallest demand. We extend these techniques to the more general case of arbitrary nondecreasing value functions, and give an O(log 3 D) approximation algorithm.