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61
Approximation Algorithms and Online Mechanisms for Item Pricing
 IN ACM CONFERENCE ON ELECTRONIC COMMERCE
, 2005
"... 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 singleminded bidders who each want at most k items. This ..."
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Cited by 60 (9 self)
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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 singleminded 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 4approximation, 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 ) .
Mechanism Design via Machine Learning
 IN PROC. OF THE 46TH IEEE SYMP. ON FOUNDATIONS OF COMPUTER SCIENCE
, 2005
"... 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 #approximation) al ..."
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Cited by 46 (10 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 #approximation) algorithm for the standard algorithmic problem, we can convert it into a (1 + #)approximation (or #(1 + #)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 comparison class of solutions. 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 and asymmetric, and the range of bidders' valuations may be large.
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 31 (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 29 (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 22 (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
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 19 (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 16 (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.
Stackelberg Network Pricing Games
, 2008
"... We study a multiplayer oneround game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of m priceable edges in a graph. The other edges have a fixed cost. Based on the leader’s decision one or more followers optimize a polynomialtime solvable combinatorial min ..."
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Cited by 13 (2 self)
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We study a multiplayer oneround game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of m priceable edges in a graph. The other edges have a fixed cost. Based on the leader’s decision one or more followers optimize a polynomialtime solvable combinatorial minimization problem and choose a minimum cost solution satisfying their requirements based on the fixed costs and the leader’s prices. The leader receives as revenue the total amount of prices paid by the followers for priceable edges in their solutions, and the problem is to find revenue maximizing prices. Our model extends several known pricing problems, including singleminded and unitdemand pricing, as well as Stackelberg pricing for certain follower problems like shortest path or minimum spanning tree. Our first main result is a tight analysis of a singleprice algorithm for the single follower game, which provides a (1+ε) log mapproximation for any ε> 0. This can be extended to provide a (1+ε)(log k +log m)approximation for the general problem and k followers. The latter result is essentially best possible, as the problem is shown to be hard to approximate within O(log ε k+log ε m). If followers have demands, the singleprice algorithm provides a (1 + ε)m 2approximation, and the problem is hard to approximate within O(m ε) for some ε> 0. Our second main result is a polynomial time algorithm for revenue maximization in the special case of Stackelberg bipartite vertex cover, which is based on nontrivial maxflow and LPduality techniques. Our results can be extended to provide constantfactor approximations for any constant number of followers.
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 13 (1 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.
How to Sell a Graph: Guidelines for Graph Retailers
, 2006
"... We consider a profit maximization problem where we are asked to price a set of m items that are to be assigned to a set of n customers. The items can be represented as the edges of an undirected (multi)graph G, where an edge multiplicity larger than one corresponds to multiple copies of the same ite ..."
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Cited by 13 (3 self)
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We consider a profit maximization problem where we are asked to price a set of m items that are to be assigned to a set of n customers. The items can be represented as the edges of an undirected (multi)graph G, where an edge multiplicity larger than one corresponds to multiple copies of the same item. Each customer is interested in purchasing a bundle of edges of G, and we assume that each bundle forms a simple path in G. Each customer has a known budget for her respective bundle, and is interested only in that particular bundle. The goal is to determine item prices and a feasible assignment of items to customers in order to maximize the total profit. When the underlying graph G is a path, we derive a fully polynomial time approximation scheme, complementing a recent NPhardness result. If the underlying graph is a tree, and edge multiplicities are one, we show that the problem is polynomially solvable, contrasting its APXhardness for the case of unlimited availability of items. However, if the underlying graph is a grid, and edge multiplicities are one, we show that it is even NPcomplete to approximate the maximum profit to within a factor n 1−ε.