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22
Approximation Algorithms and Online Mechanisms for Item Pricing
, 2007
"... Abstract: We present approximation and online algorithms for problems of pricing a collection of items for sale so as to maximize the 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 ..."
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Cited by 65 (10 self)
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Abstract: We present approximation and online algorithms for problems of pricing a collection of items for sale so as to maximize the 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 work of Briest and Krysta (2006) who achieve an O(k2) 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 wi j on the edges, find prices pi ≥ 0 for the vertices to maximize {(i, j):wi j≥pi+p j} (pi + p j). We also improve the approximation of Guruswami et al. (2005) for the “highway problem” in which all desired subsets are intervals on a line, from O(logm+ logn) to O(logn), where m is the number of bidders and n is the number of items. Our approximation algorithms can
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 32 (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
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 23 (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
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.
A theory of lossleaders: Making money by pricing below cost
 In Proc. 3rd International Workshop on Internet and Network Economics. Lecture Notes in Computer Science
, 2007
"... We consider the problem of assigning prices to goods of fixed marginal cost in such a way as to maximize revenue in the presence of singleminded customers. We focus in particular on the question of how pricing certain items below their marginal costs can lead to an improvement in overall profit, ev ..."
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Cited by 12 (3 self)
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We consider the problem of assigning prices to goods of fixed marginal cost in such a way as to maximize revenue in the presence of singleminded customers. We focus in particular on the question of how pricing certain items below their marginal costs can lead to an improvement in overall profit, even when customers behave in a fully rational manner. We develop two frameworks for analyzing this issue that we call the discount and coupon models, and examine both fundamental “profitability gaps ” (to what extent can pricing below cost help to improve profit) as well as algorithms for pricing in these models in a number of settings. To design our algorithms, we use several tools including a particular DAG representation and graph decomposition techniques which are of independent interest.
A sublogarithmic approximation for highway and tollbooth pricing
 In Proceedings of the 37th International Colloquium on Automata, Languages and Programming
, 2010
"... An instance of the tollbooth problem consists of an undirected network and a collection of singleminded customers, each of which is interested in purchasing a fixed path subject to an individual budget constraint. The objective is to assign a perunit price to each edge in a way that maximizes the c ..."
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Cited by 10 (2 self)
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An instance of the tollbooth problem consists of an undirected network and a collection of singleminded customers, each of which is interested in purchasing a fixed path subject to an individual budget constraint. The objective is to assign a perunit price to each edge in a way that maximizes the collective revenue obtained from all customers. The revenue generated by any customer is equal to the overall price of the edges in her desired path, when this cost falls within her budget; otherwise, that customer will not purchase any edge. Our main result is a deterministic algorithm for the tollbooth problem on trees whose approximation ratio is O(log m / log log m), where m denotes the number of edges in the underlying graph. This finding improves on the currently best performance guarantees for trees, due to Elbassioni et al. (SAGT ’09), as well as for paths (commonly known as the highway problem), due to Balcan and Blum (EC ’06). An additional interesting consequence is a computational separation between tollbooth pricing on trees and the original prototype problem of singleminded unlimited supply pricing, under a plausible hardness hypothesis due to Demaine et al. (SODA ’06).
Extremevalue theorems for optimal multidimensional pricing
, 2011
"... We provide a Polynomial Time Approximation Scheme for the multidimensional unitdemand pricing problem, when the buyer’s values are independent (but not necessarily identically distributed.) For all ɛ> 0, we obtain a (1 + ɛ)factor approximation to the optimal revenue in time polynomial, when th ..."
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Cited by 7 (2 self)
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We provide a Polynomial Time Approximation Scheme for the multidimensional unitdemand pricing problem, when the buyer’s values are independent (but not necessarily identically distributed.) For all ɛ> 0, we obtain a (1 + ɛ)factor approximation to the optimal revenue in time polynomial, when the values are sampled from Monotone Hazard Rate (MHR) distributions, quasipolynomial, when sampled from regular distributions, and polynomial in n poly(log r) , when sampled from general distributions supported on a set [umin, rumin]. We also provide an additive PTAS for all bounded distributions. Our algorithms are based on novel extreme value theorems for MHR and regular distributions, and apply probabilistic techniques to understand the statistical properties of revenue distributions, as well as to reduce the size of the search space of the algorithm. As a byproduct of our techniques, we establish structural properties of optimal solutions. We show that, for all ɛ> 0, g(1/ɛ) distinct prices suffice to obtain a (1+ɛ)factor approximation to the optimal revenue for MHR distributions, where g(1/ɛ) is a quasilinear function of 1/ɛ that does not depend on the number of items. Similarly, for all ɛ> 0 and n> 0, g(1/ɛ · log n) distinct prices suffice for regular distributions,
Improved hardness of approximation for Stackelberg shortestpath pricing
, 2009
"... We consider the Stackelberg shortestpath pricing problem, which is defined as follows. Given a graph G with fixedcost and pricable edges and two distinct vertices s and t, we may assign prices to the pricable edges. Based on the predefined fixed costs and our prices, a customer purchases a cheapes ..."
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Cited by 5 (2 self)
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We consider the Stackelberg shortestpath pricing problem, which is defined as follows. Given a graph G with fixedcost and pricable edges and two distinct vertices s and t, we may assign prices to the pricable edges. Based on the predefined fixed costs and our prices, a customer purchases a cheapest stpath in G and we receive payment equal to the sum of prices of pricable edges belonging to the path. Our goal is to find prices maximizing the payment received from the customer. While Stackelberg shortestpath pricing was known to be APXhard before, we provide the first explicit approximation threshold and prove hardness of approximation within 2 − o(1). 1
Pricing commodities, or how to sell when buyers have restricted valuations
 IN 5TH WORKSHOP ON APPROXIMATION AND ONLINE ALGORITHMS
, 2007
"... 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 nearoptimal prices for this problem, focusing on a commodity market, ..."
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Cited by 5 (2 self)
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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 nearoptimal 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 LProunding) easily extends to a different scenario, in which the buyers want to buy all the desired goods, as long as they are within budget.