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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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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.
Dynamic and nonuniform pricing strategies for revenue maximization
 In Foundations of Computer Science (FOCS
, 2009
"... We consider the Item Pricing problem for revenue maximization in the limited supply setting, where a single seller with n items caters to m buyers with unknown subadditive valuation functions who arrive in a sequence. The seller sets the prices on individual items, and the price of a bundle of items ..."
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Cited by 15 (5 self)
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We consider the Item Pricing problem for revenue maximization in the limited supply setting, where a single seller with n items caters to m buyers with unknown subadditive valuation functions who arrive in a sequence. The seller sets the prices on individual items, and the price of a bundle of items is the sum of the prices of the individual items in the bundle. Each buyer buys a subset of yet unsold items so as to maximize her utility, defined as her valuation of the subset minus the price of the subset. Our goal is to design pricing strategies, possibly randomized, that guarantee an expected revenue that is within a small factor α of the maximum possible social welfare – an upper bound on the maximum revenue that can be generated by any pricing mechanism. Much of the earlier work has focused on the unlimited supply setting, where selling items to some buyer does not affect their availability to the future buyers. Recently, Balcan et. al. [4] studied the limited supply setting, giving a simple randomized algorithm that assigns a single randomly chosen price to all items (uniform pricing strategy) in the beginning, and never changes it (static pricing strategy). They showed that this strategy guarantees an 2 O( √ log n log log n)
The Stackelberg Minimum Spanning Tree Game
 In Proc. of 10th WADS
, 2007
"... Abstract. We consider a oneround twoplayer network pricing game, the Stackelberg Minimum Spanning Tree game or StackMST. The game is played on a graph (representing a network), whose edges are colored either red or blue, and where the red edges have a given fixed cost (representing the competitor’ ..."
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Abstract. We consider a oneround twoplayer network pricing game, the Stackelberg Minimum Spanning Tree game or StackMST. The game is played on a graph (representing a network), whose edges are colored either red or blue, and where the red edges have a given fixed cost (representing the competitor’s prices). The first player chooses an assignment of prices to the blue edges, and the second player then buys the cheapest possible minimum spanning tree, using any combination of red and blue edges. The goal of the first player is to maximize the total price of purchased blue edges. This game is the minimum spanning tree analog of the wellstudied Stackelberg shortestpath game. We analyze the complexity and approximability of the first player’s best strategy in StackMST. In particular, we prove that the problem is APXhard even if there are only two different red costs, and give an approximation algorithm whose approximation ratio is at most min{k, 3 + 2 ln b, 1 + ln W}, where k is the number of distinct red costs, b is the number of blue edges, and W is the maximum ratio between red costs. We also give a natural integer linear programming formulation of the problem, and show that the integrality gap of the fractional relaxation asymptotically matches the approximation guarantee of our algorithm. 1
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 7 (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
Stackelberg Network Pricing is Hard to Approximate
, 812
"... In the Stackelberg network pricing problem, one has to assign tariffs to a certain subset of the arcs of a given transportation network. The aim is to maximize the amount paid by the user of the network, knowing that the user will take a shortest stpath once the tariffs are fixed. Roch, Savard, and ..."
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In the Stackelberg network pricing problem, one has to assign tariffs to a certain subset of the arcs of a given transportation network. The aim is to maximize the amount paid by the user of the network, knowing that the user will take a shortest stpath once the tariffs are fixed. Roch, Savard, and Marcotte (Networks, Vol. 46(1), 57–67, 2005) proved that this problem is NPhard, and gave an O(log m)approximation algorithm, where m denote the number of arcs to be priced. In this note, we show that the problem is also APXhard. Keywords: Combinatorial optimization; APXhardness; Network pricing; Stackelberg games 1
Welfare and Profit Maximization with Production Costs
, 2011
"... Combinatorial Auctions are a central problem in Algorithmic Mechanism Design: pricing and allocating goods to buyers with complex preferences in order to maximize some desired objective (e.g., social welfare, revenue, or profit). The problem has been wellstudied in the case of limited supply (one c ..."
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Cited by 4 (0 self)
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Combinatorial Auctions are a central problem in Algorithmic Mechanism Design: pricing and allocating goods to buyers with complex preferences in order to maximize some desired objective (e.g., social welfare, revenue, or profit). The problem has been wellstudied in the case of limited supply (one copy of each item), and in the case of digital goods (the seller can produce additional copies at no cost). Yet in the case of resources—oil, labor, computing cycles, etc.—neither of these abstractions is just right: additional supplies of these resources can be found, but at increasing difficulty (marginal cost) as resources are depleted. In this work, we initiate the study of the algorithmic mechanism design problem of combinatorial pricing under increasing marginal cost. The goal is to sell these goods to buyers with unknown and arbitrary combinatorial valuation functions to maximize either the social welfare, or the seller’s profit; specifically we focus on the setting of posted item prices with buyers arriving online. We give algorithms that achieve constant factor approximations for a class of natural cost functions—linear, lowdegree polynomial, logarithmic—and that give logarithmic approximations for arbitrary increasing marginal cost functions (along with a necessary additive loss). We show that these bounds are essentially best possible for these settings.
Sequential Item Pricing for Unlimited Supply
 In Proceedings of the Workshop on Internet and Network Economics
, 2010
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Bargaining and Pricing in Networked Economic Systems
, 2011
"... Economic systems can often be modeled as games involving several agents or players who act according to their own individual interests. Our goal is to understand how various features of an economic system affect its outcomes, and what may be the best strategy for an individual agent. In this work, w ..."
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Economic systems can often be modeled as games involving several agents or players who act according to their own individual interests. Our goal is to understand how various features of an economic system affect its outcomes, and what may be the best strategy for an individual agent. In this work, we model an economic system as a combination of many bilateral economic opportunities, such as that between a buyer and a seller. The transactions are complicated by the existence of many economic opportunities, and the influence they have on each other. For example, there may be several prospective sellers and buyers for the same item, with possibly differing costs and values. Such a system may be modeled by a network, where the nodes represent players and the edges represent opportunities. We study the effect of network structure on the outcome of bargaining among players, through theoretical
Stackelberg Pricing is Hard to Approximate within 2 − epsilon
, 2009
"... Stackelberg Pricing Games is a twolevel combinatorial pricing problem studied in the Economics, Operation Research, and Computer Science communities. In this paper, we consider the decadeold shortest path version of this problem which is the first and most studied problem in this family. The game ..."
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Stackelberg Pricing Games is a twolevel combinatorial pricing problem studied in the Economics, Operation Research, and Computer Science communities. In this paper, we consider the decadeold shortest path version of this problem which is the first and most studied problem in this family. The game is played on a graph (representing a network) consisting of fixed cost edges and pricable or variable cost edges. The fixed cost edges already have some fixed price (representing the competitor’s prices). Our task is to choose prices for the variable cost edges. After that, a client will buy the cheapest path from a node s to a node t, using any combination of fixed cost and variable cost edges. The goal is to maximize the revenue on variable cost edges. In this paper, we show that the problem is hard to approximate within 2 − ǫ, improving the previous APXhardness result by Joret [to appear in Networks]. Our technique combines the existing ideas with a new insight into the price structure and its relation to the hardness of the instances.