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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.
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 12 (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.
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
Specializations and generalizations of the Stackelberg minimum spanning tree game
 In Proc. of the 6th Workshop on Internet and Network Economics (WINE), LNCS 6484
, 2010
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On bilevel machine scheduling problems
"... Bilevel optimization is concerned with twolevel optimization problems, where there is a top level decision maker or leader, and there is one (or more) bottom level decision maker or follower. Each decision maker optimizes its own objective function and is affected by the actions of the other. The f ..."
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Cited by 2 (2 self)
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Bilevel optimization is concerned with twolevel optimization problems, where there is a top level decision maker or leader, and there is one (or more) bottom level decision maker or follower. Each decision maker optimizes its own objective function and is affected by the actions of the other. The follower makes its decisions after, and in view of, the decisions of the leader. For an
Graph Pricing Problem on Bounded Treewidth, Bounded Genus and kPartite Graphs
, 2013
"... Consider the following problem. A seller has infinite copies of n products represented by nodes in a graph. There are m consumers, each has a budget and wants to buy two products. Consumers are represented by weighted edges. Given the prices of products, each consumer will buy both products she wa ..."
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Consider the following problem. A seller has infinite copies of n products represented by nodes in a graph. There are m consumers, each has a budget and wants to buy two products. Consumers are represented by weighted edges. Given the prices of products, each consumer will buy both products she wants, at the given price, if she can afford to. Our objective is to help the seller price the products to maximize her profit. This problem is called graph vertex pricing (GVP) problem and has resisted several recent attempts despite its current simple solution. This motivates the study of this problem on special classes of graphs. In this paper, we study this problem on a large class of graphs such as graphs with bounded treewidth, bounded genus and kpartite graphs. We show that there exists an FPTAS for GVP on graphs with bounded treewidth. This result is also extended to an FPTAS for the more general singleminded pricing problem. On bounded genus graphs we present a PTAS and show that GVP is NPhard even on planar graphs.
On the Stackelberg fuel pricing problem
"... We consider the Stackelberg fuel pricing problem in which a company has to decide the fuel selling price at each of its gas stations in order to maximize its revenue, assuming that the selling prices of the competitors and the customers ’ preferences are known in advance. We show that, even in the ..."
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We consider the Stackelberg fuel pricing problem in which a company has to decide the fuel selling price at each of its gas stations in order to maximize its revenue, assuming that the selling prices of the competitors and the customers ’ preferences are known in advance. We show that, even in the basic case in which the road network is modeled by an undirected planar graph and the competitors discriminate on two different selling prices only, the problem is APXhard. On the positive side, we design a polynomial time algorithm for instances in which the number of gas stations owned by the company is constant, while, in the general case, we show that the singleprice algorithm (which provides the bestknown solutions for essentially all the Stackelberg pricing problems studied in the literature up to date) achieves an approximation ratio which is logarithmic in some parameters of the input instance. This result, in particular, is tight and holds for a much more general class of Stackelberg network pricing problems.
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.