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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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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.
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 14 (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 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
On Stackelberg pricing with computationally bounded consumers
 In Proc. 5th Intl. Workshop Internet & Network Economics (WINE
, 2009
"... In Stackelberg pricing a leader sets prices for items in order to maximize revenue from a follower purchasing a feasible subset of items. We consider computationally bounded followers who cannot optimize exactlyoverthe rangeofall feasible subsets, but who apply publicly known algorithms to determine ..."
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Cited by 3 (2 self)
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In Stackelberg pricing a leader sets prices for items in order to maximize revenue from a follower purchasing a feasible subset of items. We consider computationally bounded followers who cannot optimize exactlyoverthe rangeofall feasible subsets, but who apply publicly known algorithms to determine the items to purchase. This corresponds to general multidimensional pricing when customers cannot optimize their valuation functions efficiently but still aim to act rationally to the best of their ability. We consider two versions of this novel type of pricing problem. In the MinKnapsack variant items are weighted objects and the follower seeks to purchase a mincost selection of objects of some bounded weight. When he uses a greedy 2approximation algorithm, we provide a polynomialtime (2+ε)approximation algorithm for the leader’s revenue maximization problem based on socalled nearuniform price assignments. We also prove the problem to be strongly NPhard. In the SetCover variant items are subsets of some ground set which the follower seeks to cover. When he uses a standard primaldual approach, we prove that exact revenue maximization is possible in polynomial time when elements have frequency 2 (VertexCover variant). This stands in sharp contrast to APXhardness for the problem with elements of frequency 3. 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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Cited by 3 (0 self)
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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
Network interdiction – models, applications, unexplored directions
, 2010
"... Network interdiction is the monitoring or halting of an adversary’s activity on a network. Its models involve two players, usually called the interdictor and the evader (or, in the more general context of Stackelberg games, leader and follower). The evader operates on the network to optimize some ob ..."
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Network interdiction is the monitoring or halting of an adversary’s activity on a network. Its models involve two players, usually called the interdictor and the evader (or, in the more general context of Stackelberg games, leader and follower). The evader operates on the network to optimize some objective such as moving through the network as fast as possible (shortest path interdiction), or with as little probability of being detected as possible (most reliable path interdiction), or to maximize the amount of goods transported through the network (network flow interdiction). The interdictor has the capacity to change the structure or parameters of the network (remove vertices or edges, increase detection probabilities, or lower arc capacities) in order to minimize the evader’s objective function. The study of network interdiction began with military applications: disruption of the flow of enemy troops. More recent applications include infectious disease control, counterterrorism, interception of contraband and illegal items such as drugs, weapons, or nuclear material, and the monitoring of computer networks. A large variety of models have been proposed for different interdiction problems. These include combinatorial optimization, stochastic programming, and game theoretic approaches. In this note we attempt to collect the most researched models, match them with applications, and summarize the latest algorithmic and complexity results. In Section 1 we introduce the basic ideas and define the necessary terms. Section 2 is concerned with various models that have been proposed in the literature, as well as with algorithms and complexity bounds. In Section 3 we examine which models would be appropriate for which applications. Finally, in Section 4 we outline two promising research directions for the future. 1
Stackelberg Pricing is Hard to Approximate within 2 − ǫ
, 910
"... 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. 1
Profitmaximizing pricing for tollbooths
"... Abstract. The input to the tollbooth problem is a graph G = (V, E) and a set of m buyers Pi where each buyer is interested in buying a path Pi connecting si, ti ∈ V. Each buyer comes with a budget b(Pi), a positive real number. The problem is to set nonnegative prices to the edges E of G. A buyer b ..."
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Abstract. The input to the tollbooth problem is a graph G = (V, E) and a set of m buyers Pi where each buyer is interested in buying a path Pi connecting si, ti ∈ V. Each buyer comes with a budget b(Pi), a positive real number. The problem is to set nonnegative prices to the edges E of G. A buyer buys her path Pi if the sum of prices on Pi is at most her budget. If a buyer buys her path, we obtain a profit equal to the sum of prices on the chosen path, and otherwise we get nothing from the buyer. The objective of the problem is to find prices for the edges that maximizes the revenue obtained by selling the edges to the buyers who can afford their path. There is no restriction on the number of buyers that can simultaneously use an edge. In this paper, we consider the tollboth problem on a tree. Hence, each si, ti path is uniquely determined. A upward instance is when the tree is rooted at a vertex r, and ti is an ancestor of si for each buyer Pi, i = 1,..., m. We show that any solution to a naturally defined upward instance obtained from a tollbooth instance can be converted back to a solution to the orginial problem at a loss of at most a constant factor in the approximation. Using known results on the tollbooth problem for instances, we obtain improved algorithms for the tollbooth problem on trees. 1
Stackelberg Shortest Path Tree Game, Revisited ∗
, 2012
"... Let G(V, E) be a directed graph with n vertices and m edges. The edges E of G are divided into two types: EF and EP. Each edge of EF has a fixed price. The edges of EP are the priceable edges and their price is not fixed a priori. Let r be a vertex of G. For an assignment of prices to the edges of E ..."
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Let G(V, E) be a directed graph with n vertices and m edges. The edges E of G are divided into two types: EF and EP. Each edge of EF has a fixed price. The edges of EP are the priceable edges and their price is not fixed a priori. Let r be a vertex of G. For an assignment of prices to the edges of EP, the revenue is given by the following procedure: select a shortest path tree T from r with respect to the prices (a tree of cheapest paths); the revenue is the sum, over all priceable edges e, of the product of the price of e and the number of vertices below e in T. Assuming that k = EP  ≥ 2 is a constant, we provide a data structure whose construction takes O(m+n log k−1 n) time and with the property that, when we assign prices to the edges of EP, the revenue can be computed in (log k−1 n). Using our data structure, we save almost a linear factor when computing the optimal strategy in the Stackelberg shortest paths tree game of [D. Bilò and L. Gualà and G. Proietti and P. Widmayer. Computational aspects of a 2Player Stackelberg shortest paths tree game. Proc. WINE 2008]. 1