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An overview of Stackelberg pricing in networks. (English) Zbl 1146.91018
Summary: The Stackelberg pricing problem has two levels of decision making: tariff setting by an operator, and then selection of the cheapest alternative by customers. In the network version, an operator determines tariffs on a subset of the arcs that he owns. Customers, who wish to connect two vertices with a path of a certain capacity, select the cheapest path. The revenue for the operator is determined by the tariff and the amount of usage of his arcs. The most natural model for the problem is a (bilinear) bilevel program, where the upper level problem is the pricing problem of the operator, and the lower level problem is a shortest path problem for each of the customers.
This paper contains a compilation of theoretical and algorithmic results on the network Stackelberg pricing problem. The description of the theory and algorithms is generally informal and intuitive. We redefine the underlying network of the problem, to obtain a compact representation. Then we describe a basic branch-and-bound enumeration procedure. Both concepts are used for complexity issues and for the development of algorithms: establishing NP-hardness, approximability, special cases solvable in polynomial time, and an efficient exact branch-and-bound algorithm.

MSC:
91B24 Microeconomic theory (price theory and economic markets)
91B26 Auctions, bargaining, bidding and selling, and other market models
91A40 Other game-theoretic models
91A80 Applications of game theory
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[1] Basar, T.; Srikant, R., A Stackelberg network game with a large number of followers, Journal of optimization theory and applications, 115, 3, 479-490, (2002) · Zbl 1031.91016
[2] Bouhtou, M.; van Hoesel, S.; van der Kraaij, A.F.; Lutton, J.L., Linear tarification in multi-commodity telecommunications networks, INFORMS journal on computing, (2002)
[3] Brotcorne, L.; Labbé, M.; Marcotte, P.; Savard, G., A bilevel model and solution algorithm for a freight Tariff-setting problem, Transportation science, 34, 3, 289-302, (2000) · Zbl 0991.90573
[4] Brotcorne, L.; Labbé, M.; Marcotte, P.; Savard, G., A bilevel model for toll optimization on a multicommodity transportation network, Transportation science, 35, 4, 345-358, (2001) · Zbl 1069.90502
[5] Brotcorne, L., Labbé, M., Marcotte, P., Savard, G., 2005. Joint design and pricing on a network. Working paper.
[6] Dewez, S., 2004. On the toll setting problem. Ph.D. dissertation, Université Libre de Bruxelles.
[7] Fortz, B., Thorup, M., 2000. Internet traffic engineering by optimizing OSPF weights. In: Proceedings of the 19th IEEE Conference on Computer Communications (INFOCOM), pp. 519-528.
[8] Grigoriev, A., van Hoesel, S., van der Kraaij, A.F., Uetz, M., Bouhtou, M., in press. Pricing network edges to cross a river. Naval Research Logistics Quarterly.
[9] Jeroslow, R.G., The polynomial hierarchy and a simple model for competitive analysis, Mathematical programming, 32, 146-164, (1985) · Zbl 0588.90053
[10] Labbé, M.; Marcotte, P.; Savard, G., A bilevel model of taxation and its application to optimal highway pricing, Management science, 44, 1608-1622, (1998) · Zbl 0989.90014
[11] Roch, S.; Savard, G.; Marcotte, P., Design and analysis of an approximation algorithm for Stackelberg network pricing, Networks, 46, 1, 57-67, (2005) · Zbl 1072.90004
[12] Roughgarden, T., 2001. Designing networks for selfish users is hard. In: Proceedings of the 42nd Annual IEEE Symposium on the Foundations of Computer Science, pp. 472-482.
[13] Roughgarden, T., Tardos, E., 2000. How bad is selfish routing? In: Proceedings of the 41st Annual IEEE Symposium on the Foundations of Computer Science, pp. 93-102. · Zbl 1323.90011
[14] van der Kraaij, A.F., 2004. Pricing in networks. Ph.D. dissertation, University Maastricht.
[15] van Hoesel, S.; van der Kraaij, A.F.; Mannino, C.; Oriolo, G.; Bouhtou, M., Polynomial cases of the tarification problem, Meteor research memorandum, RM03053, (2003), under review
[16] Vicente, L.N.; Calamai, P.H., Bilevel and multilevel programming: A bibliography review, Journal of global optimization, 5, 291-306, (1994) · Zbl 0822.90127
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