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The Core of an Extended Tree Game: A New Characterisation

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  • Barbara von Schnurbein

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    Abstract

    Cost allocation problems on networks can be interpreted as cooperative games on a graph structure. In the classical standard tree game, the cost of a service delivered, by a source has to be allocated between homogeneous users at the vertices. But, modern networks have also the capacity to supply different (levels of) services. For example, a cable network that provides diff erent television standards. Users that choose different levels of service can not be treated equally. The extended tree game accounts for such differences between users. Here, players are characterised by their level of demand, consequently the implications on the cost structure of the problem can be considered. We show how an ET-game can be formulated as the sum of unanimity games. This observation enables us to directly calculate the weighted Shapley values and to identify the core of an ET-game.

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    Bibliographic Info

    Paper provided by Rheinisch-Westfälisches Institut für Wirtschaftsforschung, Ruhr-Universität Bochum, Universität Dortmund, Universität Duisburg-Essen in its series Ruhr Economic Papers with number 0212.

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    Length: 22 pages
    Date of creation: Nov 2010
    Date of revision:
    Handle: RePEc:rwi:repape:0212

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    Keywords: Cooperative game theory; extended tree game; core;

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    1. René Brink & Gerard Laan & Valeri Vasil’ev, 2007. "Component efficient solutions in line-graph games with applications," Economic Theory, Springer, vol. 33(2), pages 349-364, November.
    2. Koster, M.A.L., 1999. "Cost Sharing in Production Situations and Network Exploitation," Open Access publications from Tilburg University urn:nbn:nl:ui:12-80718, Tilburg University.
    3. Borm, P.E.M. & Hamers, H.J.M. & Hendrickx, R.L.P., 2001. "Operations research games: A survey," Open Access publications from Tilburg University urn:nbn:nl:ui:12-305110, Tilburg University.
    4. Endre Bjørndal & Maurice Koster & Stef Tijs, 2004. "Weighted allocation rules for standard fixed tree games," Computational Statistics, Springer, vol. 59(2), pages 249-270, 06.
    5. Monderer, Dov & Samet, Dov, 2002. "Variations on the shapley value," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 54, pages 2055-2076 Elsevier.
    6. Ehud Kalai & Dov Samet, 1983. "On Weighted Shapley Values," Discussion Papers 602, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    7. Sharkey, W.W., 1991. "Network Models in Economics," Papers 69, Bell Communications - Economic Research Group.
    8. Monderer, Dov & Samet, Dov & Shapley, Lloyd S, 1992. "Weighted Values and the Core," International Journal of Game Theory, Springer, vol. 21(1), pages 27-39.
    9. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
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