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Tree, Web and Average Web Value for Cycle-Free Directed Graph Games

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  • Khmelnitskaya, A.
  • Talman, A.J.J.

    (Tilburg University, School of Economics and Management)

Abstract

On the class of cycle-free directed graph games with transferable utility solution concepts, called web values, are introduced axiomatically, each one with respect to a chosen coalition of players that is assumed to be an anti-chain in the directed graph and is considered as a management team. We provide their explicit formula representation and simple recursive algorithms to calculate them. Additionally the efficiency and stability of web values are studied. Web values may be considered as natural extensions of the tree and sink values as has been defined correspondingly for rooted and sink forest graph games. In case the management team consists of all sources (sinks) in the graph a kind of tree (sink) value is obtained. In general, at a web value each player receives the worth of this player together with his subordinates minus the total worths of these subordinates. It implies that every coalition of players consisting of a player with all his subordinates receives precisely its worth. We also define the average web value as the average of web values over all management teams in the graph. As application the water distribution problem of a river with multiple sources, a delta and possibly islands is considered.
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Suggested Citation

  • Khmelnitskaya, A. & Talman, A.J.J., 2011. "Tree, Web and Average Web Value for Cycle-Free Directed Graph Games," Other publications TiSEM f38b966e-e26b-4d71-b85a-3, Tilburg University, School of Economics and Management.
    • Handle: RePEc:tiu:tiutis:f38b966e-e26b-4d71-b85a-36e863648900
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    References listed on IDEAS

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    1. Ambec, Stefan & Sprumont, Yves, 2002. "Sharing a River," Journal of Economic Theory, Elsevier, vol. 107(2), pages 453-462, December.
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    7. Anna Khmelnitskaya, 2010. "Values for rooted-tree and sink-tree digraph games and sharing a river," Theory and Decision, Springer, vol. 69(4), pages 657-669, October.
    8. Lei Li & Xueliang Li, 2011. "The covering values for acyclic digraph games," International Journal of Game Theory, Springer;Game Theory Society, vol. 40(4), pages 697-718, November.
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    Cited by:

    1. Anna Khmelnitskaya & Özer Selçuk & Dolf Talman, 2020. "The average covering tree value for directed graph games," Journal of Combinatorial Optimization, Springer, vol. 39(2), pages 315-333, February.
    2. Sylvain Béal & Amandine Ghintran & Eric Rémila & Philippe Solal, 2015. "The sequential equal surplus division for rooted forest games and an application to sharing a river with bifurcations," Theory and Decision, Springer, vol. 79(2), pages 251-283, September.
    3. Sylvain Béal & Eric Rémila & Philippe Solal, 2015. "Discounted Tree Solutions," Working Papers hal-01377923, HAL.
    4. Ata Atay & Marina Núñez, 2019. "Multi-sided assignment games on m-partite graphs," Annals of Operations Research, Springer, vol. 279(1), pages 271-290, August.
    5. Sylvain Beal & Amandine Ghintran & Eric Remila & Philippe Solal, 2013. "The River Sharing Problem: A Survey," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 15(03), pages 1-19.

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