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The average covering tree value for directed graph games

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  • Khmelnitskaya, Anna
  • Selcuktu, Ozer
  • Talman, A.J.J.

    (Tilburg University, School of Economics and Management)

Abstract

We introduce a single-valued solution concept, the so-called average covering tree value, for the class of transferable utility games with limited communication structure represented by a directed graph. The solution is the average of the marginal contribution vectors corresponding to all covering trees of the directed graph. The covering trees of a directed graph are those (rooted) trees on the set of players that preserve the dominance relations between the players prescribed by the directed graph. The average covering tree value is component efficient, and under a particular convexity-type condition it is stable. For transferable utility games with complete communication structure the average covering tree value equals to the Shapley value of the game. If the graph is the directed analog of an undirected graph the average covering tree value coincides with the gravity center solution.
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Suggested Citation

  • Khmelnitskaya, Anna & Selcuktu, Ozer & Talman, A.J.J., 2020. "The average covering tree value for directed graph games," Other publications TiSEM df006318-c4d7-4ab2-ab07-a, Tilburg University, School of Economics and Management.
    • Handle: RePEc:tiu:tiutis:df006318-c4d7-4ab2-ab07-a9c8a90a4b3a
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    References listed on IDEAS

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    1. Roger B. Myerson, 1977. "Graphs and Cooperation in Games," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 225-229, August.
    2. Khmelnitskaya, Anna & Talman, Dolf, 2014. "Tree, web and average web values for cycle-free directed graph games," European Journal of Operational Research, Elsevier, vol. 235(1), pages 233-246.
    3. Gabrielle Demange, 2004. "On Group Stability in Hierarchies and Networks," Journal of Political Economy, University of Chicago Press, vol. 112(4), pages 754-778, August.
    4. Herings, P.J.J. & van der Laan, G. & Talman, A.J.J. & Yang, Z., 2010. "The average tree solution for cooperative games with communication structure," Games and Economic Behavior, Elsevier, vol. 68(2), pages 626-633, March.
    5. Khmelnitskaya, A. & Selçuk, O. & Talman, A.J.J., 2014. "The Shapley Value for Directed Graph Games," Discussion Paper 2014-064, Tilburg University, Center for Economic Research.
    6. Herings, P. Jean Jacques & van der Laan, Gerard & Talman, Dolf, 2008. "The average tree solution for cycle-free graph games," Games and Economic Behavior, Elsevier, vol. 62(1), pages 77-92, January.
    7. Koshevoy, Gleb & Talman, Dolf, 2014. "Solution concepts for games with general coalitional structure," Mathematical Social Sciences, Elsevier, vol. 68(C), pages 19-30.
    8. Koshevoy, G.A. & Talman, A.J.J., 2011. "Solution Concepts for Games with General Coalitional Structure (Replaces CentER DP 2011-025)," Discussion Paper 2011-119, Tilburg University, Center for Economic Research.
    9. 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.
    10. Koshevoy, G.A. & Talman, A.J.J., 2011. "Solution Concepts for Games with General Coalitional Structure (Replaced by CentER DP 2011-119)," Discussion Paper 2011-025, Tilburg University, Center for Economic Research.
    11. 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. Koshevoy, G.A. & Suzuki, T. & Talman, A.J.J., 2013. "Solutions For Games With General Coalitional Structure And Choice Sets," Other publications TiSEM a831011f-430e-4e82-b6f6-5, Tilburg University, School of Economics and Management.
    2. Khmelnitskaya, A. & Selçuk, O. & Talman, A.J.J., 2014. "The Shapley Value for Directed Graph Games," Discussion Paper 2014-064, Tilburg University, Center for Economic Research.

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    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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