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The Two-Step Average Tree Value for Graph and Hypergraph Games

Author

Listed:
  • Kang, Liying
  • Khmelnitskaya, Anna
  • Shan, Erfang
  • Talman, A.J.J.

    (Tilburg University, School of Economics and Management)

  • Zhang, Guang

    (Tilburg University, School of Economics and Management)

Abstract

We introduce the two-step average tree value for transferable utility games with restricted cooperation represented by undirected communication graphs or hypergraphs. The solution can be considered as an alternative for both the average tree solution for graph games and the average tree value for hypergraph games. Instead of averaging players’ marginal contributions corresponding to all admissible rooted spanning trees of the underlying (hyper)graph, which determines the average tree solution or value, we consider a two-step averaging procedure, in which first, for each player the average of players’ marginal contributions corresponding to all admissible rooted spanning trees that have this player as the root is calculated, and second, the average over all players of all the payoffs obtained in the first step is computed. In general these two approaches lead to different solution concepts. Contrary to the average tree value, the new solution satisfies component fairness and the total cooperation equal treatment property on the entire class of hypergraph games. Moreover, the two-step average tree value is axiomatized on the class of semi-cycle-free hypergraph games, which is more general than the class of cycle-free hypergraph games by allowing the underlying hypergraphs to contain certain cycles. The two-step average tree value is also core stable on the subclass of superadditive semi-cycle-free hypergraph games.
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Suggested Citation

  • Kang, Liying & Khmelnitskaya, Anna & Shan, Erfang & Talman, A.J.J. & Zhang, Guang, 2020. "The Two-Step Average Tree Value for Graph and Hypergraph Games," Other publications TiSEM 54b390b3-2713-4a64-874c-8, Tilburg University, School of Economics and Management.
    • Handle: RePEc:tiu:tiutis:54b390b3-2713-4a64-874c-84d3974c7e5d
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    References listed on IDEAS

    as
    1. Debasis Mishra & A. Talman, 2010. "A characterization of the average tree solution for tree games," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(1), pages 105-111, March.
    2. van den Nouweland, Anne & Borm, Peter & Tijs, Stef, 1992. "Allocation Rules for Hypergraph Communication Situations," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(3), pages 255-268.
    3. Roger B. Myerson, 1977. "Graphs and Cooperation in Games," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 225-229, 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. Liying Kang & Anna Khmelnitskaya & Erfang Shan & Dolf Talman & Guang Zhang, 2023. "The two-step average tree value for graph and hypergraph games," Annals of Operations Research, Springer, vol. 323(1), pages 109-129, April.
    6. E. Algaba & J. M. Bilbao & P. Borm & J. J. López, 2001. "The Myerson value for union stable structures," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 54(3), pages 359-371, December.
    7. Liying Kang & Anna Khmelnitskaya & Erfang Shan & Dolf Talman & Guang Zhang, 2021. "The average tree value for hypergraph games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(3), pages 437-460, December.
    8. 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.
    9. Koshevoy, Gleb & Talman, Dolf, 2014. "Solution concepts for games with general coalitional structure," Mathematical Social Sciences, Elsevier, vol. 68(C), pages 19-30.
    10. Borm, P.E.M. & Owen, G. & Tijs, S.H., 1992. "On the position value for communication situations," Other publications TiSEM 5a8473e4-1df7-42df-ad53-f, Tilburg University, School of Economics and Management.
    11. Mishra, D. & Talman, A.J.J., 2009. "A Characterization of the Average Tree Solution for Cycle-Free Graph Games," Discussion Paper 2009-17, Tilburg University, Center for Economic Research.
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    1. Liying Kang & Anna Khmelnitskaya & Erfang Shan & Dolf Talman & Guang Zhang, 2023. "The two-step average tree value for graph and hypergraph games," Annals of Operations Research, Springer, vol. 323(1), pages 109-129, April.

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    JEL classification:

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

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