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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, Center For Economic Research)

  • Zhang, Guang

    (Tilburg University, Center For Economic Research)

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.
(This abstract was borrowed from another version of this item.)

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," Discussion Paper 2020-018, Tilburg University, Center for Economic Research.
    • Handle: RePEc:tiu:tiucen:54b390b3-2713-4a64-874c-84d3974c7e5d
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    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.
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    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.
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    13. 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.
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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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    More about this item

    Keywords

    TU game; hypergraph communication structure; average tree value; component fairness;
    All these keywords.

    JEL classification:

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

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