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From Hierarchies to Levels: New Solutions for Games with Hierarchical Structure

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Recently, applications of cooperative game theory to economic allocation problems have gained popularity. In many of these problems, players are organized according to either a hierarchical structure or a levels structure that restrict players’ possibilities to cooperate. In this paper, we propose three new solutions for games with hierarchical structure and characterize them by properties that relate a player’s payoff to the payoffs of other players located in specific positions in the structure relative to that player. To define each of these solutions, we consider a certain mapping that transforms any hierarchical structure into a levels structure, and then we apply the standard generalization of the Shapley Value to the class of games with levels structure. The transformations that map the set of hierarchical structures to the set of levels structures are also studied from an axiomatic viewpoint by means of properties that relate a player’s position in both types of structure.

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  • Mikel Álvarez-Mozos & Rene van den Brink & Gerard van der Laan & Oriol Tejada, 2015. "From Hierarchies to Levels: New Solutions for Games with Hierarchical Structure," CER-ETH Economics working paper series 15/215, CER-ETH - Center of Economic Research (CER-ETH) at ETH Zurich.
    • Handle: RePEc:eth:wpswif:15-215
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    18. Mikel Álvarez-Mozos & René van den Brink & Gerard van der Laan & Oriol Tejada, 2015. "From Hierarchies to Levels: New Solutions for Games," Tinbergen Institute Discussion Papers 15-072/II, Tinbergen Institute.
    19. Álvarez-Mozos, M. & van den Brink, R. & van der Laan, G. & Tejada, O., 2013. "Share functions for cooperative games with levels structure of cooperation," European Journal of Operational Research, Elsevier, vol. 224(1), pages 167-179.
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    1. M. Álvarez-Mozos & R. Brink & G. Laan & O. Tejada, 2017. "From hierarchies to levels: new solutions for games with hierarchical structure," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(4), pages 1089-1113, November.
    2. Besner, Manfred, 2018. "The weighted Shapley support levels values," MPRA Paper 87617, University Library of Munich, Germany.
    3. Sylvain Béal & Sylvain Ferrières & Philippe Solal, 2022. "The priority value for cooperative games with a priority structure," International Journal of Game Theory, Springer;Game Theory Society, vol. 51(2), pages 431-450, June.
    4. Besner, Manfred, 2020. "Values for level structures with polynomial-time algorithms, relevant coalition functions, and general considerations," MPRA Paper 99355, University Library of Munich, Germany.
    5. Besner, Manfred, 2017. "Weighted Shapley levels values," MPRA Paper 82978, University Library of Munich, Germany.
    6. Manfred Besner, 2022. "Harsanyi support levels solutions," Theory and Decision, Springer, vol. 93(1), pages 105-130, July.
    7. Encarnacion Algaba & Rene van den Brink, 2021. "Networks, Communication and Hierarchy: Applications to Cooperative Games," Tinbergen Institute Discussion Papers 21-019/IV, Tinbergen Institute.
    8. Xun-Feng Hu & Deng-Feng Li, 2021. "The Equal Surplus Division Value for Cooperative Games with a Level Structure," Group Decision and Negotiation, Springer, vol. 30(6), pages 1315-1341, December.
    9. Xianghui Li & Yang Li, 2021. "On the Structural Stability of Values for Cooperative Games," Journal of Optimization Theory and Applications, Springer, vol. 189(3), pages 873-888, June.

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    More about this item

    Keywords

    TU-game; hierarchical structure; levels structure; Shapley Value; axiomatization.;
    All these keywords.

    JEL classification:

    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General

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