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Axiomatizations of the Shapley value for cooperative games on antimatroids

Author

Listed:
  • E. Algaba
  • J. M. Bilbao
  • R. van den Brink
  • A. Jiménez-Losada

Abstract

Cooperative games on antimatroids are cooperative games restricted by a combinatorial structure which generalize the permission structure. So, cooperative games on antimatroids group several well-known families of games which have important applications in economics and politics. Therefore, the study of the rectricted games by antimatroids allows to unify criteria of various lines of research. The current paper establishes axioms that determine the restricted Shapley value on antimatroids by conditions on the cooperative game v and the structure determined by the antimatroid. This axiomatization generalizes the axiomatizations of both the conjunctive and disjunctive permission value for games with a permission structure. We also provide an axiomatization of the Shapley value restricted to the smaller class of poset antimatroids. Finally, we apply our model to auction situations. Copyright Springer-Verlag Berlin Heidelberg 2003

Suggested Citation

  • E. Algaba & J. M. Bilbao & R. van den Brink & A. Jiménez-Losada, 2003. "Axiomatizations of the Shapley value for cooperative games on antimatroids," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 57(1), pages 49-65, April.
    • Handle: RePEc:spr:mathme:v:57:y:2003:i:1:p:49-65
    DOI: 10.1007/s001860200250
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    Cited by:

    1. M. Josune Albizuri & Satoshi Masuya & José M. Zarzuelo, 2022. "Characterization of a value for games under restricted cooperation," Annals of Operations Research, Springer, vol. 318(2), pages 773-785, November.
    2. Meng, Fanyong & Chen, Xiaohong & Zhang, Qiang, 2015. "A coalitional value for games on convex geometries with a coalition structure," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 605-614.
    3. Selcuk, O. & Talman, A.J.J., 2013. "Games With General Coalitional Structure," Discussion Paper 2013-002, Tilburg University, Center for Economic Research.
    4. Encarnacion Algaba & Rene van den Brink, 2021. "Networks, Communication and Hierarchy: Applications to Cooperative Games," Tinbergen Institute Discussion Papers 21-019/IV, Tinbergen Institute.
    5. Matthew Ryan, 2010. "Mixture sets on finite domains," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 33(2), pages 139-147, November.
    6. C. Manuel & E. González-Arangüena & R. Brink, 2013. "Players indifferent to cooperate and characterizations of the Shapley value," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(1), pages 1-14, February.
    7. Encarnación Algaba & René Brink & Chris Dietz, 2017. "Power Measures and Solutions for Games Under Precedence Constraints," Journal of Optimization Theory and Applications, Springer, vol. 172(3), pages 1008-1022, March.
    8. Navarro, Florian, 2020. "The center value: A sharing rule for cooperative games on acyclic graphs," Mathematical Social Sciences, Elsevier, vol. 105(C), pages 1-13.
    9. Tobias Hiller, 2021. "Hierarchy and the size of a firm," International Review of Economics, Springer;Happiness Economics and Interpersonal Relations (HEIRS), vol. 68(3), pages 389-404, September.
    10. Guangming Wang & Zeguang Cui & Erfang Shan, 2022. "An Axiomatization of the Value α for Games Restricted by Augmenting Systems," Mathematics, MDPI, vol. 10(15), pages 1-9, August.
    11. René Brink & Ilya Katsev & Gerard Laan, 2011. "Axiomatizations of two types of Shapley values for games on union closed systems," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 47(1), pages 175-188, May.
    12. Kishimoto, Shin & Watanabe, Naoki, 2017. "The kernel of a patent licensing game: The optimal number of licensees," Mathematical Social Sciences, Elsevier, vol. 86(C), pages 37-50.
    13. van den Brink, René & González-Arangüena, Enrique & Manuel, Conrado & del Pozo, Mónica, 2014. "Order monotonic solutions for generalized characteristic functions," European Journal of Operational Research, Elsevier, vol. 238(3), pages 786-796.
    14. Rene van den Brink & Ilya Katsev & Gerard van der Laan, 2023. "Properties of Solutions for Games on Union-Closed Systems," Mathematics, MDPI, vol. 11(4), pages 1-16, February.
    15. van den Brink, René & He, Simin & Huang, Jia-Ping, 2018. "Polluted river problems and games with a permission structure," Games and Economic Behavior, Elsevier, vol. 108(C), pages 182-205.
    16. René Brink, 2012. "On hierarchies and communication," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(4), pages 721-735, October.
    17. Frank Huettner & Harald Wiese, 2016. "The Need for Permission, the Power to Enforce, and Duality in Cooperative Games with a Hierarchy," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 18(04), pages 1-12, December.
    18. Emilio Calvo & Esther Gutiérrez-López, 2015. "The value in games with restricted cooperation," Discussion Papers in Economic Behaviour 0115, University of Valencia, ERI-CES.
    19. Encarnacion Algaba & Rene van den Brink, 2019. "The Shapley Value and Games with Hierarchies," Tinbergen Institute Discussion Papers 19-064/II, Tinbergen Institute.
    20. Encarnación Algaba & René Brink & Chris Dietz, 2018. "Network Structures with Hierarchy and Communication," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 265-282, October.
    21. René Brink & Chris Dietz, 2014. "Games with a local permission structure: separation of authority and value generation," Theory and Decision, Springer, vol. 76(3), pages 343-361, March.
    22. Zhengxing Zou & Qiang Zhang, 2018. "Harsanyi power solution for games with restricted cooperation," Journal of Combinatorial Optimization, Springer, vol. 35(1), pages 26-47, January.

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