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Two Approaches to Estimate the Shapley Value for Convex Partially Defined Games

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  • Satoshi Masuya

    (Faculty of Business Administration, Daito Bunka University, 1-9-1, Takashimadaira, Itabashi-ku, Tokyo 175-8571, Japan)

Abstract

In the classical approach of von Neumann and Morgenstern to cooperative games, it was assumed that the worth of all coalitions must be given. However, in real-world problems, the worth of some coalitions may be unknown. Therefore, in this study, we consider the Shapley value for convex partially defined games using two approaches. Firstly, we introduce a polytope that includes the set of Shapley values that can be obtained from a given convex partially defined game and select one rational value in some sense from the set. The elements of this polytope are said to be the Shapley payoff vectors. Secondly, we obtain the set of Shapley values that can be obtained from a given convex partially defined game and select one rational value in some sense from the set. Moreover, we axiomatize the proposed two values.

Suggested Citation

  • Satoshi Masuya, 2023. "Two Approaches to Estimate the Shapley Value for Convex Partially Defined Games," Mathematics, MDPI, vol. 12(1), pages 1-15, December.
    • Handle: RePEc:gam:jmathe:v:12:y:2023:i:1:p:17-:d:1304332
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    References listed on IDEAS

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    1. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer;Game Theory Society, vol. 17(2), pages 89-99.
    2. Willson, Stephen J, 1993. "A Value for Partially Defined Cooperative Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(4), pages 371-384.
    3. SCHMEIDLER, David, 1969. "The nucleolus of a characteristic function game," LIDAM Reprints CORE 44, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. David Housman, 2002. "Linear and symmetric allocation methods for partially defined cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(3), pages 377-404.
    5. Satoshi Masuya & Masahiro Inuiguchi, 2016. "A fundamental study for partially defined cooperative games," Fuzzy Optimization and Decision Making, Springer, vol. 15(3), pages 281-306, September.
    6. 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.
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