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A necessary and sufficient condition for weighted Shapley values in the core

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  • Chen, Xinjuan
  • Zhan, Minghua

Abstract

In this note, we improve Skoda and Venel’s condition in Skoda et al. (2023) to be a sufficient and necessary condition. It seems to be the first result on the sufficient and necessary condition to ensure that the weighted Shapley value is in the core.

Suggested Citation

  • Chen, Xinjuan & Zhan, Minghua, 2025. "A necessary and sufficient condition for weighted Shapley values in the core," Economics Letters, Elsevier, vol. 256(C).
    • Handle: RePEc:eee:ecolet:v:256:y:2025:i:c:s0165176525004550
    DOI: 10.1016/j.econlet.2025.112618
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    References listed on IDEAS

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    1. Monderer, Dov & Samet, Dov & Shapley, Lloyd S, 1992. "Weighted Values and the Core," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(1), pages 27-39.
    2. Inarra, Elena & Usategui, Jose M, 1993. "The Shapley Value and Average Convex Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 22(1), pages 13-29.
    3. Casajus, André, 2018. "Symmetry, mutual dependence, and the weighted Shapley values," Journal of Economic Theory, Elsevier, vol. 178(C), pages 105-123.
    4. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
    5. Abe, Takaaki, 2019. "Decomposing a balanced game: A necessary and sufficient condition for the nonemptiness of the core," Economics Letters, Elsevier, vol. 176(C), pages 9-13.
    6. Lloyd S. Shapley, 1967. "On balanced sets and cores," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 14(4), pages 453-460.
    7. Skoda, Alexandre & Venel, Xavier, 2023. "Weighted average-convexity and Shapley values," Games and Economic Behavior, Elsevier, vol. 140(C), pages 88-98.
    8. Sprumont, Yves, 1990. "Population monotonic allocation schemes for cooperative games with transferable utility," Games and Economic Behavior, Elsevier, vol. 2(4), pages 378-394, December.
    9. Casajus, André, 2021. "Weakly balanced contributions and the weighted Shapley values," Journal of Mathematical Economics, Elsevier, vol. 94(C).
    10. Nowak, A.S. & Radzik, T., 1995. "On axiomatizations of the weighted Shapley values," Games and Economic Behavior, Elsevier, vol. 8(2), pages 389-405.
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    JEL classification:

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

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