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Decomposition of solutions and the Shapley value

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  • Casajus, André
  • Huettner, Frank

Abstract

We suggest foundations for the Shapley value and for the naïve solution, which assigns to any player the difference between the worth of the grand coalition and its worth after this player left the game. To this end, we introduce the decomposition of solutions for cooperative games with transferable utility. A decomposer of a solution is another solution that splits the former into a direct part and an indirect part. While the direct part (the decomposer) measures a player's contribution in a game as such, the indirect part indicates how she affects the other players' direct contributions by leaving the game. The Shapley value turns out to be unique decomposable decomposer of the naïve solution.

Suggested Citation

  • Casajus, André & Huettner, Frank, 2018. "Decomposition of solutions and the Shapley value," Games and Economic Behavior, Elsevier, vol. 108(C), pages 37-48.
    • Handle: RePEc:eee:gamebe:v:108:y:2018:i:c:p:37-48
    DOI: 10.1016/j.geb.2017.05.001
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    References listed on IDEAS

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    Cited by:

    1. Casajus, André, 2021. "Weakly balanced contributions and the weighted Shapley values," Journal of Mathematical Economics, Elsevier, vol. 94(C).
    2. Liu, Aixin & Li, Haitao & Wang, Lin, 2024. "Exploring multi-potential games in strategic form: A graph theoretic approach," Applied Mathematics and Computation, Elsevier, vol. 474(C).
    3. Takaaki Abe & Satoshi Nakada, 2023. "Potentials and solutions of cooperative games with a fixed player set," International Journal of Game Theory, Springer;Game Theory Society, vol. 52(3), pages 757-774, September.
    4. Andr'e Casajus & Yukihiko Funaki & Frank Huettner, 2024. "Random partitions, potential, value, and externalities," Papers 2402.00394, arXiv.org, revised Jun 2024.
    5. Ori Haimanko, 2020. "Generalized Coleman-Shapley indices and total-power monotonicity," International Journal of Game Theory, Springer;Game Theory Society, vol. 49(1), pages 299-320, March.
    6. André Casajus & Rodrigue Tido Takeng, 2023. "Second-order productivity, second-order payoffs, and the Owen value," Annals of Operations Research, Springer, vol. 320(1), pages 1-13, January.
    7. Rodrigue Tido Takeng & Arnold Cedrick Soh Voutsa & Kévin Fourrey, 2023. "Decompositions of inequality measures from the perspective of the Shapley–Owen value," Theory and Decision, Springer, vol. 94(2), pages 299-331, February.
    8. Sébastien Courtin & Rodrigue Tido Takeng & Frédéric Chantreuil, 2024. "Decomposition of interaction indices: alternative interpretations of cardinal–probabilistic interaction indices," Theory and Decision, Springer, vol. 97(1), pages 139-165, August.
    9. André Casajus & Rodrigue Tido Takeng, 2022. "Second-order productivity, second-order payoffs, and the Owen value," Post-Print hal-03798448, HAL.
    10. André Casajus & Frank Huettner, 2019. "The Coleman–Shapley index: being decisive within the coalition of the interested," Public Choice, Springer, vol. 181(3), pages 275-289, December.
    11. Casajus, André, 2018. "Symmetry, mutual dependence, and the weighted Shapley values," Journal of Economic Theory, Elsevier, vol. 178(C), pages 105-123.

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    More about this item

    Keywords

    Decomposition; Shapley value; Potential; Consistency; Higher-order contributions; Balanced contributions;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D60 - Microeconomics - - Welfare Economics - - - General

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