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Weak differential marginality and the Shapley value

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  • Casajus, André
  • Yokote, Koji
Abstract
The principle of differential marginality for cooperative games states that the differential of two players' payoffs does not change when the differential of these players' marginal contributions to coalitions containing neither of them does not change. Together with two standard properties, efficiency and the null player property, differential marginality characterizes the Shapley value. For games that contain more than two players, we show that this characterization can be improved by using a substantially weaker property than differential marginality. Weak differential marginality requires two players' payoffs to change in the same direction when these players' marginal contributions to coalitions containing neither of them change by the same amount.

Suggested Citation

  • Casajus, André & Yokote, Koji, 2017. "Weak differential marginality and the Shapley value," Journal of Economic Theory, Elsevier, vol. 167(C), pages 274-284.
    • Handle: RePEc:eee:jetheo:v:167:y:2017:i:c:p:274-284
    DOI: 10.1016/j.jet.2016.11.007
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    References listed on IDEAS

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    1. Perez-Castrillo, David & Wettstein, David, 2001. "Bidding for the Surplus : A Non-cooperative Approach to the Shapley Value," Journal of Economic Theory, Elsevier, vol. 100(2), pages 274-294, October.
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    5. Casajus, André & Huettner, Frank, 2014. "Weakly monotonic solutions for cooperative games," Journal of Economic Theory, Elsevier, vol. 154(C), pages 162-172.
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    8. Casajus, André & Huettner, Frank, 2013. "Null players, solidarity, and the egalitarian Shapley values," Journal of Mathematical Economics, Elsevier, vol. 49(1), pages 58-61.
    9. René Brink & Yukihiko Funaki & Yuan Ju, 2013. "Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 40(3), pages 693-714, March.
    10. André Casajus, 2011. "Differential marginality, van den Brink fairness, and the Shapley value," Theory and Decision, Springer, vol. 71(2), pages 163-174, August.
    11. Calvo, Emilio & Santos, Juan Carlos, 1997. "Potentials in cooperative TU-games," Mathematical Social Sciences, Elsevier, vol. 34(2), pages 175-190, October.
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    Citations

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    Cited by:

    1. Calleja, Pere & Llerena Garrés, Francesc, 2018. "Weak fairness and the Shapley value," Working Papers 2072/306979, Universitat Rovira i Virgili, Department of Economics.
    2. Shan, Erfang & Cui, Zeguang & Lyu, Wenrong, 2023. "Gain–loss and new axiomatizations of the Shapley value," Economics Letters, Elsevier, vol. 228(C).
    3. van den Brink, René & Núñez, Marina & Robles, Francisco, 2021. "Valuation monotonicity, fairness and stability in assignment problems," Journal of Economic Theory, Elsevier, vol. 195(C).
    4. André Casajus & Koji Yokote, 2019. "Weakly differentially monotonic solutions for cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(3), pages 979-997, September.
    5. Besner, Manfred, 2022. "The grand surplus value and repeated cooperative cross-games with coalitional collaboration," Journal of Mathematical Economics, Elsevier, vol. 102(C).
    6. Bahel, Eric, 2021. "Hyperadditive games and applications to networks or matching problems," Journal of Economic Theory, Elsevier, vol. 191(C).
    7. Shan, Erfang & Cui, Zeguang & Yu, Bingxin, 2024. "New characterizations of the Shapley value using weak differential marginalities," Economics Letters, Elsevier, vol. 238(C).
    8. Emilio Calvo Ramón & Esther Gutiérrez-López, 2022. "The equal collective gains value in cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 51(1), pages 249-278, March.
    9. Takumi Kongo, 2020. "Similarities in axiomatizations: equal surplus division value and first-price auctions," Review of Economic Design, Springer;Society for Economic Design, vol. 24(3), pages 199-213, December.
    10. Xinjuan Chen & Minghua Zhan & Zhihui Zhao, 2024. "A characterization of the Owen value via sign symmetries," Theory and Decision, Springer, vol. 97(3), pages 553-561, November.
    11. Takaaki Abe & Satoshi Nakada, 2023. "Core stability of the Shapley value for cooperative games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 60(4), pages 523-543, May.
    12. Casajus, André, 2018. "Symmetry, mutual dependence, and the weighted Shapley values," Journal of Economic Theory, Elsevier, vol. 178(C), pages 105-123.
    13. Sylvain Béal & Sylvain Ferrières & Adriana Navarro‐Ramos & Philippe Solal, 2023. "Axiomatic characterizations of the family of Weighted priority values," International Journal of Economic Theory, The International Society for Economic Theory, vol. 19(4), pages 787-816, December.
    14. C. Manuel & E. Ortega & M. del Pozo, 2023. "Marginality and the position value," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(2), pages 459-474, July.
    15. Li, Wenzhong & Xu, Genjiu & van den Brink, René, 2024. "Sign properties and axiomatizations of the weighted division values," Journal of Mathematical Economics, Elsevier, vol. 112(C).

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    More about this item

    Keywords

    TU game; Shapley value; Differential marginality; Weak differential marginality;
    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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