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A Matrix Approach To The Associated Consistency With An Application To The Shapley Value

Author

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  • GÉRARD HAMIACHE

    (Université Lille Nord de France, EQUIPPE-Lille 3, Domaine universitaire du Pont de Bois, B.P. 60149, 59653 Villeneuve d'Ascq Cedex, France)

Abstract

In an article by Hamiache (IJGT, 2001) an axiomatization of the Shapley value has been proposed. Three axioms were called on, inessential game, continuity and associated consistency. This present article proposes a new proof, based on elementary linear algebra. Games are represented by vectors. Associated games are the results of matrix operations. The eigenvalues of the involved matrices are computed and it is shown that they are diagonalizable. The present contribution offers a powerful tool allowing further generalizations of the Shapley value, which were difficult to consider on the basis of the previous proof.

Suggested Citation

  • Gérard Hamiache, 2010. "A Matrix Approach To The Associated Consistency With An Application To The Shapley Value," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 12(02), pages 175-187.
    • Handle: RePEc:wsi:igtrxx:v:12:y:2010:i:02:n:s0219198910002581
    DOI: 10.1142/S0219198910002581
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    Citations

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    Cited by:

    1. Florian Navarro, 2019. "Necessary players, Myerson fairness and the equal treatment of equals," Annals of Operations Research, Springer, vol. 280(1), pages 111-119, September.
    2. Sylvain Béal & Eric Rémila & Philippe Solal, 2016. "Characterizations of Three Linear Values for TU Games by Associated Consistency: Simple Proofs Using the Jordan Normal Form," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 18(01), pages 1-21, March.
    3. Liu Cui & Xue Juan & Gao Hongwei & Petrosian Ovanes & Wang Lei, 2018. "Irrational-Behavior-Proof Conditions Based on Limit Characteristic Functions," Journal of Systems Science and Information, De Gruyter, vol. 7(1), pages 1-16, March.
    4. Gérard Hamiache & Florian Navarro, 2020. "Associated consistency, value and graphs," International Journal of Game Theory, Springer;Game Theory Society, vol. 49(1), pages 227-249, March.
    5. Florian Navarro, 2022. "Associated consistency and the Aumann-Drèze value," Post-Print hal-03678064, HAL.
    6. Qianqian Kong & Hao Sun & Genjiu Xu & Dongshuang Hou, 2019. "Associated Games to Optimize the Core of a Transferable Utility Game," Journal of Optimization Theory and Applications, Springer, vol. 182(2), pages 816-836, August.
    7. Genjiu Xu & René van den Brink & Gerard van der Laan & Hao Sun, 2012. "Associated Consistency Characterization of Two Linear Values for TU Games by Matrix Approach," Tinbergen Institute Discussion Papers 12-105/II, Tinbergen Institute.
    8. Xu, Genjiu & Driessen, Theo S.H. & Sun, Hao & Su, Jun, 2013. "Consistency for the additive efficient normalization of semivalues," European Journal of Operational Research, Elsevier, vol. 224(3), pages 566-571.
    9. Petrosyan, Leon & Sedakov, Artem & Sun, Hao & Xu, Genjiu, 2017. "Convergence of strong time-consistent payment schemes in dynamic games," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 96-112.
    10. Florian Navarro, 2022. "Associated consistency and the Aumann-Drèze value," Working Papers hal-03678064, HAL.

    More about this item

    Keywords

    Shapley value; matrix; consistency; associated game; 91A12;
    All these keywords.

    JEL classification:

    • B4 - Schools of Economic Thought and Methodology - - Economic Methodology
    • C0 - Mathematical and Quantitative Methods - - General
    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • D5 - Microeconomics - - General Equilibrium and Disequilibrium
    • D7 - Microeconomics - - Analysis of Collective Decision-Making
    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics

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