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

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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)

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    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.

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    Bibliographic Info

    Article provided by World Scientific Publishing Co. Pte. Ltd. in its journal International Game Theory Review.

    Volume (Year): 12 (2010)
    Issue (Month): 02 ()
    Pages: 175-187

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    Handle: RePEc:wsi:igtrxx:v:12:y:2010:i:02:p:175-187

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    Related research

    Keywords: Shapley value; matrix; consistency; associated game; 91A12;

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    Cited by:
    1. Genjiu Xu & Ren� 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.
    2. Genjiu Xu & Ren� 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.
    3. 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.

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