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Consistency for the additive efficient normalization of semivalues

  • Xu, Genjiu
  • Driessen, Theo S.H.
  • Sun, Hao
  • Su, Jun
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    This paper contributes to consistency for the additive efficient normalization of semivalues. Motivated from the additive efficient normalization of a semivalue being a B-revision of the Shapley value, we introduce the B-reduced game which is an extension of Sobolev’s reduced game. Then the additive efficient normalization of a semivalue is axiomatized as the unique value satisfying covariance, symmetry, and B-consistency. Furthermore, by means of the path-independently linear consistency together with the standardness for two-person games, the additive efficient normalization of semivalues is also characterized. Accessorily, the additive efficient normalization of semivalues is directly verified as the linear consistent least square values (see Ruiz et al., 1998).

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    Article provided by Elsevier in its journal European Journal of Operational Research.

    Volume (Year): 224 (2013)
    Issue (Month): 3 ()
    Pages: 566-571

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    Handle: RePEc:eee:ejores:v:224:y:2013:i:3:p:566-571
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    1. Maschler, Michael, 1992. "The bargaining set, kernel, and nucleolus," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 18, pages 591-667 Elsevier.
    2. Klaus Kultti & Hannu Salonen, 2007. "Minimum norm solutions for cooperative games," International Journal of Game Theory, Springer, vol. 35(4), pages 591-602, April.
    3. Dragan, Irinel, 1996. "New mathematical properties of the Banzhaf value," European Journal of Operational Research, Elsevier, vol. 95(2), pages 451-463, December.
    4. Irinel Dragan, 2006. "The least square values and the shapley value for cooperative TU games," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 14(1), pages 61-73, June.
    5. L. Hernández-Lamoneda & R. Juárez & F. Sánchez-Sánchez, 2007. "Dissection of solutions in cooperative game theory using representation techniques," International Journal of Game Theory, Springer, vol. 35(3), pages 395-426, February.
    6. Carreras, Francesc & Giménez, José Miguel, 2011. "Power and potential maps induced by any semivalue: Some algebraic properties and computation by multilinear extensions," European Journal of Operational Research, Elsevier, vol. 211(1), pages 148-159, May.
    7. Ruiz, Luis M. & Valenciano, Federico & Zarzuelo, Jose M., 1998. "The Family of Least Square Values for Transferable Utility Games," Games and Economic Behavior, Elsevier, vol. 24(1-2), pages 109-130, July.
    8. Thomson, W., 1996. "Consistent Allocation Rules," RCER Working Papers 418, University of Rochester - Center for Economic Research (RCER).
    9. Theo Driessen, 2010. "Associated consistency and values for TU games," International Journal of Game Theory, Springer, vol. 39(3), pages 467-482, July.
    10. Theo Driessen & Elena Yanovskaya, 2002. "Note On linear consistency of anonymous values for TU-games," International Journal of Game Theory, Springer, vol. 30(4), pages 601-609.
    11. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
    12. 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.
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