Axiomatization of Shapley Values of Fagle and Kern Type on Set Systems

Aoi Honda; Yoshiaki Okazaki

doi:10.20965/jaciii.2008.p0409

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JACIII Vol.12 No.5 pp. 409-415

doi: 10.20965/jaciii.2008.p0409

(2008)

Paper:

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Axiomatization of Shapley Values of Fagle and Kern Type on Set Systems

Aoi Honda and Yoshiaki Okazaki

Department of Systems Design and Informatics, Kyushu Institute of Technology,
Iizuka, Fukuoka 820-8502, Japan

Received:

October 10, 2007

Accepted:

February 15, 2008

Published:

September 20, 2008

Keywords:

cooperative game, shapley value, Faigle and Kern's Shapley value, normal set system

Abstract

We propose axiomatizing a generalized Shapley value of games for potential application to games on set systems satisfying the condition of normality. This encompasses both the original Shapley value and Faigle and Kern's Shapley value, which is generalized for a cooperative game defined on a subcoalition.

Cite this article as:

A. Honda and Y. Okazaki, “Axiomatization of Shapley Values of Fagle and Kern Type on Set Systems,” J. Adv. Comput. Intell. Intell. Inform., Vol.12 No.5, pp. 409-415, 2008.

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References

[1]
L. Shapley, “A value for n-person games,” In H. Kuhn and A. Tucker (Eds.), Contributions to the Theory of Games, Vol.2, 28, in Annals of Mathematics Studies, pp. 307-317, Princeton University Press, 1953.
E. Algaba and J. M. Bilbao, R. van den Brink, and A. Jime'nez-Losada, “Aximatizations of the Shapley value for cooperative games on antimatroids,” Math. Meth Oper. Res. 57, pp. 49-65, 2003.
U. Faigle and W. Kern, “The Shapley value for cooperative games under precedence constraints,” Int. J. of Game Theory, 21, pp. 249-266, 1992.
A. Honda and M. Grabisch, “Entropy of capacities on lattices,” Information Sciences, 176, pp. 3472-3489, 2006.
A. Honda and M. Grabisch, “An axiomatization of entropy of capacities on set systems,” European Journal of Operational Research, in press.
J. Aczel, “Lectures on functional equations and their applications,” Academic Press, 1966.
B. A. Davey and H. A. Priestley, “Introduction to lattices and order,” Cambridge University Press, 1990.
A. Dukhovny, “General entropy of general measures,” Int. J. Uncertain. Fuzziness Knowledge-Based Systems, 10, pp. 213-225, 2002.
M. Grabisch, “An axiomatization of the Shapley value and interaction index for games on lattices,” SCIS-ISIS 2004, 2nd Int. Conf. on Soft Computing and Intelligent Systems and 5th Int. Symp. on Advanced Intelligent Systems, Yokohama, Japan, 2004.
C. R. Hsiao and T. E. S. Raghavan, “Shapley value for multichoice cooperative games I,” Games and Economic Behavior, 5, pp. 240-256, 1993.
I. Kojadinovic, J.-L. Marichal, and M. Roubens, “An axiomatic approach to the definition of the entropy of a discrete Choquet capacity,” Information Sciences, 172, pp. 131-153, 2005.
J.-L. Marichal and M. Roubens, “Entropy of discrete fuzzy measure,” Int. J. Uncertain. Fuzziness Knowledge-Based Systems, 8, pp. 625-640, 2000.
C. E. Shannon, “A mathematical theory of communication,” Bell System Tech. Journ, 27, pp. 374-423, pp. 623-656, 1948.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1]
L. Shapley, “A value for n-person games,” In H. Kuhn and A. Tucker (Eds.), Contributions to the Theory of Games, Vol.2, 28, in Annals of Mathematics Studies, pp. 307-317, Princeton University Press, 1953.
E. Algaba and J. M. Bilbao, R. van den Brink, and A. Jime'nez-Losada, “Aximatizations of the Shapley value for cooperative games on antimatroids,” Math. Meth Oper. Res. 57, pp. 49-65, 2003.
U. Faigle and W. Kern, “The Shapley value for cooperative games under precedence constraints,” Int. J. of Game Theory, 21, pp. 249-266, 1992.
A. Honda and M. Grabisch, “Entropy of capacities on lattices,” Information Sciences, 176, pp. 3472-3489, 2006.
A. Honda and M. Grabisch, “An axiomatization of entropy of capacities on set systems,” European Journal of Operational Research, in press.
J. Aczel, “Lectures on functional equations and their applications,” Academic Press, 1966.
B. A. Davey and H. A. Priestley, “Introduction to lattices and order,” Cambridge University Press, 1990.
A. Dukhovny, “General entropy of general measures,” Int. J. Uncertain. Fuzziness Knowledge-Based Systems, 10, pp. 213-225, 2002.
M. Grabisch, “An axiomatization of the Shapley value and interaction index for games on lattices,” SCIS-ISIS 2004, 2nd Int. Conf. on Soft Computing and Intelligent Systems and 5th Int. Symp. on Advanced Intelligent Systems, Yokohama, Japan, 2004.
C. R. Hsiao and T. E. S. Raghavan, “Shapley value for multichoice cooperative games I,” Games and Economic Behavior, 5, pp. 240-256, 1993.
I. Kojadinovic, J.-L. Marichal, and M. Roubens, “An axiomatic approach to the definition of the entropy of a discrete Choquet capacity,” Information Sciences, 172, pp. 131-153, 2005.
J.-L. Marichal and M. Roubens, “Entropy of discrete fuzzy measure,” Int. J. Uncertain. Fuzziness Knowledge-Based Systems, 8, pp. 625-640, 2000.
C. E. Shannon, “A mathematical theory of communication,” Bell System Tech. Journ, 27, pp. 374-423, pp. 623-656, 1948.