Matrix approach to the Shapley value and dual similar associated consistency

G. Xu, Theo Driessen

Research output: Book/ReportReportProfessional

Abstract

Replacing associated consistency in Hamiache's axiom system by dual similar associated consistency, we axiomatize the Shapley value as the unique value verifying the inessential game property, continuity and dual similar associated consistency. Continuing the matrix analysis for Hamiache's axiomatization of the Shapley value, we construct the dual similar associated game and introduce the dual similar associated transformation matrix $M_\lambda^{DSh}$ as well. In the game theoretic framework we show that the dual game of the dual similar associated game is Hamiache's associated game of the dual game. For the purpose of matrix analysis, we derive the similarity relationship $M_\lambda^{DSh}=QM_\lambda Q^{-1}$ between the dual similar associated transformation matrix $M_\lambda^{DSh}$ and associated transformation matrix $M_\lambda$ for Hamiache's associated game, where the transformation matrix $Q$ represents the duality operator on games. This similarity of matrices transfers associated consistency into dual similar associated consistency, and also implies the inessential property for the limit game of the convergent sequence of repeated dual similar associated games. We conclude this paper with three tables summarizing all matrix results.
Original language Undefined Enschede Toegepaste Wiskunde 12 Published - Apr 2006

Publication series

Name Applied Mathematics Memoranda Department of Applied Mathematics, University of Twente 1797 0169-2690

• EWI-8052
• IR-66578
• METIS-237574
• MSC-15A18
• MSC-91A12

Cite this

Xu, G., & Driessen, T. (2006). Matrix approach to the Shapley value and dual similar associated consistency. (Applied Mathematics Memoranda; No. 1797). Enschede: Toegepaste Wiskunde.
Xu, G. ; Driessen, Theo. / Matrix approach to the Shapley value and dual similar associated consistency. Enschede : Toegepaste Wiskunde, 2006. 12 p. (Applied Mathematics Memoranda; 1797).
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Xu, G & Driessen, T 2006, Matrix approach to the Shapley value and dual similar associated consistency. Applied Mathematics Memoranda, no. 1797, Toegepaste Wiskunde, Enschede.

Matrix approach to the Shapley value and dual similar associated consistency. / Xu, G.; Driessen, Theo.

Enschede : Toegepaste Wiskunde, 2006. 12 p. (Applied Mathematics Memoranda; No. 1797).

Research output: Book/ReportReportProfessional

TY - BOOK

T1 - Matrix approach to the Shapley value and dual similar associated consistency

AU - Xu, G.

AU - Driessen, Theo

N1 - eemcs-eprint-8052

PY - 2006/4

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N2 - Replacing associated consistency in Hamiache's axiom system by dual similar associated consistency, we axiomatize the Shapley value as the unique value verifying the inessential game property, continuity and dual similar associated consistency. Continuing the matrix analysis for Hamiache's axiomatization of the Shapley value, we construct the dual similar associated game and introduce the dual similar associated transformation matrix $M_\lambda^{DSh}$ as well. In the game theoretic framework we show that the dual game of the dual similar associated game is Hamiache's associated game of the dual game. For the purpose of matrix analysis, we derive the similarity relationship $M_\lambda^{DSh}=QM_\lambda Q^{-1}$ between the dual similar associated transformation matrix $M_\lambda^{DSh}$ and associated transformation matrix $M_\lambda$ for Hamiache's associated game, where the transformation matrix $Q$ represents the duality operator on games. This similarity of matrices transfers associated consistency into dual similar associated consistency, and also implies the inessential property for the limit game of the convergent sequence of repeated dual similar associated games. We conclude this paper with three tables summarizing all matrix results.

AB - Replacing associated consistency in Hamiache's axiom system by dual similar associated consistency, we axiomatize the Shapley value as the unique value verifying the inessential game property, continuity and dual similar associated consistency. Continuing the matrix analysis for Hamiache's axiomatization of the Shapley value, we construct the dual similar associated game and introduce the dual similar associated transformation matrix $M_\lambda^{DSh}$ as well. In the game theoretic framework we show that the dual game of the dual similar associated game is Hamiache's associated game of the dual game. For the purpose of matrix analysis, we derive the similarity relationship $M_\lambda^{DSh}=QM_\lambda Q^{-1}$ between the dual similar associated transformation matrix $M_\lambda^{DSh}$ and associated transformation matrix $M_\lambda$ for Hamiache's associated game, where the transformation matrix $Q$ represents the duality operator on games. This similarity of matrices transfers associated consistency into dual similar associated consistency, and also implies the inessential property for the limit game of the convergent sequence of repeated dual similar associated games. We conclude this paper with three tables summarizing all matrix results.

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Xu G, Driessen T. Matrix approach to the Shapley value and dual similar associated consistency. Enschede: Toegepaste Wiskunde, 2006. 12 p. (Applied Mathematics Memoranda; 1797).