Matrix Approach to Cooperative Game Theory

G. Xu Genjiu

Research output: ThesisPhD Thesis - Research UT, graduation UT

89 Downloads (Pure)

Abstract

In this monograph, the algebraic representation and the matrix approach are applied to study linear operators on the game space, more precisely, linear transformations on games and linear values. In terms of the essential notion of a coalitional matrix, these linear operators are represented algebraically by products of the corresponding coalitional matrix and the worth vector (representing the game). We perform a matrix analysis in the setting of cooperative game theory, to study axiomatizations of linear values, by investigating appropriate properties of these representation matrices. Particularly, the Shapley value is the most important representative. In summary, the concepts of eigenvalues, eigenvectors, null space, the diagonalization procedure and the similarity property for matrices, the system of linear equations and its solution set, the Moebius transformation and the complementary Moebius transformation, the basis for a linear space and so on, can be applied successfully to cooperative game theory. We conclude that the matrix analysis is a new and powerful technique for research in the field of cooperative game theory.
Original languageUndefined
Awarding Institution
  • University of Twente
Supervisors/Advisors
  • Uetz, Marc Jochen, Supervisor
  • Li, Xueliang, Supervisor
  • Driessen, Theo, Advisor
  • Li, Xueliang, Supervisor
Award date28 Aug 2008
Place of PublicationZutphen
Publisher
Print ISBNs978-90-365-2710-1
DOIs
Publication statusPublished - 28 Aug 2008

Keywords

  • IR-59410
  • EWI-13839
  • METIS-252073

Cite this

Xu Genjiu, G. (2008). Matrix Approach to Cooperative Game Theory. Zutphen: University of Twente. https://doi.org/10.3990/1.9789036527101
Xu Genjiu, G.. / Matrix Approach to Cooperative Game Theory. Zutphen : University of Twente, 2008. 165 p.
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Xu Genjiu, G 2008, 'Matrix Approach to Cooperative Game Theory', University of Twente, Zutphen. https://doi.org/10.3990/1.9789036527101

Matrix Approach to Cooperative Game Theory. / Xu Genjiu, G.

Zutphen : University of Twente, 2008. 165 p.

Research output: ThesisPhD Thesis - Research UT, graduation UT

TY - THES

T1 - Matrix Approach to Cooperative Game Theory

AU - Xu Genjiu, G.

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Y1 - 2008/8/28

N2 - In this monograph, the algebraic representation and the matrix approach are applied to study linear operators on the game space, more precisely, linear transformations on games and linear values. In terms of the essential notion of a coalitional matrix, these linear operators are represented algebraically by products of the corresponding coalitional matrix and the worth vector (representing the game). We perform a matrix analysis in the setting of cooperative game theory, to study axiomatizations of linear values, by investigating appropriate properties of these representation matrices. Particularly, the Shapley value is the most important representative. In summary, the concepts of eigenvalues, eigenvectors, null space, the diagonalization procedure and the similarity property for matrices, the system of linear equations and its solution set, the Moebius transformation and the complementary Moebius transformation, the basis for a linear space and so on, can be applied successfully to cooperative game theory. We conclude that the matrix analysis is a new and powerful technique for research in the field of cooperative game theory.

AB - In this monograph, the algebraic representation and the matrix approach are applied to study linear operators on the game space, more precisely, linear transformations on games and linear values. In terms of the essential notion of a coalitional matrix, these linear operators are represented algebraically by products of the corresponding coalitional matrix and the worth vector (representing the game). We perform a matrix analysis in the setting of cooperative game theory, to study axiomatizations of linear values, by investigating appropriate properties of these representation matrices. Particularly, the Shapley value is the most important representative. In summary, the concepts of eigenvalues, eigenvectors, null space, the diagonalization procedure and the similarity property for matrices, the system of linear equations and its solution set, the Moebius transformation and the complementary Moebius transformation, the basis for a linear space and so on, can be applied successfully to cooperative game theory. We conclude that the matrix analysis is a new and powerful technique for research in the field of cooperative game theory.

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Xu Genjiu G. Matrix Approach to Cooperative Game Theory. Zutphen: University of Twente, 2008. 165 p. https://doi.org/10.3990/1.9789036527101