Matrix analysis for associated consistency in cooperative game theory

Abstract

Hamiache's recent axiomatization of the well-known Shapley value for TU games states that the Shapley value is the unique solution verifying the following three axioms: the inessential game property, continuity and associated consistency. Driessen extended Hamiache's axiomatization to the enlarged class of efficient, symmetric, and linear values, of which the Shapley value is the most important representative. In this paper, we introduce the notion of row (resp. column)-coalitional matrix in the framework of cooperative game theory. Particularly, both the Shapley value and the associated game are represented algebraically by their coalitional matrices called the Shapley standard matrix $M^{Sh}$ and the associated transformation matrix $M_\lambda,$ respectively. We develop a matrix approach for Hamiache's axiomatization of the Shapley value. The associated consistency for the Shapley value is formulated as the matrix equality $M^{Sh}=M^{Sh}·M_\lambda.$ The diagonalization procedure of $M_\lambda$ and the inessential property for coalitional matrices are fundamental tools to prove the convergence of the sequence of repeated associated games as well as its limit game to be inessential. In addition, a similar matrix approach is applicable to study Driessen's axiomatization of a certain class of linear values. Matrix analysis is adopted throughout both the mathematical developments and the proofs. In summary, it is illustrated that matrix analysis is a new and powerful technique for research in the field of cooperative game theory