Matrix Approach to Cooperative Game Theory

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.