A matrix approach to associated consistency of the Shapley value for games in generalized characteristic function form

The paper is devoted to the Shapley value for cooperative games in generalized characteristic function form in which the players of coalitions are supposed to be ordered. An axiomatization of the generalized Shapley value is presented in terms of three properties, namely continuity, associated consistency, and inessential game property. The proof of the characterization of the Shapley value is based on a matrix representation of so-called associated games. We analyze the dimension of the eigenspaces of this matrix and show that the matrix is diagonalizable.