Characterizations of Three Linear Values for TU Games by Associated Consistency: Simple Proofs Using the Jordan Normal Form

This paper studies values for cooperative games with transferable utility. Numerous such values can be characterized by axioms of Psi(epsilon)-associated consistency, which require that a value is invariant under some parametrized linear transformation Psi(epsilon) on the vector space of cooperative games with transferable utility. Xu et al. [(2008) Linear Algebr. Appl. 428, 1571-1586; (2009) Linear Algebr. Appl. 430, 2896-2897] Xu et al. [(2013) Linear Algebr. Appl. 439, 2205-2215], Hamiache [(2010) Int. Game Theor. Rev. 12, 175-187] and more recently Xu et al. [(2015) Linear Algebr. Appl. 471, 224-240] follow this approach by using a matrix analysis. The main drawback of these articles is the heaviness of the proofs to show that the matrix expression of the linear transformations is diagonalizable. By contrast, we provide quick proofs by relying on the Jordan normal form of the previous matrix.