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

This article studies values for cooperative games with transferable utility. Numerous such values can be characterized by axioms of Ψε-associated consistency, which require that a value is invariant under some parametrized linear transformation Ψε on the vector space of cooperative games with transferable utility. Xu et al. (2008, 2009, Linear Algebra Appl.), Xu et al. (2013, Linear Algebra Appl.), Hamiache (2010, Int. Game Theory Rev.) and more recently Xu et al. (2015, Linear Algebra Appl.) 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.