The Shapley value (Shapley, 1953) has been axiomatically characterized from different points of view. van den Brink (2001) proposes a characterization by means of efficiency, fairness and the null player property. In this paper, we characterize the family of single-valued solutions obtained by relaxing fairness into weak fairness. To point out the Shapley value, we impose the additional axiom of weak self consistency and strengthen the null player property into the dummy player property. Remarkably, impossibility results emerge when replacing self consistency by a large set of consistency properties.