Set-valued nonlinear analogues of the Shapley value

One of Shapley's classic axioms deals with the addition of the games. We consider the family of axiom systems depending on various operations over the set of cooperative games. The existence and the uniqueness of the corresponding a priori set-valued solutions are studied. The family of solutions (satisfying the proposed system of axioms) contains, as particular cases, the Shapley value and the [epsilon]-core. The modification of the proposed system of axioms is studied, prekernel being the maximal solution satisfying this modification.