Transferable utility games with an additional power structure on the coalitions are considered. This power structure is not given explicitly, but only implicitly via a value; a value is a map that assigns an N-vector to every game with player set N. The implicit power structure is described by the concept of effectiveness of a coalition for a given value. The effectiveness of coalitions is constrained by axioms; in particular, the collection of effective coalitions is assumed to be closed under taking unions. Other axioms concern efficiency and consistency in a sense related to the consistency axiom of Hart and Mas-Colell. The main result of the paper is an axiomatic characterization of a class of restricted Shapley values, with the effective coalitions forming a lattice.