The Shapley value for arbitrary families of coalitions

We address the problem of finding a suitable definition of a value similar to that of Shapley’s, when the games are defined on a subfamily of coalitions with no structure. We present two frameworks: one based on the familiar efficiency, linearity and null player axioms, and the other on linearity and the behavior on unanimity games. We give several properties and examples in each case, and give necessary and sufficient conditions on the family of coalitions for the approaches to coincide.