Every game form and effectivity function (EF) generates a family of characteristic functions (CFs) via feasible utility function profiles. We investigate the extent this family characterizes the underlying structure in which agents interact. The strategic structure CFs characterize is found to be limited to that which is implicit in the representative EF. If the dependency between CF and utility function profile is observable and coalitions pursue only pure strategies, then EFs are fully characterizable by their CF progeny. When mixed strategies are viable, CFs are only sufficient for the “effective surface” of an EF. A number of important EF properties are CF characterizable even when dependency between CF and utility function profile is unobservable. Even so, radically different EFs may have the same CF lineage.