Our concern is the extension of the theory of the Shapley value to games with externalities. Using the standard axiom systems behind the Shapley value for an arbitrary exogenous coalition structure leads to the identification of bounds on players’ payoffs around an \"externality-free\" value....

We first observe that two of Maskin's results do not extend beyond three players: we construct a four-player partition function with non-positive externalities whose unique solution is inefficient, as well as a four-player characteristic function that has a unique efficient solution for each...