Efficiency and collusion neutrality in cooperative games and networks

In the literature various axiomatizations of the Shapley and Banzhaf value can be found where the only difference is the Shapley value satisfying efficiency and the Banzhaf value satisfying collusion neutrality. Both properties seem to be desirable. In this paper, we show that there is no solution satisfying efficiency, collusion neutrality and the null player property. After that we show that this impossibility is resolved when cooperation among players is restricted by a cycle-free communication network such that only connected coalitions are feasible. In particular, all hierarchical outcomes introduced by Demange (2004) and their convex combinations satisfy these three properties.