Hierarchical Outcomes and Collusion Neutrality

We consider collusive agreements in (superadditive TU-)games that bind collusion members to act as a single player in such a way that the collusion performs its role in the original game if all members are together; otherwise, the collusion plays no role, acting like the empty set. Collusive agreements are feasible only for a set of players who are connected on a tree. Collusion neutrality requires that no feasible collusive agreement in uences the total payo for the collusion mem- bers. On the domain of all games, tree-restricted or unrestricted, there is a solution satisfying collusion neutrality, eciency and null-player property if and only if the tree is a line. Either replacing null-player property with very-null-player property or restricting the domain to the subdomain of tree-restricted games, we show that ane combina- tions of hierarchical solutions (Demange 2004, van den Brink 2012) are the only solutions satisfying the three axioms together with linearity. All these results hold replacing collusion neutrality with the combi- nation of pairwise neutrality and non-bossiness. Adding a mild equal treatment axiom, we obtain characterizations of the average tree solu- tion (the average of hierarchical solutions, i.e., the ane combination with equal weights).