Comparable Axiomatizations of the Myerson Value, the Restricted Banzhaf Value, Hierarchical Outcomes and the Average Tree Solution for Cycle-Free Graph Restricted Games

Abstract

We consider cooperative transferable utility games, or simply TU-games, with a limited communication structure in which players can cooperate if and only if they are connected in the communication graph. A difference between the restricted Banzhaf value and the Myerson value (i.e. the Shapley value of the restricted game) is that the restricted Banzhaf value satisfies collusion neutrality, while the Myerson value satisfies component efficiency. Requiring both efficiency and collusion neutrality for cycle-free graph games yields other solutions such as the hierarchical outcomes and the average tree solution. Since these solutions also satisfy the superfluous player property, this also `solves' an impossibility for TU-games since there is no solution for these games that satisfies efficiency, collusion neutrality and the null player property. We give axiomatizations of the restricted Banzhaf value, the hierarchical outcomes and the average tree solution that are comparable with axiomatizations of the Myerson value in case the communication graph is cycle-free. Finally, we generalize these solutions to classes of solutions for cycle-free graph games using network power measures.