Comparable Characterizations of Four Solutions for Permission Tree Games

In the field of cooperative games there is an extensive literature that studies situations of restricted cooperation. Myerson (1979) introduced communication graph games, where players can only cooperate if they are connected in an undirected graph representing the communication possibilities. The Myerson value of such a game is obtained by taking the Shapley value of the corresponding restricted game. For the special case that the graph is cycle-free and connected, Demange (2004) introduced for each player the corresponding hierarchical outcome, being the marginal contribution vector for a particular permutation of the player set induced by the graph. Gilles, Owen and van den Brink (1992) introduced games with a (hierarchical) permission structure modeled by a directed graph on the set of players. In the conjunctive (disjunctive) approach, a coalition is said to be feasible, if for every player in the coalition also all (at least one of) its predecessors (if any) belong(s) to the coalition. The conjunctive (disjunctive) permission value is obtained by taking the Shapley value of the associated conjunctive (disjunctive) restricted game. The two approaches coincide when the permission structure is given by a rooted tree. In this paper we consider games with a hierarchical permission structure given by a rooted tree and modify the Myerson value to a value for such games. We also consider for these games the hierarchical outcome with respect to the root of the tree (top player), along with a new solution that assigns all payoff to the unique top player in the hierarchy. Then comparable characterizations are given of these three solutions and the (conjunctive) permission value.