The cg-average tree value for games on cycle-free fuzzy communication structures

The main goal in a cooperative game is to obtain a fair allocation of the profit due the cooperation of the involved agents. The most known of these allocations is the Shapley value. This allocation considers that the communication among the players is complete. The Myerson value is a modification of the Shapley value considering a communication structure which determines the feasible bilateral relationships among the agents. This allocation of the profit is not always a stable solution. Another payoff allocation for games with a communication structure from the definition of the Shapley value is the average tree value. This one is a stable solution for any game using a cycle-free communication structure. Later fuzzy communication structures were introduced. In a fuzzy communication structure, the membership of the agents and the relationships among them are leveled. The Myerson value was extended in several different ways depending on the behavior of the agents. In this paper, the average tree value is extended to games with fuzzy communication structures taking one particular version: the Choquet by graphs (cg). We present an application to the management of an electrical network with an algorithmic implementation.