Abstract: In this paper, the generalization of the average tree solution is considered. It is the unique solution satisfying component fairness and component efficiency when the crisp graph game is cycle-free. We propose the fuzzy average tree solution that is an imputation for the graph game with fuzzy coalitions. Moreover, on a complete graph game with fuzzy coalitions the fuzzy average tree solution coincides with the fuzzy Shapely value. Finally, we give the relationship between the fuzzy average tree solution and the fuzzy core.

Key words: graph game, fuzzy average tree solution, fuzzy Shapley value, fuzzy coalition, link-convex