A modified position value for communication situations and its fuzzification

Myerson treated various situations of cooperation in the field of cooperative games and proposed the communication structure. In this paper, we define and characterize an allocation rule in terms of the position value, called an average spanning tree solution, for communication situations by introducing a concept of cooperation relationship which says that two players are deemed to possess this relationship if there is a communication path through them. Considering the fact that the extent to what players participate in a coalition may be partially restricted for uncertain possibilities, we construct a graph game in multilinear extension form and continue to explore the fuzzy average spanning tree solution on a framework of communication situation with fuzzy coalition.