Integral Representations of Cooperative Game with Fuzzy Coalitions

In this paper, the cooperative games with fuzzy coalitions were discussed. Several integral representations which described the fuzzy character function of the cooperative game were given, and the important solution –-the Shapley function was also investigated. For the cooperative game where the rate of participation of every player in a coalition is allowed to range within interval [0,1], the multi-linear extension form and the Choquet integral representation were proposed. The Choquet integral representation can be regarded as a general form of cooperative game with fuzzy coalitions. The explicit formulas and some properties of fuzzy characteristic function which was determined by coalition structure were discussed in detail.