Cooperative games with fuzzy coalitions and fuzzy characteristic functions

In this paper, an extension of a cooperative fuzzy game is proposed, which admits the representation of the rate of participation of every player in a coalition and also associates fuzziness with the value of the game. Games, subject to fuzzy coalitions as well as those pertaining to fuzzy characteristic functions or vague expectations are separately studied in the literature. We propose an extension of a fuzzy game with fuzzy coalitions and vague expectations together and obtain some interesting properties. It has been observed that most of the properties satisfied by a crisp game hold good in the fuzzy sense in this extension. A practical application of the proposed model in Investment Theory is being provided. Further, a Shapley function in the fuzzy sense has been proposed as a solution concept to this class of games. The notion of a fuzzy population monotonic allocation function (FPMAF) is defined and established that the proposed Shapley function is an FPMAF also.