Solution concepts in cooperative fuzzy games

We deal with some solution concepts in cooperative fuzzy games, games with fuzzy coalition, which admit the representation of players´ participation degree in each coalition. In our previous research, we have introduced a natural class of fuzzy games and a natural definition of the Shapley function. Furthermore, we have given a Shapley function in explicit form on the class. In this paper we introduce core function and dominance core function as functions which map a pair of a fuzzy game and a fuzzy coalition to the corresponding core and dominance core, respectively. It is shown that they coincide if υ is monotone nondecreasing with respect to each player´s participation degree. Balancedness is also defined. We show that the core of a fuzzy game is nonempty if the game is balanced, as in a crisp game. Furthermore, we show that the barycentre of the extreme points of the core coincides with the Shapley value in a convex game in our proposed class. Finally, an illustrative example is given