Solution concepts in cooperative fuzzy games

Abstract

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 /spl upsi/ 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.