The Core and the Related Solution Concepts in Cooperative Fuzzy Games

Abstract

In this paper, we deal with some solution concepts in cooperative fuzzy games, games with fuzzy coalitions, which admit the representation of rates of players' participation in each coalition. In our previous research, we have introduced a natural class of fuzzy games and a definition of Shapley function. Furthermore, we have given an explicit form of the Shapley function on the class. In this paper we introduce a core function and a dominance core function as functions which derive the core and the dominance core from a given pair of a fuzzy game and a fuzzy coalition. It is shown that they coincide if v is superadditive and monotone nondecreasing with respect to rates of players' participation. Balancedness is also defined. We show that the core of a fuzzy game is nonempty if and only if the game is balanced, as in a crisp game. Furthermore, we show that the center of gravity of the core is the Shapley value for a convex game in our proposed class. Finally, an illustrative example is given.