A real Shapley value for cooperative games with fuzzy characteristic function

Abstract

There are cooperative situations in which the players have only imprecise expectations about the profit that can be obtained by each coalition. In order to model these situations, several families of games have been introduced in the literature. Games with fuzzy characteristic function are among them. The main problem that arises when dealing with one of these games is how to allocate among the players the total profit derived from the cooperation. In this regard it seems reasonable that the vagueness in the payments of the coalitions will cause vagueness in the payoffs of the players. In fact, the values introduced for these games assign a fuzzy payoff to each player in the game. However, in some of these situations it might be necessary to assign a precise payoff to each player. With this purpose, in this paper we use a well known ranking for fuzzy numbers to introduce a real Shapley value for games with fuzzy characteristic function.