The measure of interaction among players in games with fuzzy coalitions

In this paper, the derivative among participation levels of players in games with fuzzy coalitions, which measures the simultaneous interaction among players with certain participation levels, is defined. Some properties of the derivative of fuzzy game with k-monotonicity are studied. Furthermore, the interaction index among participation levels is established. Based on the assumption that fuzzy characteristic functions are continuous, the integral index of (absolute) interaction among levels is proposed and the integral index of (absolute) interaction among players is defined as well. At last, the Shapley total integral index of interaction among players is constructed, and a series of properties relative to the index are proved.