Characteristic functions for cooperative interval games

Games where players' payoffs are given by intervals, instead of scalars, provide a conceptually attractive framework to account for uncertainty and vagueness in decision-making processes. In a cooperative game, one needs to determine the characteristic function values for all possible coalitions. In this paper, we extend the classical $ \alpha $ and $ \beta $ characteristic functions (CFs), initially defined for scalar payoffs, to cooperative interval games. Both characteristic functions are based on a solution of zero-sum interval games with the coalition being the maximizer player and the anti-coalition the minimizer player. We propose an algorithm to define the interval values of a cooperative game in the form of $ \alpha $ and $ \beta $ CFs and illustrate its use on an example with three players. Further, we discuss some properties of cooperative interval games and calculate an interval Shapley value. As expected, different characteristic functions lead to different Shapley values.