The Shapley value, one of the most common solution concepts in Operations Research applications of cooperative game theory, is defined and axiomatically characterized in different game-theoretical models. In this paper, we focus on the Shapley value for cooperative games where the set of players is finite and the coalition values are compact intervals of real numbers. In this study, we study the properties of the interval Shapley value on the class of size monotonic interval games, and axiomatically characterize its restriction to a special subclass of cooperative interval games by using fairness property, efficiency and the null player property. Further, we introduce the interval Banzhaf value and the interval egalitarian rule. Finally, the paper ends with a conclusion and an outlook to future studies.