On the symmetry relation between different characteristic functions for additively separable cooperative games

We analyze 4 characteristic functions \begin{document}$ V^\alpha $\end{document}, \begin{document}$ V^\delta $\end{document}, \begin{document}$ V^\zeta $\end{document}, and \begin{document}$ V^\eta $\end{document}, and give a necessary condition for these functions to satisfy the relation \begin{document}$ V^\alpha - V^\delta = V^\zeta - V^\eta $\end{document} for all coalitions \begin{document}$ S $\end{document}. To do so, we define and formally analyze the class of additively separable games. It is shown that many important types of games, both static and dynamic, belong to this class.