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

Abstract

<p style='text-indent:20px;'>We analyze 4 characteristic functions <inline-formula><tex-math id="M1">\begin{document}$ V^\alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ V^\delta $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ V^\zeta $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M4">\begin{document}$ V^\eta $\end{document}</tex-math></inline-formula>, and give a necessary condition for these functions to satisfy the relation <inline-formula><tex-math id="M5">\begin{document}$ V^\alpha - V^\delta = V^\zeta - V^\eta $\end{document}</tex-math></inline-formula> for all coalitions <inline-formula><tex-math id="M6">\begin{document}$ S $\end{document}</tex-math></inline-formula>. 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.</p>