Characterizations of the $$\textbf{u}$$ u -prenucleolus by dually- $$\textbf{u}$$ u -essential coalitions

We extend the theory of TU-games with utility functions, which is a generalization of TU-games with restricted cooperation, to include dual games. By using the theory of dual games, we define dually- $$\textbf{u}$$ u -essential coalitions and show that they characterize the $$\textbf{u}$$ u -prenucleolus of $$\textbf{u}$$ u -balanced games. Additionally, we demonstrate that the intersection of $$\textbf{u}$$ u -essential and dually- $$\textbf{u}$$ u -essential coalitions also forms a characterization set for the $$\textbf{u}$$ u -prenucleolus, provided that the $$\textbf{u}$$ u -least-core is a proper subset of the $$\textbf{u}$$ u -core.