Characterizations of the \(\textbf{u}\)-prenucleolus by dually-\(\textbf{u}\)-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-\(\textbf{u}\)-essential coalitions and show that they characterize the \(\textbf{u}\)-prenucleolus of \(\textbf{u}\)-balanced games. Additionally, we demonstrate that the intersection of \(\textbf{u}\)-essential and dually-\(\textbf{u}\)-essential coalitions also forms a characterization set for the \(\textbf{u}\)-prenucleolus, provided that the \(\textbf{u}\)-least-core is a proper subset of the \(\textbf{u}\)-core.