TU-games with utilities: the prenucleolus and its characterization set

TU-games with utility functions are considered. Generalizations of the prenucleolus, essential coalitions and the core: the \({\textbf{u}}\) $\mathbf{u}$ -prenucleolus, \({\textbf{u}}\) $\mathbf{u}$ -essential coalitions and the \({\textbf{u}}\) $\mathbf{u}$ -core respectively are introduced. We show that \({\textbf{u}}\) $\mathbf{u}$ -essential coalitions form a characterisation set for the \({\textbf{u}}\) $\mathbf{u}$ -prenucleolus in case of games with nonempty \({\textbf{u}}\) $\mathbf{u}$ -core.