The prenucleolus of games with restricted cooperation

A cooperative game with restricted cooperation is a triple $(N,v,\Omega)$, where $N$ is a finite set of players, $\Omega\subset2^N$, $N\in\Omega$ is a collection of feasible coalitions, $v\colon\Omega\to\mathbb R$ is a characteristic function. The definition implies that if $\Omega=2^N$, then the game $(N,v,\Omega)=(N,v)$ is a classical cooperative game with transferable utilities (TU). The class of all games with restricted cooperation $\mathcal G^r$ with an arbitrary universal set of players is considered. The prenucleolus for the class is defined in the same way as for classical TU games. Necessary and sufficient conditions on a collection $\Omega$ providing existence and singlevaluedness of the prenucleoli for the class $\mathcal G^r$ are found Axiomatic characterizations of the prenucleolus for games with two-type collections $\Omega$ generated by coalitional structures are given.