The prenucleolus for games with restricted cooperation
A game with restricted cooperation is a triple (N,v,Ω), where N is a finite set of players, Ω⊂2N is a nonempty collection of feasible coalitions such that N∈Ω, andv:Ω→R is a characteristic function. The definition implies that if Ω=2N, then the game(N,v,Ω)=(N,v) is the classical transferable utility (TU) cooperative game.
The class of all games with restricted cooperation Gr with an arbitrary universal set of players is considered. The prenucleolus and the prekernel for games with restricted cooperation are defined in the same way as the prenucleolus and the prekernel for classical TU games. Necessary and sufficient conditions for the collection Ω to imply the single-valuedness of the prenucleolus are obtained. Axiomatic characterizations of the prenucleolus and of the prekernel for the class with a balanced collection of feasible coalitions Ω are given.
In the collection Ω there may be identical players belonging to the same coalitions. In that case, the set of symmetric preimputations is defined as those where identical players have equal payoffs. The symmetric prenucleolus, being the nucleolus w.r.t. the set of symmetrical preimputations, is defined and characterized.