Generalized Kernels and Bargaining Sets for Families of Coalitions

For a fixed collection of subsets of the player set, two generalizations of Aumann–Maschler theory of the bargaining set for cooperative TU-games, where objections and counter-objections are permitted only between elements of this collection, and corresponding generalizations of the kernel are considered. We describe conditions on the fixed collection of coalitions that ensure existence of corresponding sets of imputations for all $n$-person games.
All sufficient conditions are based on a generalization of [Peleg]. Here relations are defined not on the player set, but on the set of coalitions, and acyclicity is not assumed. Obtained sufficient conditions are also necessary for both generalized bargaining sets if the number of players is no more than five and for one of generalized kernels.