Cooperative games on intersection closed systems and the Shapley value

A situation in which a finite set of agents can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. In the literature, various models of games with restricted cooperation can be found, in which only certain subsets of the agent set are allowed to form. In this article, we consider such sets of feasible coalitions that are closed under intersection, i.e., for any two feasible coalitions, their intersection is also feasible. Such set systems, called intersection closed systems, are a generalization of the convex geometries. We use the concept of closure operator for intersection closed systems and we define the restricted TU-game taking into account the limited possibilities of cooperation determined by the intersection closed system. Next, we study the properties of this restricted TU-game. Finally, we introduce and axiomatically characterize a family of allocation rules for games TU-games on intersection closed systems, which contains a natural extension of the Shapley value.