On Scarf's theorem for generalized cooperative games with external relations

In this paper, we consider a generalization of cooperative games to the case where a coalition can distribute the earned utility not only among its members but also to other players. In particular, we consider an example where coalitions are required to share their winnings with non-contributing players. For these generalized games, we also provide an analogue of Scarf’s theorem. It turns out that in this generalization, the existence of a non-empty core is closely related to a homotopy invariant of covers defined by the cooperative game.