Some solutions for generalized games with restricted cooperation

We consider generalizations of Transferable Utility games with restricted cooperation in partition function form and propose their interpretation as allocation problems with several public resources. Either all resources are goods or all resources are bads. Each resource is distributed between points of its set and permissible coalitions are subsets of the union these sets. Each permissible coalition estimates each allocation of resources by its gain/loss function, that depends on the restriction of the allocation on that coalition. Moreover, we define objections at an allocation between permissible coalitions and their feasibility is described by a directed graph $$\varGamma $$ Γ , where permissible coalitions are its vertices. We define new solution concepts (positive envy stable solution w.r.t. $$\varGamma $$ Γ for gain functions and negative envy stable solution w.r.t. $$\varGamma $$ Γ for loss functions). These solutions are simplifications of the generalized kernel of cooperative games and generalize the equal sacrifice solution for claim problems. An allocation belongs to these solutions if there do not exist objections at this allocation between permissible coalitions. We describe completely conditions on $$\varGamma $$ Γ that ensure the existence of these envy stable solutions and conditions that ensure the inclusion of the generalized nucleolus, the generalized anti-nucleolus, and the Wardrop equilibria in these envy stable solutions.