The bounded core for games with restricted cooperation

A game with restricted cooperation is a triple $(N,v,\Omega)$, where $N$ is a finite set of players, $\Omega\subset2^N$ is a non-empty collection of feasible coalitions such that $N\in\Omega$, and $v\colon\Omega\to\mathbb R$ is a characteristic function. Unlike the classical TU games the cores for games with restricted cooperation may be unbounded. Recently Grabisch and Sudhölter [9] proposed a new concept – the bounded core – that for assigns to a game $(N,v,\Omega)$ the union of all bounded faces of the core. The bounded core can be empty even the core is not empty. An axiomatization of the bounded core for the class $\mathcal G^r$ with restricted cooperation is given with the help of axioms efficiency, boundedness, bilateral consistency, a weakening of converse consistency, and ordinality. The last axiom states that the property of a payoff vector to belong to a solution only depends on the signs of the corresponding components of the excess vectors, but not on their values. Another axiomatization of the core is given for the subclass $\mathcal G^r_{bc}\subset\mathcal G^r$ of games with non-empty bounded cores. The characterizing axioms are non-emptiness, covariance, boundedness, bilateral consistency, and superadditivity.