This paper presents two characterizations of the core on the domain of all NTU games. One is based on consistency with respect to ‘complement‐reduced game’ and converse consistency with respect to ‘max‐reduced game’. The other is based on consistency with respect to ‘max‐reduced game’ and weak converse consistency with respect to ‘complement‐reduced game’. Besides, we introduce an alternative definition of individual rationality, we name conditional individual rationality, which is compatible with non‐emptiness. We discuss axiomatic characterizations involving conditional individual rationality for the core.
