Nonsymmetric variants of the prekernel and the prenucleolus

A solution on a class of TU games that satisfies the axioms of the pre-nucleolus or -kernel except the equal treatment property and is single valued for two-person games, is a nonsymmetric pre-nucleolus (NSPN) or -kernel (NSPK). In this paper we investigate the NSPKs and NSPNs and their relations to the positive prekernel and to the positive core. It turns out that any NSPK is a subsolution of the positive prekernel. Moreover, it is shown that an arbitrary NSPK, when applied to a TU game, intersects the set of preimputations whose dissatisfactions coincide with the dissatisfactions of an arbitrary element of any other NSPK applied to this game. This result also provides a new proof of sufficiency of the characterizing condition for NSPKs due to the first author in his PhD thesis published in 1994 as a discussion paper. Any NSPN belongs to "its" NSPK. Several classes of NSPNs are presented, all of them are subsolutions of the positive core. It is shown that any NSPN is a subsolution of the positive core provided that it satisfies the equal treatment property on an infinite universe of potential players. Moreover, we prove that, for any game that has a nonempty anticore, any NSPN selects its prenucleolus as its unique element.