Undominated nonnegative excesses and core extensions of transferable utility games

•The core extension of cooperative games is studied.•Undominated nonnegative excesses are defined for the core extension.•Two new extended cores are introduced.•Their relationships with the least core and the positive core are investigated.•Procedures for calculating the new extended cores are provided with examples. The extension of the core for cooperative games with transferable utility is studied. By considering only nonnegative coalitional excesses, we introduce the concept of undominated nonnegative excess vectors and demonstrate that some well-known extended cores can be defined based on this concept. Moreover, we propose two new core extensions: the min-max tax core derived by minimizing the maximal tax paid by all players and the lexicographical min-max tax core derived by lexicographically minimizing the taxes paid by all players in all feasible coalition structures for the stabilization of the grand coalition. Both of the new extended cores coincide with the core when the latter is not empty. We demonstrate that the min-max tax core is different from the least core but coincides with it for super-additive games with empty core, and the lexicographical min-max tax core is different from the positive core but coincides with the latter for all super-additive games. Our study provides a new and taxation interpretation of the least core and the positive core for super-additive games.