Nucleolus

When the imputation set is nonempty, the nucleolus is the unique imputation lexicographically minimizing the vector of excesses of a coalition sorted decreasingly. It minimizes the largest complaint first, then subsequent complaints among ties. It can be computed through successive linear programs that minimize the current maximum excess and retain constraints fixed by prior stages. Its uniqueness is the standard nucleolus theorem. If the core of a cooperative game is nonempty, the nucleolus belongs to it; an empty core does not preclude a nucleolus.

The prenucleolus uses the same sorted excesses of a coalition but minimizes over all efficient payoff vectors, without imposing individual rationality. It can differ from the nucleolus when coalition values are not superadditive. Any computation must state which domain is used rather than silently relaxing imputation inequalities.