Excess of a coalition ($e(S,x)=v(S)-{\sum }_{i\in S}{x}_i$)

The excess measures a coalition's complaint against an allocation: a positive excess means its own attainable value exceeds its assigned total. The core of a cooperative game requires every excess to be nonpositive at an efficient allocation. The nucleolus lexicographically minimizes the decreasingly sorted excess vector over imputations.