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For an imputation $y$, the excess of a coalition is $e(S,y)=v(S)-{\sum }_{i\in S}{y}_i$. Arrange these excesses in decreasing order. The nucleolus is the imputation that lexicographically minimizes this ordered vector: minimize the largest complaint first, then the next largest among ties, and so on. The empty and grand-coalition excesses are fixed at zero for every efficient allocation, so they can be omitted without changing the lexicographic comparison.

For every efficient $y$ the three pair excesses have sumHence at least one pair has excess at least $-4/3$, and the largest proper-coalition excess is at least $-4/3$. At the proposed allocation all three pair excesses equal $-4/3$, while all three singleton excesses equal $-8/3$. It therefore attains the best possible largest proper-coalition excess.

Moreover, any allocation attaining this bound must have all three pair excesses equal $-4/3$, since they are individually at most that value and sum to $-4$. This forceswhose unique solution is $y=x$. It is already an imputation, so the first nonconstant lexicographic stage has a unique minimizer; there are no later choices to resolve. Therefore

For a transferable utility game with $v(\varnothing )=0$, an efficient allocation satisfies ${\sum }_{i\in N}{x}_i=v(N)$. An imputation is efficient and individually rational, $x_i\ge v(\{i\})$. Write $x(S)={\sum }_{i\in S}{x}_i$ and define the excess of a coalition as $e(S,x)=v(S)-x(S)$.

The core of a cooperative game isIt consists of allocations immune to a coalition's blocking: no coalition can obtain more for its members by leaving. The singleton inequalities imply individual rationality. The core may be empty.

The nucleolus, when the imputation set is nonempty, is the unique imputation that lexicographically minimizes the list of coalition excesses arranged from largest to smallest. It first minimizes the largest complaint, then the second largest among ties, and so on. Including the empty and grand coalitions adds constant zeros and does not change the solution. Minimization on efficient allocations without individual rationality instead defines the prenucleolus, a different convention that matters for this paper's game.

The Shapley value is the average marginal contribution of each player over all uniformly ordered player arrivals:Exactly $|S|!(n-|S|-1)!$ orderings have $S$ as the predecessor set of $i$. This is an average-contribution fairness rule, rather than a blocking-stability condition; it need not be individually rational for an arbitrary game.