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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.

An imputation is an efficient, individually rational allocation in a transferable utility game. Its total is the grand-coalition value and each player receives at least its singleton value. The set is nonempty exactly when $v(N)\ge {\sum }_{i}v(\{i\})$, and is then compact. This individual-rationality condition distinguishes the nucleolus from the prenucleolus.

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.

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.

For three players, the predecessor-set weights are $1/3$ for sizes zero and two, and $1/6$ for size one. ThusTherefore the Shapley value is $\boxed{(19/3,17/6,17/6)}$, whose coordinates sum to $12$.

For the nucleolus, use imputations $x_1+x_2+x_3=12$, $x_1\ge 4$, $x_2\ge 3$, $x_3\ge 2$. The excess of $\{1,3\}$ is $11-(12-x_2)=x_2-1\ge 2$. Thus the smallest possible largest excess is at least $2$. It is attained when $x_2=3$, $x_3=9-x_1$, and $5\le x_1\le 7$: the remaining proper-coalition excesses then areall at most $2$, while $e_{13}=2$. Conversely, a largest excess of $2$ forces $x_2=3$ and precisely this interval for $x_1$.

On this first-stage face the top excess $2$ is fixed. The next largest excess is $7-x_1\ge 0$, because the other varying excesses are nonpositive and $e_2=0$ is fixed. Its unique minimum is zero at $x_1=7$. No further lexicographic tie remains. Hence the nucleolus is $\boxed{(7,3,2)}$. Its proper-coalition excesses, sorted decreasingly, are $(2,0,0,0,0,-3)$.

Finally, a core allocation would require $x_2\ge 3$ and $x_1+x_3\ge 11$, whose sum contradicts the efficient total $12$. Thus the core of a cooperative game is empty. The game is not superadditive, so neither core nonemptiness nor individual rationality of the Shapley value should be presumed. Under the distinct prenucleolus convention the answer would be $(15/2,2,5/2)$. Balancing $e_2=3-x_2$ and $e_{13}=x_2-1$ gives a first-stage maximum of $1$ and forces $x_2=2$. The remaining first-stage constraints restrict $x_1$ to $[7,8]$. The next largest complaints are $e_{12}=8-x_1$ and $e_{23}=x_1-7$, balanced at $x_1=15/2$, with $x_3=5/2$. This is not an imputation and is not the nucleolus under the definition in (a).

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.