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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.