= Bankruptcy game
{title2=$v(S)=\max\{0,a-\sum_{i\notin S}c_i\}$}
With nonnegative claims $c_i$ and an estate $0\leq a\leq\sum_i c_i$, a coalition's guaranteed worth is the estate left after all outside claims are met, or zero if those claims exhaust it. Writing the total deficit $d=\sum_i c_i-a$ gives $v(S)=(\sum_{i\in S}c_i-d)_+$. For disjoint coalitions, $(p+q-d)_+\geq(p-d)_++(q-d)_+$, so this is a <superadditive coalitional game>. Its <core of a cooperative game> consists exactly of efficient allocations with $0\leq x_i\leq c_i$: the upper bounds are equivalent to the complementary-coalition constraints, and the lower bounds together with efficiency imply every other coalition constraint.
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