Bankruptcy game ($v(S)=\max \{0,a-{\sum }_{i\notin S}{c}_i\}$)

With nonnegative claims $c_i$ and an estate $0\le a\le {\sum }_i{c}_i$, 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, $(p+q-d{)}_{+}\ge (p-d{)}_{+}+(q-d{)}_{+}$, so this is a superadditive coalitional game. Its core of a cooperative game consists exactly of efficient allocations with $0\le x_i\le c_i$: the upper bounds are equivalent to the complementary-coalition constraints, and the lower bounds together with efficiency imply every other coalition constraint.