 Solution

In a transferable utility game, the characteristic function of a coalitional game assigns each coalition $S$ the total payoff $v(S)$ that its members can attain or guarantee together, normalized by $v(\varnothing )=0$. If disjoint coalitions $S,T$ can combine their separate strategies without interference, their union can secure both payoffs, givingThis is a superadditive coalitional game. The pooling assumption explains the intended assertion, but superadditivity is not automatic for an abstract characteristic function: the two-player values $v(\{1\})=v(\{2\})=v(\{1,2\})=1$, with empty value zero, are a counterexample. Thus the word “always” requires either that pooling assumption or a definition restricting the class of games to superadditive ones.

For the given bankruptcy game, the total deficit is $19-15=4$, so the coalition value is $({\sum }_{i\in S}{c}_i-4{)}_{+}$. Its computed values areIt is indeed superadditive: for disjoint claim sums $p,q\ge 0$, $(p+q-4{)}_{+}\ge (p-4{)}_{+}+(q-4{)}_{+}$. If both right-hand terms are positive the difference is $4$; if only one is positive, adding the other claim sum cannot reduce the value; if neither is positive the right side is zero.