I'm studying a first course in Game Theory, I don't understand why in superadditive games, the grand coalition has the highest payoff among all coalitional structures.
If by contraddiction we assume that in a superadditive game $(G,v)$ there exist $S\subset N$ such that $v(S)>v(N)$, then by superadditivity $v(N)\geq v(S)+v(N - S)>v(N)+v(N-S).$
Is there something that force $v(N-S)$ to be positive or null?
In the first answer, it was claimed that a super-additive game has only a non-negative range and that the grand coalition must have the greatest value. Both claims are not correct. Of course, if the game has non-negative coalition values, then the grand coalition must have greatest value. And yes, almost all relevant economical games are mapping into the non-negative range. However, the definition of super-additivity allows that the value of a coalition can be negative, but then the grand coalition cannot have anymore the greatest value. As a counter-example consider the following three-person game
$v(\emptyset)=0,v(\{1\})=-1,v(\{2\})=0,v(\{3\})=0,v(\{1,2\})=7,v(\{1,3\})=7,v(\{2,3\})=13,v(N)=12.$
This game is super-additive and the greatest value of $13$ is assigned to coalition $\{2,3\}$. Furthermore, the only convention imposed on a TU game is that the empty set will be assigned with the zero value.