I need to show that
A game is weakly super additive iff it is strategically equivalent to a zero-normalized monotonic game.
A weakly super additive game $\langle N,v \rangle$ is one that for which $$ v(S) + v(\{j\}) \leq v(S \cup \{j\}); \quad \forall S\subseteq N, \, j\notin S.$$
A game $\langle N,u \rangle$ is zero-normalized monotonic whenever
$$ u(S) \leq u(T) \quad \text{for $S \subseteq T \subseteq N$, and} \quad u(i)=0,\, \forall i\in N $$
Now we call two games $\langle N,v \rangle$ and $\langle N,u \rangle$ strategically equivalent if there is a $q\in \mathbb{R}$ and $\alpha \in \mathbb{R}^N$ such that $$ v(S) = q\,u(S) +\sum_{i\in S} \alpha(i).$$ So I think our problem reduces to finding such $\alpha$ and $q$ but so far I have no clue on how to do this. Any help will be appreciated!