Two games $(N,v)$ and $(N,w)$ are strategically equivalent if there exists $\alpha$ and $\beta \in \mathbb{R}^N$ such that $$w(S)=\alpha v(S)+\beta(S)\text{ for all } S\subseteq N.$$ I do not follow in the snippet below why the $0-$monotonic strategically equivalent to $(N,v)$ is unique as they claim below:
The $0$-normalized game is obtained by subtracting from each $v(S)$ the number $\sum_{i\in S} v(i)$. This leads to a unique game. Of course, for every game with all $v(i)=0$ there are many games that have this particular game as their $0$-normalized version...