I'm currently doing some exercises on cooperative games. To be honest I can't really find any examples of (0-1)-normalization of games. Consider the game with $N=\lbrace I,II,III \rbrace$ and characteristic function as follows: $$v(\emptyset)=0,\ v(I)=1,\ v(II)=2,\ v(III)=3,$$ $$v(\lbrace I,II\rbrace)=5,\ v(\lbrace I, III\rbrace)=7,\ v(\lbrace II,III\rbrace)=9$$ $$v(\lbrace I,II,II\rbrace)=12$$
So the (0-1)-normalization of such game should satisfy following conditions:
- $v(N)=1$
- $0\leq v(S)\leq 1$ for every $S\subseteq N$
- $v(i)=0$ for $i\in\lbrace1,2,3\rbrace$
So we know that in this normalized game $$v'(\emptyset)=v'(I)=v'(II)=v'(III)=0,$$ $$v'(\lbrace I,II,II\rbrace)=1$$
EDIT: Okay I think I figured this out using formula: $$v'(S)=\frac{v(S)-\sum_{i\in S } v(i)}{v(N)-\sum_{i\in N } v(i)}.$$ By this I calculated: $$v'(\lbrace I,II\rbrace)=1/3,\ v'(\lbrace I, III\rbrace)=1/2,\ v'(\lbrace II,III\rbrace)=2/3$$ Is that correct?