A game \(v\in G^N\) is: 0-1 normalized if \(v(i)=0\) for all \(i\in N\) and \(v(N)=1\);
0-0 normalized if \(v(i)=0\) for all \(i\in N\) and \(v(N)=0\);
and 0-(-1) normalized if \(v(i)=0\) for all \(i\in N\) and \(v(N)=-1\).
If \(v(N)>\sum_{i\in N}v(i)\), the 0-1 normalized game of \(v\), \(v_{0,1}\in G^N\), is defined by
$$v_{0,1}(S)=\frac{v(S)-\sum_{i\in S}v(i)}{v(N)-\sum_{i\in N}v(i)}$$ for all \(S\in 2^N\).
If \(v(N)<\sum_{i\in N}v(i)\), the 0-(-1) normalized game of \(v\), \(v_{0,-1}\in G^N\), is defined by
$$v_{0,-1}(S)=-\frac{v(S)-\sum_{i\in S}v(i)}{v(N)-\sum_{i\in N}v(i)}$$ for all \(S\in 2^N\).
If \(v(N)=\sum_{i\in N}v(i)\), the 0-0 normalized game of \(v\), \(v_{0,0}\in G^N\), is defined by
$$v_{0,0}(S)=v(S)-\sum_{i\in S}v(i)$$ for all \(S\in 2^N\).