Normalized Banzhaf index (${\beta }_i^{\prime }(v)=v(N){\beta }_i(v)/{\sum }_j{\beta }_j(v)$)

The normalized Banzhaf index rescales the probabilistic Banzhaf indices to sum to the grand-coalition value, whenever the denominator is nonzero. It preserves cooperative-game symmetry and assigns zero to a null player, but generally fails cooperative-game additivity. For example, add the three-player majority game to a dictatorship of player one. Their normalized indices are $(1/3,1/3,1/3)$ and $(1,0,0)$, whereas the sum game has normalized index $(6/5,2/5,2/5)$, not their vector sum. The sum game has grand-coalition value two, which must be retained in the normalization.