= Normalized Banzhaf index
{title2=$\beta'_i(v)=v(N)\beta_i(v)/\sum_j\beta_j(v)$}
= Normalised Banzhaf index
{synonym}
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.
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