 Solution

Where its denominator is nonzero, normalization enforces cooperative-game efficiency and preserves cooperative-game symmetry and the null player axiom. The violated axiom is cooperative-game additivity.

For an exact counterexample on three players, let $v$ be the majority game, and let $w$ be the dictatorship in which a coalition wins precisely when it contains player one. Their probabilistic Banzhaf indices arewith $v(N)=w(N)=1$. HenceUnnormalized indices add, so $\beta (v+w)=(3/2,1/2,1/2)$. Their sum is $5/2$, while $(v+w)(N)=2$. ThereforeThe sum game is a valid transferable utility game, although it is not binary; that is the natural domain of the additivity axiom. The failure comes from a game-dependent scaling factor. A zero denominator, as for the zero game, makes the displayed normalization undefined unless a separate convention is supplied; none of the counterexample's denominators vanish.