 Solution

The probabilistic Banzhaf index satisfies cooperative-game symmetry, the null player axiom and cooperative-game additivity, but not cooperative-game efficiency.

For cooperative-game symmetry, interchangeable players have $v(Ti)=v(Tj)$ for every $T$ omitting them. The Banzhaf difference identity therefore gives equal indices. For a null player, every marginal contribution in the defining average is zero, so its index is zero.

For cooperative-game additivity, each marginal contribution is additive:Averaging both sides over the same $2^{n-1}$ coalitions gives ${\beta }_i(v+w)={\beta }_i(v)+{\beta }_i(w)$.

For failure of cooperative-game efficiency, the indices computed in part (b) sum to $5/4$, whereas the grand-coalition value is one. This is already a counterexample. Independently, a three-player majority game gives each player index $1/2$, also exceeding the total value when summed. Pivotality averaged over independently chosen coalitions does not telescope in the way the ordering-average Shapley value does.