= Solution
The <probabilistic Banzhaf index> satisfies \b[<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:
$$
(v+w)(S\cup\{i\})-(v+w)(S)=[v(S\cup\{i\})-v(S)]+[w(S\cup\{i\})-w(S)].
$$
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.
Back to article page
