Probabilistic Banzhaf index (${\beta }_i(v)=2^{-(n-1)}{\sum }_{S\subseteq N\setminus \{i\}}[v(S\cup \{i\})-v(S)]$)

The probabilistic Banzhaf index averages a player's marginal contribution over all coalitions not containing that player, each given equal weight. In a simple cooperative game, it is the probability of changing a losing coalition into a winning one when every other player joins independently with probability $1/2$. Its sum over players need not equal the grand-coalition value: in a three-player majority game each player has index $1/2$, giving total $3/2$ despite $v(N)=1$. This convention differs from both the raw count of pivotal coalitions and the normalized Banzhaf index.

The name refers to John Banzhaf, whose 1965 paper addresses the mathematical analysis of weighted voting.^[ref]

For distinct players $i,j$, expand each probabilistic Banzhaf index by whether the sampled coalition contains the other player. The terms $v(T)$ and $v(T\cup \{i,j\})$ cancel on subtraction, leaving the displayed identity. In a weighted voting game with $w_i\le w_j$, every summand is nonnegative, proving that the index is monotone in the player's weight. Equal weights give equal indices, although different weights can also have equal indices.

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.