Banzhaf difference identity (${\beta }_j-{\beta }_i=2^{-(n-2)}{\sum }_{T\subseteq N\setminus \{i,j\}}[v(T\cup \{j\})-v(T\cup \{i\})]$)

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.