= Banzhaf difference identity
{c}
{title2=$\beta_j-\beta_i=2^{-(n-2)}\sum_{T\subseteq N\setminus\{i,j\}}[v(T\cup\{j\})-v(T\cup\{i\})]$}
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 $w_i\leq w_j$, 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.
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