= Probabilistic Banzhaf index
{title2=$\beta_i(v)=2^{-(n-1)}\sum_{S\subseteq N\setminus\{i\}}[v(S\cup\{i\})-v(S)]$}
= Banzhaf index
{c}
{synonym}
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.<Weighted Voting Doesn't Work: A Mathematical Analysis by John Banzhaf>[Author's reference page, citation to Rutgers Law Review 19 (1965), starting p. 317]{ref}
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