# The Banzhaf power index

Banzhaf's (1965) index is also concerned with the fraction of possibilities in which a voter is pivotal, but only considers the *combinations* of such possibilities, rather than *permutations.* In other words, Banzhaf's approach does not worry about the order in which voters are arranged. To distinguish between the Banzhaf approach and the Shapley-Shubik approach, consider any winning coalition *S* u {i}. If *i* leaves this group and *S* turns into a losing coalition, then *i* is said to be *critical* (the ways in which the coalition *S* can be arranged are ignored). The (*normalised* ) *Banzhaf power index* of voter *i* is the number of coalitions in which *i *is critical, divided by the total number of all such coalitions in which some voter is critical. Mathematically, we have:

where *6j* is the number of coalitions in which *i* is critical (also known as the number of *swings* for i). Returning to our previous example, the coalitions in which some player *i* is critical are {*A,*B},{B,C} and {C, A}.

Each shareholder is critical in two coalitions. Therefore, *У ^{N} в.* = 6 and we have:

which, in this particular example, is the same as ф® (this need not always be the case).