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).

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