EXAMPLE 15 Banzhaf Power in Alice’s Committee

Alice’s Banzhaf index has already been determined: It is .

Bill, a weight-1 participant, will be a critical voter in any winning coalition with exactly 6 votes. He will need, in addition to his own vote, Alice’s vote and the votes of two other participants, or, without Alice, he will need the votes of five other participants. He must get the two or the five votes from the seven other weight-1 participants—he can’t vote twice!

There are winning coalitions in which Bill casts a critical vote and Alice is included, and winning coalitions where Bill casts a critical vote and Alice is not included. Bill’s Banzhaf power index is . The other participants have the same index.

By the Banzhaf model, Alice is times as powerful as Bill. This is reasonably close to the Shapley–Shubik model’s assessment that Alice is 4 times as powerful as Bill.