EXAMPLE 7 A Nine-Person Committee

Alice is chairperson of a committee. She has 3 votes, and there are eight other members, each with 1 vote. The quota for passing a measure is a simple majority, 6 of the 11 votes. in our notation, this voting system is .

Each weight-1 member has the same power index. Our strategy is to compute Alice’s Shapley–Shubik power index first. By subtracting her index from 1, we will get the share of power for the remaining members of the committee. Because there are eight of them, and they are equally powerful, we can find the index of each weight-1 member by dividing by 8. Thus, we’ll avoid having to examine all voting permutations.

Alice will be the pivotal voter in any voting permutation where she is in the fourth position, when her vote would bring the total voting weight in favor from 3 to 6; the fifth position, when she would increase the total weight in favor from 4 to 7; or the sixth position, when the total would increase from 5 to 8 with her vote. if she is in any other position in a permutation, another member of the committee will be the pivot.

Because Alice is pivotal in three of the nine positions of a permutation, her Shapley–Shubik index is . The remaining of the voting power is shared equally by the eight other voters. Therefore, each has of the power.

According to the Shapley–Shubik model, Alice is 4 times as powerful as a weight-1 member, although her voting weight is only 3. (Divide her Shapley–Shubik power index by that of a weight-1 voter: .)