EXAMPLE 9 Alice’s Power After the Revolt

Alice will be pivot in any permutation where the following occurs:

  • She is second, with Bill first. There are permutations of this type, since the 1-vote members, who come after Alice, can be in any order.
  • She is third, with Bill and a 1-vote member first. There are of these permutations, because Bill could be in one to two positions, and the 1-vote members can be in any order.
  • She is fourth, with Bill and a 1-vote member following. Again, there are 48 permutations in this category.
  • She is fifth, with Bill last. There are of these permutations, because there are ways to order the four weight-1 voters who came before Alice.

All told, Alice is pivot in 144 permutations, and her Shapley–Shubik power index is . The revolt has reduced her power considerably. The four remaining 1-vote members of the committee control of the power, so the remaining of the power—twice Alice’s—is Bill’s.