21. Agnes, Boris, and Carla, weight-1 voters in the committee described in Exercise 20, cede their votes to a fourth weight-1 voter, Essie. Use the following steps to recalculate the power indices.
21.
(a)
(b) Three voters must precede Gerry in the permutation. One must be Essie or Franklin, and two must be chosen from the four weight-1 voters other than Gerry, who already has been placed in the middle of the permutation. The number of ways to select these voters is . Once selected, there are ways to order them. The remaining three voters, who will be after Gerry in the permutation, can also be ordered in 6 ways, so the number of permutations in which Gerry is pivotal is .
(c) Gerry’s Shapley-Shubik index is , and each of the other four weight-1 voters has the same index. The sum of the indices of the weight-1 voters is . The remaining of the power is shared equally by Essie and Franklin, so the Shapley-Shubik power index of each is .
A-28
(d) Such a coalition would include either Essie or Franklin and two weight-1 voters in addition to Gerry. There are such coalitions. That is Gerry’s Banzhaf power index.
(e) To be a winning coalition, at least 3 votes are needed in addition to Franklin’s. For Franklin to be a critical voter, the coalition must not include more than 6 votes in addition to Franklin’s. There are ways to assemble three weight-1 voters to get 3 more votes to join Franklin. To get 4 more votes, Franklin could be joined by Essie alone or by four weight-1 voters; there are of these coalitions. To get 5 more votes, one needs either all of the weight-1 voters or Essie and four weight-1 voters. There are 6 ways to assemble a coalition of this type. To get 6 more votes, we would need Essie and two weight-1 voters; there are such coalitions.
(f) Franklin’s Banzhaf power index, which is the same as Essie’s, is , and we have noted that Gerry, and each other weight-1 voter, has an index of 12.