Question 11.67

37. A nine-member committee has a chairperson and eight ordinary members. A motion can pass if and only if it has the support of the chairperson and at least two other members, or if it has the support of all eight ordinary members.

  1. image Find an equivalent weighted voting system.
  2. Determine the Banzhaf power index.
  3. Determine the Shapley-Shubik power index.
  4. image Compare the results of parts (b) and (c). Do the power indices agree on how power is shared in this committee?

37.

(a)

(b) The Banzhaf power index of the chairperson is 246, and each ordinary member’s index is 8. There are critical votes in all. According to the Banzhaf model, the chair has of the power, and each of the other members has 2.90% of the power.

(c) The chair is pivot in a permutation when located in positions 3 through 8 of a permutation; that is, in 6 of the 9 locations of a permutation. Hence the Shapley-Shubik power index of the chair is . The other members share the remaining of the power, so the Shapley-Shubik power index of each is .

(d) The two models do not agree closely in this case.