Question 11.47

image 17.This exercise is intended for a group of 2 to 4 students. If the quota for the voting system in Exercise 16 increases, you can quickly modify the table you made by reducing the extra votes—ignore a coalition when its extra votes become negative. As the number of extra votes decreases, more of a coalition’s voters will be critical—until the coalition switches from winning to losing.

For example, is a winning coalition with 0 extra votes when the quota . If , it is a losing coalition, so and are no longer credited with critical votes for that coalition. On the other hand, and gain critical votes in and , respectively.

Determine the Banzhaf index for this system with the following quotas:

  1. 52
  2. 55
  3. 56
  4. 57
  5. 71
  6. 76
  7. 80

17.

For each quota, the winning coalitions, with the number of extra votes, are listed. Asterisks identify the critical voters. The notation for Banzhaf power indices is such that (5,3,3,1) designates 5 for , 3 for and , and 1 for .

The abbreviation “e.v.” means “ extra votes.”

(a)

(b)

(c)

(d)

(e)

(f)

(g)