Question 11.75
45. An alumni committee consists of 3 rich alumni and 12 recent graduates. To pass a measure, a majority, including at least 2 of the rich alumni, must approve.
- Describe the minimal winning coalitions.
- Suppose this is a weighted voting system. Give the recent graduates each a weight of 1, and let be the weight of each of the rich alumni. Find the total weight of each minimal winning coalition. For example, the coalition with all three rich alumni and five of the recent graduates would have weight 3r + 5.
- Compare the total weight of the losing coalition consisting of all recent graduates and one rich alumna with the total weight of one of the minimal winning coalitions to show that .
- Compare the total weight of the largest losing coalition that includes all three rich alumni with the same winning coalition you used in part (c). Does any weight that works here also satisfy the inequality in part (c)?
- Is this voting system equivalent to a weighted voting system?
45.
(a) The winning coalitions must have a total of at least 8 members, including at least 2 rich alumni. The minimal winning coalitions will have exactly 8 members: either 3 rich alumni and 5 recent graduates or 2 rich alumni and 6 recent graduates.
(b) The total weight of a coalition including 3 rich alumni and 5 recent graduates would be , and the total weight of a coalition with 2 rich alumni and 6 recent graduates would be .
(c) This losing coalition has total weight of . Its total weight is less than the total weight of the winning coalition with 2 rich almuni and 6 recent graduates. Therefore, . Subtract from both sides to get .
(d) The largest losing coalition with 3 rich alumni would include 4 recent graduates, and its total weight would be ; thus . Subtract from both sides to get ; in other words, is less than 2. But the inequality from part (c) says is greater than 6. No number can be both more than 6 and less than 2, so the two inequalities are inconsistent.
(e) No