MGF 1107/ Classroom examples/ Chapter 11

1. Suppose a county commission consists of three members, one representing each of the three cities in the county. Voting power on the commission is proportional to the population of the cities with the commissioner from city A getting 49 votes, the commissioner from city B getting 40 votes, and the commissioner from city C getting 11 votes. Since there are 100 total votes, 51 votes are required to carry any measure. Which commissioner has the most power?

2. What is wrong with the following weighted voting system? [21: 6, 5, 5, 3]

3. What is wrong with the following weighted voting system? [8: 6, 5, 5, 3]

4. A weighted voting system has 5 voters with weights 1, 1, 2, 2, and 3. Find all possible values of q.

5. A corporation has four partners with weights 10, 8, 7 and 1. The bylaws of the corporation require that two-thirds of the votes are needed to pass a motion. What is the quota?

6. Consider the weighted voting system [10: 11, 4, 3]. Describe the power of each voter.

7. Consider the weighted voting system [3: 2, 1, 1]. Find all coalitions with enough weight to prevent a measure from passing.

8. For the weighted voting system [21: 11, 10, 9, 1], list:
a) all winning coalitions containing the first voter

b) all blocking coalitions containing the first voter

c) all voters that have veto power

d) all dummy voters

9. Returning to the weighted voting system [3: 2, 1, 1] we studied two examples ago, find the number of winning and blocking coalitions for which each voter is a critical voter.

10. Recall the first example of a weighted voting system we looked at [51: 49, 40, 11]. We observed that all three voters had equal power since no vote can be won without support of any 2 voters. Let’s make this notion of power more precise by calculating its Banzhaf power index.

11. Suppose a fourth voter with weight 4 is added to the voting system in the previous problem. Find the Banzhaf power index of this new 4-voter system if a majority of votes is still needed to win a vote.

12. In which of the 5 types of 3-voter weighted systems do all the voters have equal power?

13. Show that [4: 2, 2, 1] and [6: 3, 3, 2] are equivalent.

14. Referring to the last example, which are the critical voters in each winning coalition?

15. Which of the five possible 3-voter weighted systems is [10: 9, 4, 4] equivalent to?

16. The United Nations Security Council consists of 15 voting nations. Five of them are permanent members: Great Britain, Russia, France, China, and the United States. The other 10 member nations serve two-year terms on a rotating basis. To pass a motion in the Security Council all 5 permanent members must vote yes in addition to at least 4 of the 10 non-permanent members. A winning coalition consists of all 5 permanent plus 4 or more non-permanent members.

a) How many different winning coalitions are there?

b) How many different minimal winning coalitions are there?

c) What is the total number of times all voters are critical?

d) What is the Banzhaf power index of a permanent member, expressed as a percentage?

e) What is the Banzhaf power index of a non-permanent member, expressed as a percentage?

17. Compute the Shapley-Shubik power index for the weighted voting system [4: 3, 2, 1].

18. Calculate the Shapley-Shubik index for the weighted voting system [6: 4, 2, 2, 2].

19. The North Central Florida Regional Planning Council has 36 members, representing both counties and municipalities. The city of Gainesville is directly represented by 8 members and indirectly by another 5 members who represent Alachua County and live in Gainesville. Assume these 13 members vote as a bloc. A majority is required to win a vote.

a) What percent of the council votes does this bloc have?

b) According to the Shapley-Shubik index, what percent of the power does this bloc have?

c) According to the Shapley-Shubik index, what percent of the power does each of the remaining voters have?