MGF 1107

SUPPLEMENTAL HOMEWORK/ CHAPTER 11

1. Give an example of a weighted voting system with 4 voters that can be described by:

a) dictatorship

b) unanimity

c) majority rules

2. For each of the following weighted voting systems with 3 voters, determine if the system is equivalent to a dictatorship, unanimity, majority, clique, or chair veto (see the top of p. 432).

a) [8: 5, 3, 2]    b) [7: 8, 4, 2]    c) [8: 4, 2, 2]    d) [8: 7, 4, 3]    e) [8: 4, 4, 4]

3. Consider the weighted voting system [q: 16, 8, 2]. 

a) What are the possible values of q?

b) What values of q result in a dictatorship?

4.  A weighted voting system has a Shapley-Shubik power index of .  Find x.

5. The Electoral College is a weighted voting system with 51 voters (the 50 states and the District of Columbia).  Write an expression that could be used to calculate each of the following.  You do not have to multiply out.

a) the number of possible coalitions for calculating the Banzhaf power index.

b) the number of sequential coalitions for calculating the Shapley-Shubik power index.

6. A weighted voting system has 3 voters with weights 3, 2, and 2.

a) Find the Banzhaf power index, expressed as percentages, if a majority of votes is needed to win a vote.

b) A fourth voter is added to the system whose weight is one.  If a majority is still needed to win a vote, find the Banzhaf power index, expressed as percentages, of this new 4-person weighted voting system.

c) Parts (a) and (b) of this problem illustrate what surprising characteristic of weighted voting bodies?

7. A corporation has 120 shares of stock outstanding. There are 100 shareholders who own one share each, and one shareholder who owns 20 shares. To pass an issue, owners representing 61 shares must vote yes. Determine the Shapley-Shubik index of each shareholder.

8. Compute the Shapley-Shubik power index for each of the following weighted voting systems.

a) [7: 5, 2, 2]    b) [7: 4, 4, 2]    c) [7: 4, 4, 3]    d) [7: 8, 2, 1]    e) [7: 4, 4]

9. You are asked to find the Banzhaf power index of a 3-voter weighted voting system and you come up with the answer (4, 1, 1). Without even knowing the quotas or weights, how does one know your answer must be wrong?

10. A weighted voting system has 5 voters. Write the Banzhaf power index of the system as percentages if the system is:
a) a dictatorship
b) majority rules
c) unanimity

Answers:

1a) Answers vary.  Some possibilities are [4: 4, 1, 1, 1], [4: 5, 1, 1, 1] and [8: 9, 3, 2, 1].

1b) Answers vary.  Some possibilities are [4: 1, 1, 1, 1], [8: 2, 2, 2, 2] and [7: 2, 2, 2, 2].

1c) Answers vary.  Some possibilities are [3: 1, 1, 1, 1], [5: 2, 2, 2, 2] and [12: 6, 5, 4, 4].

3a) 14, 15, 16, 17, ...., 25, 26                  3b) 14, 15, and 16

4)

5a) 2⁵¹              2b) 51!

6a) (33 1/3%, 33 1/3%, 33 1/3%)       

6b) (41 2/3%, 25%, 25%, 8 1/3%)

6c) The paradox of new members.

7) The large shareholder has an index of . Each small shareholder has an index of .

8a) 8b)   8c)   8d) (1, 0, 0)     8e)

9) The numbers in a Banzhaf power index must always be even.

10a) (100%, 0%, 0%, 0%, 0%) since, in a dictatorship, the dictator has all the power and all the other voters are dummies.
10b) (20%, 20%, 20%, 20%, 20%) since, when majority rules, all the voters have equal power.
10c) (20%, 20%, 20%, 20%, 20%) since, when a unanimous vote is required, all the voters have equal power.