Math 1116, Homework #2, Spring 2013
Name ________________________________________
Instructions: Answer the following questions on separate numbered page(s) and attach work to this
page. Indicate for each problem on this sheet which page the answer to the question can be found, and
indicate the answers clearly in your work (you can circle the answer, for instance).
1. Consider the weighted voting systems listed below. In each case answer the following
questions:
i.
How many players are there?
ii.
What is the total weight of all votes?
iii.
What is the quota?
iv.
What is the smallest value that the quota could possibly take?
v.
What would be the quota is 2/3 of the votes were needed for passage?
vi.
Are there any dictators? If so, which player?
vii.
Does anyone have veto power? If so, which player(s)?
viii.
Are there any dummies? If so, which player(s)?
a.
b.
c.
d.
[31: 10, 10, 8, 7, 6, 4, 1, 1]
[11: 9, 6, 3, 1]
[17: 9, 6, 3, 1]
[30: 10, 9, 8, 8, 8, 6]
2. Consider the weighted voting system in 1c above.
a. List all possible voting coalitions.
b. Of these, which ones are winning coalitions? Identify all critical players in each system by
underlining the critical player(s).
c. Calculate the Banzhaf power distribution for this voting system.
d. List all the sequential coalitions.
e. In each sequential coalition, determine which players are pivotal in each coalition.
f. Find the Shapley-Shubik power distribution for this voting system.
g. Compare your results. Do the two methods predict similar power distributions?
3. The United Nations Security Council consists of 15 members, 10 of which are elected, and 5 of
which are permanent members (this includes the US, France, Britain, Russia and China). For a
resolution to pass, 9 members must support it, and this nine must include all the permanent
members. Set up a weighted voting system to represent the UN Security Council.
4. Set up a weighted voting system to represent the Electoral College vote for President in the
United States. In this system each state gets 1 vote per Senator (every state has 2), and one
vote for each Congressional Representative (apportioned by population, but must add up to
435). In addition, Washington, D.C. gets three votes. A winner of the election must obtain a
majority to win. [Hint: You can obtain a list of how many electoral votes each state has by going
to Google. Be sure that you are using the 2012 electoral count based on the 2010 census and
not older values. Wikipedia should be useful for this.]