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5 problems, 4 points each, total 20 points
Problem 1. Consider an election with 37 voters and two candidates A and B. In all of the
questions below show your work and explain. (a) How many votes (least number) would A need
to win with the simple majority method? (b) If B is the status quo candidate and the method is
2/3 majority with status quo, what is the least number of votes A would need to win? (c) Same
question for B? (d) In a weighted voting system, with simple majority, where 5 voters get 3
votes, 10 voters get 2 votes and the rest get 1 vote each, and where there is no status quo, what
is the smallest number of voters who need to vote for B in order for B to be the unique winner?
Problem 2. Consider a weighted voting method with 5 voters assigned weights 17, 15, 14, 12,
and 7, respectively, with a simple majority of the weighted votes sufficient for victory. Explain
why this method is, in effect the (unweighted) simple majority method. (Comment: This is
V(33;17,15,14,12,7) in the terminology of Chapter 19).
Problem 3. A small country of uses an Electoral College system (bloc method in language of
Chapter 1) to elect its leader. The chart below shows the population of each province, and each
province’s number of electoral votes. A simple majority of electoral votes is needed to win.
Assume that everyone votes. What is the smallest number of voters who can elect the leader?
What percentage of the total vote is this?
Hint: This problem is similar to part (d) of problem 1. It is also related to the following
“famous” problem: Look at the number of votes each state has in the actual US electoral college
(you can use the “microvotes” spreadsheet posted with class 5. Assume that everybody in each
state votes (of course in reality, not everybody is eligible, and the turnout even among eligible
voters is typically 40% – 60%). Assume winner (of the popular election in each state) takes all its
electoral votes. What is the smallest number of voters necessary to elect a president? What
percentage of the total (population of the US) is this? I will post this problem as an extra credit
Problem 4. Compute the Banzhaf power for (a) V(13;9,7,5,3,1), (b) V(14;9,7,5,3,1), and (c)
V(24;9,7,5,3,1). Compare. I would recommend the method of Alan Taylor.
Problem 5. Consider a “country” with five states A, B, C, D, E that conducts presidential
elections by a system that works like the US Electoral College. The five states have populations
9, 7, 5, 3, and 1, respectively. Compute the probability that a typical voter in each state is a
critical voter in a winning coalition. This is BCPv(1), BCPv(3), BCPv(5), BCPv(7), BCPv(9) in the
terminology of Class 5. There are two ways to do this calculation: (i) an approximate way using
Problem 6. In this problem reconsider Electoral College in the five state country from Problem
4. Use the answer to V(13;9,7,5,3,1) from Problem 4 above to find BCPs(A), BCPs(B), BCPs(C),
BCPs(D), BCPs(E) in the terminology of Class 5. Then combine this with the numbers from
Problem 4 to estimate the Banzhaf power of a voter in each state.
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