math

Math Assignment

Weighted Voting

3. Consider the weighted voting system [q: 7,5,3,1,1] a. What is the smallest value that the quota q can take? b. What is the largest value that the quota q can take? c. What is the value of the quota if at least two-thirds of the votes are required to pass a motion?

5. Consider the weighted voting system [13: 13, 6, 4, 2] a. Identify the dictators, if any. b. Identify players with veto power, if any c. Identify dummies, if any.

12. Find the Banzhaf power distribution of the weighted voting system [33: 18, 16, 15, 2]

15. Consider the weighted voting system [17: 13, 9, 5, 2]. In the sequential coalition P3, P2, P1, P4 which player is pivotal.

17. Find the Shapley-Shubik power distribution for the system [24: 17, 13, 11]

Voting Theory Assignment

1. To decide on a new website design, the designer asks people to rank three designs that have been created (labeled A, B, and C). The individual ballots are shown below. Create a preference table. ABC, ABC, ACB, BAC, BCA, BCA, ACB, CAB, CAB, BCA, ACB, ABC

3. The planning committee for a renewable energy trade show is trying to decide what

city to hold their next show in. The votes are shown below.

Number of voters 9 19 11 8

1st choice Buffalo Atlanta Chicago Buffalo

2nd choice Atlanta Buffalo Buffalo Chicago

3rd choice ChicagoChicago Atlanta Atlanta

a. How many voters voted in this election?

b. How many votes are needed for a majority? A plurality?

4. A non-profit agency is electing a new chair of the board. The votes are shown below.

Number of voters 11 5 10 3

1st choice AtkinsCortez Burke Atkins

2nd choice Cortez Burke Cortez Burke

3rd choice Burke AtkinsAtkinsCorte

c. Find the winner under the plurality method.

d. Find the winner under the Borda Count Method

5. The student government is holding elections for president. There are four candidates

(labeled A, B, C, and D for convenience). The preference schedule for the election is:

Number of voters 120 50 40 90 60 100

1st choice C B D A A D

2nd choice D C A C D B

3rd choice B A B B C A

4th choice A D C D B C

e. Find the winner under the Instant Runoff Voting method.

f. Find the winner under Copeland’s method. method.

7. Consider an election with 129 votes. a. If there are 4 candidates, what is the smallest number of votes that a plurality candidate could have? b. If there are 8 candidates, what is the smallest number of votes that a plurality candidate could have?