# Chapter 1 Practice Problems

## Part I Multiple Choice

Use the preferece schedule below to answer Questions 1-4.

 Number of voters 8 6 2 3 5 1st choice A B C D E 2nd choice B D A E A 3rd choice C E E A D 4th choice D C B C B 5th choice E A D B C

1. Using the Borda Count method the winner of this election is:

1. A
2. B
3. E
4. D
5. None of the above

2. Using the Plurality method the winner of the election is:

1. A
2. B
3. C
4. E
5. None of the above

3. Using the Pairwise Comparisons method the winner of the election is:

1. A
2. B
3. a tie between B and A
4. D
5. None of the above

4. Using the Plurality-with-elimination method the winner of the election is:

1. A
2. E
3. B
4. D
5. None of the above

5. What is the total number of pairwise comparisons possible in an election among 20 candidates:

1. 20
2. 210
3. 150
4. 190
5. None of the above

### Questions 6-11 refer to the following preference schedule. The Mathematics For All Club is having an election for a president. The candidates are Amber, Bill and Jeniere. Each of the members is asked to submit a preference ballot. Here is the result:

 Number of Voters 5 4 4 2 First Choice Amber Bill Jeniere Bill Second Choice Jeniere Jeniere Amber Amber Third Choice Bill Amber Bill Jeniere

6. How many members of the Mathematics for All Club submitted their ballots?

1. 10
2. 4
3. 15
4. 20
5. None of the above

7. Who is the winner under the plurality method?

1. Jeniere
2. Amber
3. Bill
4. No winner
5. None of the above

8. Who is the winner under the plurality-with-elimination method?

1. Amber
2. Bill
3. a tie between Bill and Jeniere
4. Jeniere
5. None of the above

9. Using the Borda Count method the winner of this election is:

1. Bill
2. Amber
3. a tie between Jeniere and Amber
4. Jeniere
5. None of the above

10. Do we have a Condorcet winner under the pairwise comparison method?

1. Yes
2. No
3. It is impossible to know
4. None of the above

11. Which candidate comes in second place under the extended plurality-with-elimination method of ranking?

1. Amber
2. Bill
3. a tie between Bill and Amber
4. Jenier
5. None of the above

## Part II Show all your work

Given the following preference schedule:

 Number of voters 5 3 8 7 3 1st choice C D E B A 2nd choice B A B A D 3rd choice D C A C C 4th choice E B C D E 5th choice A E D E B

12. Find the winner of the election using the Borda count method which assigns 5, 4, 3, 2, 1 point(s) for a first, second, third, fourth, and fifth choice in that order. If there is a tie indicate so.

13. Find the winner of the election using the pairwise comparisons method, if there is a tie indicate so.

14. Find the winner of the election using the plurality-with-elimination method, if there is a tie indicate so.

## Part III Discussion Problems:

15. Address the problems/inconsistencies, positive aspects and a typical situation when the Borda Count method may be used.

Are you sure you want to see the soultions? Think about it. May be it is wise if you do it next time. At least that is what I think and recommend. But, if you insist, well, what can I do! Here it is, see it at your own risk.