Deck 1: The Mathematics of Elections: the Paradoxes of Democracy

Question 1

Your college radio station is holding auditions to find the next Campus Idol. There are four people competing in the auditions: Beth (B), Frank (F), Helen (H), and Sam (S). After all of the competitors have auditioned, the audience members rank each contestant in order of preference and submit their ballots. The results are given below.

How many preference ballots were cast in this competition?

Answer 1

95

Question 2

According to the preference schedule from problem 1, who would be the next Campus Idol if the Method of Pairwise Comparisons was used to choose the winner?

Answer 2

Frank or F

Question 3

The Student Government at your college consists of 184 voting members. A motion regarding funding
for a new club activity is brought to the floor. The four possible activities for which a member could
vote are the Karaoke Club (K), Hiking Club (H), Jousting Club (J) and Dog-Lovers Club (D). If the
Karaoke Club (K) receives a total of 48 votes, then what is the minimum number of votes needed by
one of the remaining clubs to ensure that no club can receive a majority of all votes?

Answer 3

92

Question 4

Find the smallest value of X so that a violation of the Condorcet Criterion occurs using the Plurality Method. Hint: B will be the winner using Plurality and C will be Condorcet.

Question 5

According to the preference schedule from problem 1, rank the five students using the Plurality Method.

Question 6

Since 2011, a total of 68 college basketball teams quality for the NCAA March Madness Tournament. These teams play each other in a series of games in a bracket style competition resulting in a total of 67 games. However, suppose that each team was required to play one game against everyone other participating team in a round-robin style tournament. How many games would this entail?

Question 7

Use the scenario and preference schedule from problem 7. Suppose that Dr. Davis (D) has to withdraw her name from consideration. Remove D from the preference table and shift all votes up one slot. Who will be chosen as the candidate to head up the Committee on Voting Standards, if the trustees use the Plurality with Elimination Method with the new preference table?

Question 8

There are 280 voting members in your school's faculty union. Resort Alpha (A), Resort Beta (B), Resort Gamma (G) and Resort Delta (D) are the four possible resorts at which the union could hold its annual meeting. Each faculty member ranks the four resorts, and the Method of Plurality is being used to determine which resort will be chosen. Suppose that 30 % of all members vote for Resort Alpha (A) as their 1^st choice. Assuming that all 280 members vote, then what is the smallest number of the remaining 1^st choice votes needed in order to guarantee that Resort Beta (B) is the winner?

Question 9

According to the preference schedule from problem 1, which candidate, if any, will receive a majority
of 5th choice votes?

Question 10

According to the preference schedule from problem 1, rank the five students using the Plurality with
Elimination Method.

Question 11

According to the preference schedule from problem 1, rank the five students using the Method of Pairwise Comparisons.

Question 12

Five students are taking part in the Campus Talent Show. Each of the five students will perform their talent in front of an audience and once all the performers have finished the audience members will rank each participant in order of preference. The candidate who has the highest ranking will move on to the national competition. The final preference schedule is given below.

Rank the five students using the Borda Count Method.

Question 13

A new president is being chosen by your college's Math Club. The candidates are Alex (A), Betsy (B), Charlie (C) , and Dana (D). The 27 members of the Math Club rank each candidate in order of preference and the results are tabulated below. Who would be chosen as the president of the Math Club using the Plurality with Elimination Method?

Question 14

Which Fairness Criterion is being violated by the actions shown in problems 7 and 8?

Question 15

Use the scenario and preference schedule from problem 7. Suppose that Dana is asked to be president of the Latin Club and has to withdraw from the race before the results are made public. Realign the preference table by removing D and shifting all of the votes up one slot. Now who would be chosen as the president of the Math Club using the Plurality with Elimination Method?

Question 16

Which Fairness Criterion is being violated by the actions shown in problems 7 and 8?

Question 17

The Board of Trustees at your college is holding elections to determine which candidate from the faculty will be chosen to head up the new Committee on Voting Standards. The choices are Dr. Abel, Dr. Barnes, Dr. Corza, and Dr. Davis or A, B, C, and D, respectively. The trustees have ranked the candidates from their first choice to their fourth choice, and the results are given below in the preference table.

Who will be chosen as the candidate to head up the Committee on Voting Standards if the trustees use the Plurality with Elimination Method?

Question 18

Which of the four Voting Methods (Plurality, Borda Count, Plurality with Elimination or Pairwise Comparison) will never violate the Majority Criterion and the Condorcet Criterion?

Question 19

According to the preference schedule from problem 1, Sam would be voted the next Campus Idol using the Plurality Method. Would this result be a violation of the Condorcet Criterion? Explain.

Question 20

According to the preference schedule from problem 1, who would be the next Campus Idol if the Borda Count Method was used to choose the winner?

Question 21

According to the preference schedule from problem 1, who is the winner of the election using the Borda Count Method?

A) A
B) B
C) C
D) D
E) There is a tie.

Question 22

According to the preference schedule from problem 1, who is the winner of the election using the Method of Pairwise Comparisons?

A) A
B) B
C) C
D) D
E) There is a tie.

Question 23

Using the Plurality with Elimination Method, rank the contestants from the following election.

A) Ali 1^st , Cat 2^nd , Bob 3^rd , Deb 4^th
B) Ali 1^st , Bob 2^nd , Cat 3^rd , Deb 4^th
C) Ali 1^st , Cat 2^nd , Deb 3^rd , Bob 4^th
D) Ali 1^st , Deb 2^nd , Cat 3^rd , Bob 4^th
E) None of the above.

Question 24

Using the preference schedule from problem 1 , who is the winner of the election using the Plurality with Elimination Method?

A) A
B) B
C) C
D) D
E) There is a tie.

Question 25

As of 2012 , there are 435 voting members in the US House of Representatives. At the opening session of the House, it is customary to have each Representative shake hands once with every other Representative. How many handshakes would this account for in 2012?

A) 870
B) 94395
C) 94830
D) 189225
E) None of the above.

Question 26

An election with four candidates (A, B, C, and D) and 180 voters will use the Plurality Method to choose a winner. After 120 ballots have been recorded, A has 26 1^st choice votes, B has 18 1^st choice votes, C has 42 1^st choice votes, and D has 34 1^st choice votes. What is the minimum number of the remaining 60 1^st choice votes that A must receive in order to guarantee that no candidate will receive a majority of 1^st choice votes?

A) 12
B) 15
C) 16
D) 24
E) None of the above.

Question 27

According to the preference schedule from problem 1 , who is the winner of the election using the Borda Count Method?

A) A
B) B
C) C
D) D
E) There is a tie.

Question 28

According to the preference schedule from problem 1, who is the winner of the election using the Plurality Method?

A) A
B) B
C) C
D) D
E) There is a tie.

Question 29

According to the preference schedule from problem 9 , who is ranked second using the Borda Count Method?

A) Ali
B) Bob
C) Cat
D) Deb
E) None of the above.

Question 30

According to the preference schedule from problem 1 , who is the winner of the election using the Plurality Method?

A) A
B) B
C) C
D) D
E) There is a tie.

Question 31

According to the preference schedule from problem 1, who is the winner of the election using the Plurality with Elimination Method?

A) A
B) B
C) C
D) D
E) There is a tie.

Question 32

According to the preference schedule from problem 9 , if Cat drops out of the race and all votes shift up one slot, then who will win the race using the Borda Count Method?

A) Ali
B) Bob
C) Cat
D) Deb
E) None of the above.

Question 33

The situations from problems (9) and (10) show an example of which of the Fairness Criterion being violated?

A) Majority Criterion
B) Condorcet Criterion
C) Monotonicity Criterion
D) Independence of Irrelevant Alternatives Criterion
E) There is no criterion being violated.

Question 34

An election with four candidates (A, B, C, and D) and 150 voters will use the Plurality Method to choose a winner. After 120 ballots have been recorded A has 26 1^st choice votes, B has 18 1^st choice votes, C has 42 1^st choice votes, and D has 34 1^st choice votes. What is the smallest number of the remaining 30 1^st choice votes that A must receive in order to guarantee a win?

A) 17
B) 24
C) 27
D) 30
E) None of the above.

Question 35

An election consists of four candidates A, B, C , and D . The candidates are ranked in order of preference by each of the voters. The preference schedule for this election is given below.
How many votes are required in order to obtain a majority of votes?

A) 18
B) 27
C) 36
D) 71
E) None of the above.

Question 36

Using the Borda Count Method, rank the contestants from the following election.

A) Ali 1^st , Deb 2^nd , Cat 3^rd , Bob 4^th
B) Cat 1^st , Ali 2^nd , Deb 3^rd , Bob 4^th
C) Ali 1^st , Cat 2^nd , Deb 3^rd , Bob 4^th
D) Deb 1^st , Bob 2^nd , Ali 3^rd , Cat 4^th
E) None of the above.

Question 37

Using the preference schedule from problem 1 , who is the winner of the election using the Method of Pairwise Comparisons?

A) A
B) B
C) C
D) D
E) There is a tie.

Question 38

The Plurality with Elimination method can possibly violate which of the Fairness Criteria?

A) Condorcet Criterion
B) Monotonicity Criterion
C) Independence of Irrelevant Alternatives Criterion
D) Majority Criterion
E) (a), (b) and (c)

Question 39

An election is held among five candidates (A, B, C, D, and E). There are 37 voters. Using the Method of Pairwise Comparisons it is found that A, B , and C each win two pairwise comparison while D wins three. If E wins the remaining comparisons in this election, then which statement is true?

A) D is the Condorcet candidate.
B) E is the Condorcet candidate.
C) D has a Majority of 1^st Choice Votes.
D) E has a Majority of 1^st Choice Votes.
E) None of the above is true.

Question 40

An election consists of four candidates A , B , C , D . The candidates are ranked in order of preference by each of the voters. The preference schedule for this election is given below.

Which candidate, if any, received a majority of 1^st choice votes?

A) A
B) B
C) C
D) D
E) None of the above.