8
B (Black Hole)
7
C (Comet)
6
D (Dark Matter)
5
E (Eclipse)
3
Identify the Plurality Winner and evaluate if they hold a true majority:
The Plurality Winner is Candidate A (Asteroid) with 8 first-place votes.
Majority? NO. 8 votes is only 27.6% of the 29 total votes, which is well below the 15-vote majority threshold. A wins purely because of fragmented support for other choices.
Topic: Social Choice Theory (Mathematics of Elections) Page 1 of 3 (Answer Key)
Teacher Answer Key & Facilitation Guide
Mascot Election Case Study
Page 2: Top-Two Runoff Analysis
Under the Top-Two Runoff method (sometimes called Plurality with Runoff), if no candidate secures a majority of first-place votes in the initial round, a final head-to-head election is held between the top two candidates. All other candidates are eliminated, and voters' ballots are redirected to whichever of the top two they rank higher.
1. Identify the Top Two Candidates from Round 1:
Which two candidates received the highest numbers of first-place votes on Page 1?
The two candidates with the most 1st-place votes are: Candidate A (8 votes) and Candidate B (7 votes).
2. Head-to-Head Preference Comparison:
For each of the five voter groups, determine which of the top two runoff candidates is preferred on their ballots.
Runoff Candidate 1: Candidate A
List the voter groups (and voter counts) that rank this candidate higher than the other runoff candidate.
• Group 1 (8 voters): Ranks A 1st, B 5th.
(Prefers A over B).
No other groups rank A over B. All other groups (2, 3, 4, 5) rank B higher than A.
Runoff Candidate 2: Candidate B
List the voter groups (and voter counts) that rank this candidate higher than the other runoff candidate.
• Group 2 (7 voters): Ranks B 1st, A 5th.
• Group 3 (6 voters): Ranks B 3rd, A 5th.
• Group 4 (5 voters): Ranks B 2nd, A 5th.
• Group 5 (3 voters): Ranks B 4th, A 5th.
3. Runoff Winner Calculation:
Sum the votes for each candidate from the reallocation above to determine the winner of the Top-Two Runoff.
Runoff Candidate A Final Votes
8
Runoff Candidate B Final Votes
21 (7 + 6 + 5 + 3)
State the Top-Two Runoff Winner and explain why they won:
The Top-Two Runoff Winner is Candidate B (Black Hole) with 21 votes (72.4% of the electorate).
Reasoning: Although A had more 1st-place votes initially, A is highly polarizing and ranked dead-last by all other voter groups. When given a head-to-head choice, the vast majority of voters (21 vs 8) preferred B over A.
Topic: Social Choice Theory (Mathematics of Elections) Page 2 of 3 (Answer Key)
Teacher Answer Key & Facilitation Guide
Mascot Election Case Study
Page 3: Sequential Runoff & Reflection
Under Sequential Runoff, the candidate with the fewest first-place votes is eliminated in each round. Ballots cast for that candidate are redistributed to the next preferred candidate listed on each ballot. This repeats until a candidate achieves an absolute majority of active votes.
Round 1 Elimination
Candidate with fewest 1st-place votes (eliminated):
E (3 votes)
To which active candidate do these 3 votes transfer?
D (Next choice on Group 5's ballots)
Round 2 Counts (Remaining: A, B, C, D):
A: 8
B: 7
C: 6
D: 8 (5 + 3)
Round 2 Elimination
Candidate with fewest active votes (eliminated):
C (6 votes)
To which active candidate do these 6 votes transfer?
D (C's ballots next active choice is D)
Round 3 Counts (Remaining: A, B, D):
A: 8
B: 7
D: 14 (8 + 6)
Round 3 Elimination & Final Winner
Candidate with fewest active votes (eliminated):
B (7 votes)
To which active candidate do these 7 votes transfer?
D (B's next active choice is D)
Final Round 4 Matchup and Vote Counts:
A: 8 votes (Group 1)
D: 21 votes (14 + 7 from B)
| Voting Method | Winning Candidate | Final Votes Received | Did Winner Have an Outright Majority? |
|---|---|---|---|
| Plurality | A (Asteroid) | 8 / 29 (27.6%) | No (Required: 15) |
| Top-Two Runoff | B (Black Hole) | 21 / 29 (72.4%) | Yes (Runoff Match) |
| Sequential Runoff (IRV) | D (Dark Matter) | 21 / 29 (72.4%) | Yes (Sequential) |
Critical Thinking Reflection Question:
Describe the paradox of this election. How is it possible that the same electorate with the exact same ballots produced three different winners? Which method do you believe reflects the true "will of the people"?
This paradox reveals that rules matter as much as votes. Plurality favors the choice with strong, narrow support (A), but ignores strong opposition. Top-Two Runoff forces a direct compromise between the top choices, favoring B. Sequential Runoff (IRV) transfers votes iteratively, allowing a consensus dark-horse candidate (D) to win because they are widely liked as a secondary option, even though they had few 1st-place votes initially. There is no single mathematical definition of the "will of the people."
Topic: Social Choice Theory (Mathematics of Elections) Page 3 of 3 (Answer Key)
Mascot Extension Case Study
Page 2: Top-Two Runoff Analysis
Under the Top-Two Runoff method (sometimes called Plurality with Runoff), if no candidate secures a majority of first-place votes in the initial round, a final head-to-head election is held between the top two candidates. All other candidates are eliminated, and voters' ballots are redirected to whichever of the top two they rank higher.
1. Identify the Top Two Candidates from Round 1:
Which two candidates received the highest numbers of first-place votes on Page 1?
2. Head-to-Head Preference Comparison:
For each of the five voter groups, determine which of the top two runoff candidates is preferred on their ballots.
Runoff Candidate 1: _________
List the voter groups (and voter counts) that rank this candidate higher than the other runoff candidate.
Runoff Candidate 2: _________
List the voter groups (and voter counts) that rank this candidate higher than the other runoff candidate.
3. Runoff Winner Calculation:
Sum the votes for each candidate from the reallocation above to determine the winner of the Top-Two Runoff.
Runoff Candidate 1 Final Votes
Runoff Candidate 2 Final Votes
State the Top-Two Runoff Winner and explain why they won:
Topic: Social Choice Theory (Mathematics of Elections) Page 2 of 3
Mascot Extension Case Study
Page 3: Sequential Runoff & Reflection
Under Sequential Runoff, the candidate with the fewest first-place votes is eliminated in each round. Ballots cast for that candidate are redistributed to the next preferred candidate listed on each ballot. This repeats until a candidate achieves an absolute majority of active votes.
Round 1 Elimination
Candidate with fewest 1st-place votes (eliminated):
To which active candidate do these 3 votes transfer?
Round 2 Counts (Remaining: A, B, C, D):
A: _____
B: _____
C: _____
D: _____
Round 2 Elimination
Candidate with fewest active votes (eliminated):
To which active candidate do these 5 votes transfer?
Round 3 Counts (Remaining: A, B, C):
A: _____
B: _____
C: _____
Round 3 Elimination & Final Winner
Candidate with fewest active votes (eliminated):
To which active candidate do these 9 votes transfer?
Final Round 4 Matchup and Vote Counts:
A: _____
B: _____
| Voting Method | Winning Candidate | Final Votes Received | Did Winner Have an Outright Majority? |
|---|---|---|---|
| Plurality | |||
| Top-Two Runoff | |||
| Sequential Runoff (IRV) |
Critical Thinking Reflection Question:
Describe the paradox of this election. How is it possible that the same electorate with the exact same ballots produced three different winners? Which method do you believe reflects the true "will of the people"?
Topic: Social Choice Theory (Mathematics of Elections) Page 3 of 3
A (Asteroid)
8
B (Black Hole)
10
C (Comet)
9
D (Dark Matter)
5
E (Eclipse)
3
Identify the Plurality Winner and evaluate if they hold a true majority:
The Plurality Winner is Candidate B (Black Hole) with 10 first-place votes.
Majority? NO. 10 votes is only 28.6% of the 35 total votes, which is well below the 18-vote majority threshold. B wins the initial round but lacks consensus support.
Topic: Social Choice Theory (Mathematics of Elections) Page 1 of 3 (Answer Key)
Teacher Answer Key & Facilitation Guide
Mascot Extension Case Study
Page 2: Top-Two Runoff Analysis
Under the Top-Two Runoff method (sometimes called Plurality with Runoff), if no candidate secures a majority of first-place votes in the initial round, a final head-to-head election is held between the top two candidates. All other candidates are eliminated, and voters' ballots are redirected to whichever of the top two they rank higher.
1. Identify the Top Two Candidates from Round 1:
Which two candidates received the highest numbers of first-place votes on Page 1?
The two candidates with the most 1st-place votes are: Candidate B (10 votes) and Candidate C (9 votes).
2. Head-to-Head Preference Comparison:
For each of the five voter groups, determine which of the top two runoff candidates is preferred on their ballots.
Runoff Candidate 1: Candidate B
List the voter groups (and voter counts) that rank this candidate higher than the other runoff candidate.
• Group 1 (10 voters): Ranks B 1st, C 5th.
(Prefers B over C).
No other groups prefer B over C. All other groups (2, 3, 4, 5) rank C higher than B.
Runoff Candidate 2: Candidate C
List the voter groups (and voter counts) that rank this candidate higher than the other runoff candidate.
• Group 2 (9 voters): Ranks C 1st, B 5th.
• Group 3 (8 voters): Ranks C 3rd, B 5th.
• Group 4 (5 voters): Ranks C 3rd, B 5th.
• Group 5 (3 voters): Ranks C 3rd, B 5th.
3. Runoff Winner Calculation:
Sum the votes for each candidate from the reallocation above to determine the winner of the Top-Two Runoff.
Runoff Candidate B Final Votes
10
Runoff Candidate C Final Votes
25 (9 + 8 + 5 + 3)
State the Top-Two Runoff Winner and explain why they won:
The Top-Two Runoff Winner is Candidate C (Comet) with 25 votes (71.4% of the electorate).
Reasoning: Although B had the slight plurality in the first round, B is overwhelmingly unpopular with non-Group-1 voters, who all rank B dead-last (5th). In a head-to-head match, C unites the remaining 71.4% of the voters.
Topic: Social Choice Theory (Mathematics of Elections) Page 2 of 3 (Answer Key)
Teacher Answer Key & Facilitation Guide
Mascot Extension Case Study
Page 3: Sequential Runoff & Reflection
Under Sequential Runoff, the candidate with the fewest first-place votes is eliminated in each round. Ballots cast for that candidate are redistributed to the next preferred candidate listed on each ballot. This repeats until a candidate achieves an absolute majority of active votes.
Round 1 Elimination
Candidate with fewest 1st-place votes (eliminated):
E (3 votes)
To which active candidate do these 3 votes transfer?
A (Next choice on Group 5's ballots: E > A > C)
Round 2 Counts (Remaining: A, B, C, D):
A: 11 (8 + 3)
B: 10
C: 9
D: 5
Round 2 Elimination
Candidate with fewest active votes (eliminated):
D (5 votes)
To which active candidate do these 5 votes transfer?
A (Next active choice on Group 4's ballots: D > A)
Round 3 Counts (Remaining: A, B, C):
A: 16 (11 + 5)
B: 10
C: 9
Round 3 Elimination & Final Winner
Candidate with fewest active votes (eliminated):
C (9 votes)
To which active candidate do these 9 votes transfer?
A (Next active choice on Group 2's ballots: C > A)
Final Round 4 Matchup and Vote Counts:
A: 25 votes (16 + 9 from C)
B: 10 votes (Group 1)
| Voting Method | Winning Candidate | Final Votes Received | Did Winner Have an Outright Majority? |
|---|---|---|---|
| Plurality | B (Black Hole) | 10 / 35 (28.6%) | No (Required: 18) |
| Top-Two Runoff | C (Comet) | 25 / 35 (71.4%) | Yes (Runoff Match) |
| Sequential Runoff (IRV) | A (Asteroid) | 25 / 35 (71.4%) | Yes (Sequential) |
Critical Thinking Reflection Question:
Describe the paradox of this election. How is it possible that the same electorate with the exact same ballots produced three different winners? Which method do you believe reflects the true "will of the people"?
In this rematch, Plurality selected B. Top-Two Runoff eliminated A (the 3rd place option) in round 1, leaving B vs C, where C won. However, in Sequential Runoff, E and D were eliminated first, and their votes transferred to A, which boosted A above B and C, allowing A to win the final round! This demonstrates how early elimination rules dramatically shift outcomes—excluding a popular secondary candidate like A early (as in Top-Two) vs keeping them alive (as in IRV).
Topic: Social Choice Theory (Mathematics of Elections) Page 3 of 3 (Answer Key)
Topic: Social Choice Theory (Mathematics of Elections) Page 1 of 2
Senior Class Gift Case Study
Page 2: Borda & Condorcet Methods
The Borda Count method assigns points to rankings. For an election with 5 candidates, a first-place vote is worth 5 points, second is 4 points, third is 3 points, fourth is 2 points, and fifth is 1 point. Points are multiplied by the number of voters in each group and summed.
Calculate the total points scored by each candidate (Show your work):
A (Astronomy)
Score: ______
B (Botanical)
Score: ______
C (Computer)
Score: ______
D (Digital)
Score: ______
E (Energy)
Score: ______
Which candidate wins under Borda Count?
The Condorcet Winner is the candidate who defeats every other candidate in a series of head-to-head (1-on-1) matchups. To compare two candidates, determine how many voters rank one candidate higher than the other.
Conduct head-to-head match-ups between Candidate B and all others:
Matchup 1: Candidate B vs Candidate A
Votes preferring B: ______ | Votes preferring A: ______
Winner of Matchup:
Matchup 2: Candidate B vs Candidate C
Votes preferring B: ______ | Votes preferring C: ______
Winner of Matchup:
Matchup 3: Candidate B vs Candidate D
Votes preferring B: ______ | Votes preferring D: ______
Winner of Matchup:
Matchup 4: Candidate B vs Candidate E
Votes preferring B: ______ | Votes preferring E: ______
Winner of Matchup:
Identify the Condorcet Winner based on these matchups:
Candidate E received zero 1st-place votes but scored higher in the Borda Count than Candidate D, who won Group 4's first-place rankings outright. Explain how Borda Count and Condorcet capture different dimensions of public preference than Plurality.
Topic: Social Choice Theory (Mathematics of Elections) Page 2 of 2
Tally of 1st-Place Votes:
A (Astronomy)
28
B (Botanical)
32 (22 + 10)
C (Computer)
18
D (Digital Media)
12
E (Energy)
0
Identify the Plurality Winner and explain if they represent a consensus (>50% of electorate):
The Plurality Winner is Candidate B (Botanical Greenhouse) with 32 votes.
Consensus? NO. 32 votes is only 35.6% of the 90 total votes, which is well below the majority threshold of 46 votes. B wins the plurality because the electorate's remaining votes are split.
Topic: Social Choice Theory (Mathematics of Elections) Page 1 of 2 (Answer Key)
Teacher Answer Key & Facilitation Guide
Senior Class Gift Case Study
Page 2: Borda & Condorcet Methods
The Borda Count method assigns points to rankings. For an election with 5 candidates, a first-place vote is worth 5 points, second is 4 points, third is 3 points, fourth is 2 points, and fifth is 1 point. Points are multiplied by the number of voters in each group and summed.
Calculate the total points scored by each candidate (Show your work):
A (Astronomy)
(28*5)+(22*1)+
(18*1)+(12*1)+
(10*2)
Score: 212
B (Botanical)
(28*4)+(22*5)+
(18*3)+(12*4)+
(10*5)
Score: 374
C (Computer)
(28*3)+(22*4)+
(18*5)+(12*3)+
(10*3)
Score: 328
D (Digital)
(28*2)+(22*2)+
(18*2)+(12*5)+
(10*1)
Score: 206
E (Energy)
(28*1)+(22*3)+
(18*4)+(12*2)+
(10*4)
Score: 230
Which candidate wins under Borda Count?
Candidate B (Botanical Greenhouse) wins Borda Count with 374 points.
The Condorcet Winner is the candidate who defeats every other candidate in a series of head-to-head (1-on-1) matchups. To compare two candidates, determine how many voters rank one candidate higher than the other.
Conduct head-to-head match-ups between Candidate B and all others:
Matchup 1: Candidate B vs Candidate A
Votes preferring B: 22+18+12+10 = 62 | Votes preferring A: 28
Winner of Matchup: Candidate B (62 to 28)
Matchup 2: Candidate B vs Candidate C
Votes preferring B: 28+22+12+10 = 72 | Votes preferring C: 18
Winner of Matchup: Candidate B (72 to 18)
Matchup 3: Candidate B vs Candidate D
Votes preferring B: 28+22+18+10 = 78 | Votes preferring D: 12
Winner of Matchup: Candidate B (78 to 12)
Matchup 4: Candidate B vs Candidate E
Votes preferring B: 28+22+12+10 = 72 | Votes preferring E: 18
Winner of Matchup: Candidate B (72 to 18)
Identify the Condorcet Winner based on these matchups:
Candidate B (Botanical Greenhouse) is the Condorcet Winner because B defeats all other candidates head-to-head.
Candidate E received zero 1st-place votes but scored higher in the Borda Count than Candidate D, who won Group 4's first-place rankings outright. Explain how Borda Count and Condorcet capture different dimensions of public preference than Plurality.
Plurality only measures top-tier (1st place) enthusiasm, ignoring compromise options. Candidate D had a dedicated minority (12 voters) but was ranked last by almost everyone else, yielding only 206 Borda points. Conversely, Candidate E had zero 1st-place votes but was ranked highly (2nd or 3rd) on 50 ballots, accumulating 230 Borda points. Borda and Condorcet capture broad public utility by factoring in full rankings.
Topic: Social Choice Theory (Mathematics of Elections) Page 2 of 2 (Answer Key)
