Plurality method
Count only the 1st places. Whoever has the most wins.
Who wins the student council election?
30 students voted for student council. Which of the 4 candidates won?
- A Action
- B Bold
- C Connect
- D Dialogue
The votes
Each student ranked the 4 candidates, from their favorite (1st) to the one they want least (4th). Students who voted the same way are in the same group. This list of groups is the class's preference summary.
Start with the task panel: make a guess, then count the votes four ways. After that, the groups unlock: change how many students are in each group, and tap a candidate to move it up one place in that group.
The groups stay locked until you count the votes with all four methods, in tasks 1 to 4 of the panel.
Winners now
Which candidate won?
Look at the four groups and make a guess before counting.
Make your guess first. Count the votes unlocks once you lock it in.
The plurality method card appeared right below the groups. There are other ways to count the same votes: tap Next.
Two-round runoff
Nobody has more than half of the 1st places (more than 15), so the top two, A and B, go to a second round. In it, each student votes for whichever of A and B they ranked higher.
How many of the 30 students vote for B in the second round?
Borda count
In the Borda count, each student gives points to all 4 candidates: 3 for 1st place, 2 for 2nd, 1 for 3rd, and 0 for 4th. The one with the most points wins.
How many points does candidate C get?
In each group, find C's place and multiply that place's points by the number of students. In group 1, C is 3rd: 12 × 1 = 12. Then add up the four groups.
Pairwise comparison
In pairwise comparison, the candidates face off two at a time, and each student votes for whichever of the two they ranked higher. The winner is the one who beats every other candidate.
When C faces D, how many students vote for each one?
C's campaign
Moving a candidate up one place in a group can change the winner. Which of these swaps, made in the starting votes, makes C win under 3 of the 4 methods?
This task unlocks after you count with all four methods, in tasks 1 to 4.
No one beats everyone
Find a preference summary where no candidate beats all the others in pairwise comparison, with no tie in any matchup. You can leave groups with 0 students.
Four ways to count
Two-round runoff
With more than half of the 1st places, a candidate wins outright. If not, the top two go to a second round.
Borda count
1st place is worth 3 points, 2nd is worth 2, 3rd is worth 1, and 4th is worth 0. The one with the most points wins.
Pairwise comparison
Compare the candidates two at a time. Whoever beats every other one head-to-head wins.
See all the numbers in a table
| Candidate | 1st places | Borda points | Matchups won |
|---|
One-on-one matchups
| Matchup | Votes | Result |
|---|
What's going on
This part unlocks when you finish counting with all four methods, in tasks 1 to 4 of the panel.
The votes were the same on all four cards. What changed was the counting rule. With the starting preference summary (groups of 12, 8, 5, and 5 students), each rule elected a different candidate:
- Plurality method. A has 12 first places, B has 8, and C and D have 5 each. A wins, even without half of the votes.
- Two-round runoff. More than half of 30 is more than 15, and nobody got there. A and B go to the second round, and each student picks whichever of the two they ranked higher. 18 students prefer B to A, so B wins 18 to 12. Brazil uses this system to elect its president and state governors.
- Borda count. It gives points by position: 3 for 1st place, 2 for 2nd, 1 for 3rd, and 0 for 4th. C gets 53 points, D gets 52, B gets 39, and A gets 36. C is the first choice of only 5 students, but it's in 2nd or 3rd place for everyone else.
- Pairwise comparison. 18 students rank D above A, 17 rank D above B, and 17 rank D above C. D beats the other three head-to-head. A candidate like that is the Condorcet candidate.
Look at candidate A again. It has more first places than any other candidate, and yet 18 of the 30 students put A last. In pairwise comparison, A loses to all of them.
So which method is the fair one? In 1951, the economist Kenneth Arrow proved that, with three or more candidates, no method that uses the voters' rankings follows every fairness rule that seems like common sense, all at the same time. Choosing a method means choosing which rule you're willing to break.
The challenge is in the task panel, next to the groups.
For teachers
Objective
By the end, students can count the same preference summary with the plurality method, a two-round runoff, the Borda count, and pairwise comparison, and use an example to explain why the choice of method can change the winner. The idea is to move students past "the result is what the people want" and show that the counting rule also decides.
Prerequisites
Adding and multiplying small whole numbers, comparing numbers, knowing what "more than half" means, and reading a table. There is no formula.
How to use in class
Project the preference summary and have each student write down which candidate won and why (5 minutes). Have the class vote by show of hands for A, B, C, or D and note the most popular guess; lock in that guess in task 1 of the panel and tap Count the votes, which opens only the plurality method card. Tasks 2 to 4 ask for one calculation each (B's votes in the second round, C's Borda points, the matchup between C and D): let the class work them out on paper, in pairs, and only then type in the most common answer (10 minutes). Each method's card opens when the answer is right, or through Show the math after one attempt. Read the explanation together and finish with task 5, where the class picks a swap before testing it (5 minutes). The Going further task, on the Condorcet paradox, fits high school students and anyone who finishes early.
Common mistakes
Thinking there is a true winner and the count only confirms it: this shows up in the guess, when a student defends their candidate as "the right one." Thinking that more first places means the majority's favorite: ask how many students put A last. Thinking that in a two-round runoff the first-round leader always wins. Mixing up points and votes in the Borda count: the student says C "got 53 votes," or answers 8 in task 3 after adding the place points without multiplying by the number of students. In pairwise comparison, counting only the 1st places (5 to 5). Thinking that pairwise comparison always has a winner: the Going further task leads to a cycle where each candidate loses to another one, like rock, paper, scissors. Forgetting that "more than half" depends on the total when the numbers change.
In the book, the second method is ranked-choice voting (the last-place candidate is cut, round by round). Here it became Brazil's two-round runoff, where the second round is recounted from the same rankings.
Discussion questions
- Which method would you use for our school's student council? What argument would you make to the class?
- Candidate A has the most first places and is last for the majority. Does A deserve to win?
- In the Borda count, why 3, 2, 1, and 0? What would change with 10, 2, 1, and 0?
- If you knew the method before the election, would you change your vote to help your candidate?
- Why does Brazil use a two-round runoff for president instead of the plurality method?
Based on Contemporary Mathematics, OpenStax, sections 11.1 and 11.2, under the CC BY-NC-SA 4.0 license. This activity is under the same license. Access the book for free at openstax.org.