Ranked ballots and plurality
A ranked ballot lists the candidates from best to worst for one voter. It tells us much more than "one name". With 3 candidates there are 6 possible lists.
Plurality uses only the first choice. The candidate with the most first-choice votes wins. This is "most votes wins", used in many elections. A candidate has a majority only with more than half of all votes. In our 9-voter story, A wins with 4 votes. That is a plurality, not a majority, and 5 of 9 voters put A last.
Runoff (ranked-choice) voting
Runoff fixes part of the problem. Step 1: count first choices. Step 2: if nobody has more than half, drop the candidate with the fewest. Step 3: move each dropped voter's ballot to the next name on that list. Step 4: count again. Repeat until someone has a majority.
In the story B is dropped and the 2 ballots move to C. C gets 3 + 2 = 5 of 9 and wins. The winner changed from A to C. The votes did not change. Only the rule did.
Borda count and the Condorcet winner
Borda count gives points for every place. With n candidates, first place gets n − 1 points, the next gets n − 2, and the last gets 0. Add the points of all voters. It rewards a candidate who is liked by many, even if not first. In the story: B scores 4×1 + 3×1 + 2×2 = 11.
A Condorcet winner beats every other candidate in a one-to-one vote. B beats A by 5 to 4 and beats C by 6 to 3, so B is the Condorcet winner. Sometimes none exists. If three groups prefer A over B, B over C and C over A, the choices go round in a circle (a cycle).
Is any method fair? Kenneth Arrow proved in 1951 that with three or more candidates no ranking method can satisfy every fairness rule at once. Voters can also vote strategically (not honestly) to help a favourite. Choose the method that fits the purpose, and tell everyone the rule before the vote.
Try it: run your own election
At home: ask 9 family members or friends to rank 3 snacks on paper. Count by plurality, runoff and Borda. Do you get the same winner? Then in the 3D lab, predict the winner first, press the method, and check. Try to make all three methods agree, then try to make them disagree.
Key formulas and definitions
- Majority = more than half of all votes (more than n ÷ 2)
- Plurality: the most first-choice votes wins
- Runoff: drop the fewest, move their ballots, recount
- Borda points with n candidates: n−1, n−2, …, 1, 0
- Total Borda points = voters × (0 + 1 + … + (n−1))
- Condorcet winner: beats every other candidate one-to-one
Worked examples
1. 5 voters: 3 pick X first, 2 pick Y first. Who wins by plurality? Is it a majority?
X has 3 first-choice votes, Y has 2. X wins. Half of 5 is 2.5 and 3 is more than 2.5, so it is also a majority.
2. 4 voters rank A,B,C and 3 voters rank C,B,A and 2 voters rank B,C,A. Find the Borda points of A and C.
A: first for 4 voters (4×2 = 8), last for the rest (0). A = 8. C: last for 4 voters (0), first for 3 (3×2 = 6), second for 2 (2×1 = 2). C = 8.
3. 100 voters. First choices: P 42, Q 33, R 25. All 25 R voters rank Q next for 20 of them and P next for 5 of them. Who wins the runoff?
No one has more than 50. R is dropped. Q gets 33 + 20 = 53. P gets 42 + 5 = 47. Q wins with 53.
4. In the 9-voter story find the Condorcet winner.
A vs B: 4 prefer A, 5 prefer B, so B wins. B vs C: B is above C for 4 + 2 = 6 voters, C above B for 3. B wins. B beats both, so B is the Condorcet winner.
5. Three groups of 3 voters rank A>B>C, B>C>A and C>A>B. Is there a Condorcet winner?
A beats B 6 to 3. B beats C 6 to 3. C beats A 6 to 3. It is a circle, so no candidate beats all others. There is no Condorcet winner.
Common mistakes
- Thinking "most votes" means "most people like this". In the story A wins with 4 of 9, yet 5 of 9 rank A last.
- In a runoff, forgetting to move the dropped candidate's ballots to their next choice. They are not thrown away.
- Giving Borda points the wrong way round. The best rank must get the most points, and the last rank gets 0.
- Believing a perfect voting method exists. Arrow's theorem says that for 3 or more candidates every method breaks some fairness rule.