Voting Methods, Explained
Read this and you will know why the same voters and the same candidates can crown five different winners, how each counting rule works arithmetically, and which pathologies to watch for when you pick a method for a real election.
What the lab shows in one example
Take three candidates and 100 voters. The winner is not a fact about the voters alone. It is a fact about the voters plus the counting rule. Change the rule and the winner changes, with no ballot altered.
Here is a small case you can hold in your head. Sixty voters split their first choice between two similar candidates, A and B, who sit close together on the left. Forty voters back C on the right. First-preference counts: A gets 31, B gets 29, C gets 40.
Under plurality, C wins with 40 out of 100, even though 60 voters prefer either left candidate to C. That is the spoiler effect: A and B split a majority and hand victory to the side that a majority ranks last. Instant-runoff eliminates B (fewest first choices among the top two of the left, here 29), transfers those votes to A, and A wins 60 to 40. Same ballots, opposite result.
The lab uses spatial voters. Every voter is a point on a 2D opinion map and prefers whichever candidate sits closer. All five methods then count the exact same rankings, so any disagreement comes from the rule, never from different data.
The five rules, stated plainly
Every method starts from the same ranked ballots. They differ in what they do with those rankings.
- Plurality
- Count only each voter's first choice. Most first choices wins. Ignores everything below the top rank.
- Instant-runoff (IRV)
- Count first choices. If no candidate has more than half, eliminate the last-place candidate and transfer their ballots to the next choice still standing. Repeat until someone has a majority.
- Borda
- With m candidates, a first place is worth m-1 points, second is worth m-2, down to 0 for last. Sum the points. Highest total wins.
- Condorcet
- Compare every pair head to head. A candidate who beats every other candidate one on one is the Condorcet winner. Sometimes no such candidate exists.
- Approval
- Each voter approves every candidate within some acceptance radius. Most approvals wins. This uses a threshold, not a full ranking.
The Borda formula and why it rewards broad support
Borda is the only method here that reduces to a clean sum, so it is the easiest to compute by hand.
Here B(x) is candidate x's Borda score, n is the number of voters, m is the number of candidates, and r_v(x) is the rank voter v gives x, counting from 0 for first place. A first-place ballot adds m-1, a last-place ballot adds 0.
With three candidates m=3, so first place is worth 2 points, second 1, third 0. A candidate everyone ranks second scores 1 \times n. A candidate half the voters love and half the voters hate scores 2 \times (n/2) + 0 \times (n/2), which is the same total. Borda treats a solid second choice as equal to a polarizing split. That is its character: it favors the broadly acceptable candidate over the narrowly beloved one.
A worked count on the demo data
Three candidates, 100 voters, five rules
Use the spoiler setup above. The ballots come in three blocks, sorted by who each block ranks first, second and third.
| Block | Voters | 1st | 2nd | 3rd |
|---|---|---|---|---|
| Left near A | 31 | A | B | C |
| Left near B | 29 | B | A | C |
| Right | 40 | C | B | A |
- Plurality. First choices: A 31, B 29, C 40. Winner: C.
- Instant-runoff. No majority (top is 40). Eliminate B, the lowest at 29. B's 29 ballots list A second, so they go to A. New totals: A 60, C 40. Winner: A.
- Borda. A scores 2(31) + 1(29) + 0(40) = 91. B scores 2(29) + 1(31) + 1(40) = 129. C scores 2(40) + 0(60) = 80. Winner: B.
- Condorcet. A vs C: A beats C 60 to 40. B vs C: B beats C 60 to 40. A vs B: A wins 71 to 29 (the 40 right voters rank A above B). B loses to A, so B is not the Condorcet winner. A beats both. Winner: A.
- Approval. Depends on the radius. If each voter approves both left candidates but not C, A gets 60, B gets 60, C gets 40, and A and B tie. Approval results move with the threshold, which is exactly why the lab lets you drag it.
Three different winners from one set of ballots: C under plurality, B under Borda, A under IRV and Condorcet. This is not a trick. It is the ordinary behavior of these rules.
Yee diagrams and the moving-candidate paradox
A Yee diagram colors every point of the opinion map by who would win if the cloud of voters were centered there. Under a well-behaved method, each candidate's win region is a convex blob that contains the candidate. Plurality and IRV break this.
Watch what happens when you drag a candidate toward the center of the voters under plurality. You would expect getting closer to the median voter to help. Sometimes it makes the candidate lose, because the shift changes which rival splits the vote. The win regions come out non-convex, with holes and islands. That visual is the whole point of the Yee construction: a sane rule should never punish a candidate for moving toward the electorate.
When each method is a reasonable choice
No rule is best for every setting. Match the rule to what you care about.
Use plurality only when candidates are few and clearly distinct, so vote splitting is unlikely. It is simple to count and simple to explain, and that is its entire case. With three or more similar candidates it invites spoilers, as the worked example shows.
Use instant-runoff when you want a majority winner and voters can rank comfortably. It removes most spoilers but is not monotone: in rare configurations, ranking a candidate higher can cause that candidate to lose. Use Condorcet when a beats-all winner matters and you can tolerate the occasional cycle. Use Borda when you want the broadly acceptable choice, such as a compromise chair, and you accept that it can be gamed by insincere rankings. Use approval for large fields where ranking is a burden and a simple yes-no per candidate is enough.
If you are choosing a rule for a club or league, decide first which failure you least want to see: a spoiler, a non-monotone flip, or no clear winner at all. That single choice narrows the field faster than any general argument about fairness.
Common mistakes when reading results
Do not treat one method's winner as the "true" winner and the others as errors. Arrow's impossibility theorem proves that every ranked method violates at least one basic fairness criterion. The disagreements the lab flags are not bugs. They are the price of aggregating rankings at all.
Three errors show up most often. First, reading approval results as fixed: they shift with the acceptance radius, so a tie at one threshold becomes a clear win at another. Second, assuming a Condorcet winner always exists: with the block A beats B, B beats C, C beats A, no candidate beats all others, and the lab lists the cycle explicitly. Third, expecting IRV and Borda to agree with plurality: in the worked example all three disagree, which is normal, not a sign of a mistake.
Related tools
The Voting Methods Lab picks a single winner. When you instead need to fill several seats in proportion to vote shares, that is apportionment, a different problem with its own rules. See the Seat Apportionment Calculator for D'Hondt, Sainte-Laguë and largest remainder side by side.
Frequently asked questions
Why do plurality and instant-runoff give different winners?
Plurality counts first choices only. IRV keeps transferring votes from eliminated candidates. In the demo, C leads first choices 40 to 31 to 29, so plurality picks C, but after B is eliminated the 29 B ballots move to A, giving A 60 and the win. The lower rankings that plurality throws away decide the IRV result.
What is a Condorcet cycle and what happens then?
A cycle is when A beats B head to head, B beats C, and C beats A, so no candidate beats every rival. No Condorcet winner exists. The lab prints the cycle so you can see it, and you fall back to another rule to break it.
Is Borda easy to manipulate?
Yes. Because Borda sums positional points, a voter can bury a strong rival at the bottom to drain that rival's score. Ranking your favorite first and the main threat last, regardless of true opinion, is a known strategy. That is one reason Borda is used more for friendly rankings than high-stakes elections.
Which method has no spoilers?
None is completely spoiler-free, but the ranked methods resist spoilers far better than plurality. In the worked case, plurality lets B spoil the left bloc and elect C, while IRV, Borda and Condorcet all avoid handing victory to the candidate a majority ranks last.
Why are voters shown as points on a map?
The spatial model is the standard tool in social-choice theory. Placing each voter at a point and assuming they prefer nearer candidates lets rankings fall out of distance, so you can move candidates and watch every method recount from the same underlying preferences.
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