Voting Systems Explorer
The same set of ranked ballots can elect different winners under different rules. Enter voter blocs once, then watch plurality, ranked-choice, Borda count, and approval voting each choose a winner.
Ballots
Each bloc is a group of voters who share the same ranking. The number is how many people vote that way. Edit them, or load a scenario below.
The classic example. 100 voters in four cities each prefer their own city. The same ballots elect three different winners depending on the method.
- 1Memphisapproved
- 2Nashvilleapproved
- 3Chattanooga
- 4Knoxville
- 1Nashvilleapproved
- 2Chattanoogaapproved
- 3Knoxville
- 4Memphis
- 1Chattanoogaapproved
- 2Knoxvilleapproved
- 3Nashville
- 4Memphis
- 1Knoxvilleapproved
- 2Chattanoogaapproved
- 3Nashville
- 4Memphis
Different methods elect different winners
These ballots do not have one obvious winner. Plurality picks Memphis, ranked-choice picks Knoxville, Borda picks Nashville, and approval picks Nashville. The outcome depends on the rule, not just the votes.
Plurality
Most first-choice votes wins. Ignores everything below first place.
- ✓ Memphis42 votes
- Nashville26 votes
- Knoxville17 votes
- Chattanooga15 votes
Instant-runoff (ranked-choice)
Eliminate the last-place candidate round by round until someone has a majority.
- Memphis. 42
- Nashville. 26
- Knoxville. 17
- Chattanooga. 15
- Memphis. 42
- Knoxville. 32
- Nashville. 26
- Knoxville. 58
- Memphis. 42
Borda count
Each rank earns points. With n candidates, first place earns n-1 down to 0 for last.
- ✓ Nashville194 pts
- Chattanooga173 pts
- Memphis126 pts
- Knoxville107 pts
Approval (top 2)
Each voter approves their top 2 candidates. Most approvals wins.
- ✓ Nashville68 approvals
- Chattanooga58 approvals
- Memphis42 approvals
- Knoxville32 approvals
Condorcet check
The Condorcet winner beats every other candidate head to head. Many methods do not always elect this candidate.
First-choice votes
Borda scores
Approvals (top 2)
How Voting Methods Differ
Plurality and the spoiler effect
Plurality counts only first choices, and the candidate with the most wins. It is simple but it can split the vote. When two similar candidates run, they divide their shared supporters and a less-preferred candidate can win.
That split is called the spoiler effect. Try the Spoiler scenario to see two like-minded candidates hand the win to a third.
Instant-runoff and ranked-choice
Instant-runoff voting, also called ranked-choice, uses the full ranking. If no candidate has a majority, the last-place candidate is eliminated and each of their ballots moves to its next choice. This repeats until someone passes 50 percent.
Because dropped votes transfer to a backup, ranked-choice often avoids the spoiler problem that plurality has.
Borda count and approval voting
Borda count gives points by rank. With n candidates, a first place earns n minus 1 points, second place earns n minus 2, and so on down to 0 for last. These reward broadly liked candidates, not just first choices.
Approval voting lets each voter approve any number of candidates. Here each bloc approves its top few choices, and the most approvals wins.
Why the same ballots elect different winners
The Tennessee scenario is the classic case. 100 voters in four cities each prefer their own city. Plurality elects Memphis, ranked-choice elects Knoxville, and Borda and approval elect Nashville. Nothing about the votes changed, only the rule did.
- A Condorcet winner beats every rival head to head.
- No single method satisfies every fairness goal at once.
- Use the bar charts to see each distribution side by side.
