Instant-Runoff Voting and the Condorcet Paradox
Instant-runoff voting (IRV) can fail to elect the Condorcet winner—the candidate preferred by a majority over every other candidate—as demonstrated in the 2009 Burlington mayoral election. Additionally, IRV can exhibit monotonicity failure, where receiving more first-place votes can cause a candidate to lose.
THE IDEA
And yet—
Let’s take a look at what happened in the 2009 mayoral race in Burlington, Vermont, one of the only U.S. municipalities to use the instant-runoff system.* Get ready—a lot of numbers are about to come flying at your face.
The three main candidates were Kurt Wright, the Republican; Andy Montroll, the Democrat; and the incumbent, Bob Kiss, from the left-wing Progressive Party. (There were other minor candidates in the race, but in the interest of brevity I’m going to ignore their votes.) Here’s the ballot count:
| Montroll, Kiss, Wright | 1332 |
| Montroll, Wright, Kiss | 767 |
| Montroll | 455 |
| Kiss, Montroll, Wright | 2043 |
| Kiss, Wright, Montroll | 371 |
| Kiss | 568 |
| Wright, Montroll, Kiss | 1513 |
| Wright, Kiss, Montroll | 495 |
| Wright | 1289 |
(Not everyone was on board with the avant-garde voting system, as you can see: some people just marked their first choice.)
Wright, the Republican, gets 3297 first-place votes in all; Kiss gets 2982; and Montroll gets 2554. If you’ve ever been to Burlington, you probably feel safe in saying that a Republican mayor was not the people’s will. But in the traditional American system, Wright would have won this election, thanks to vote splitting between the two more liberal candidates.
What actually happened was entirely different. Montroll, the Democrat, had the fewest first-place votes, so he was eliminated. In the next round, Kiss and Wright each kept the first-place votes they already had, but the 1332 ballots that used to say “Montroll, Kiss, Wright” now just said “Kiss, Wright,” and they counted for Kiss. Similarly, the 767 “Montroll, Wright, Kiss” votes counted for Wright. Final vote: Kiss 4314, Wright 4064, and Kiss is reelected.
Sounds good, right? But wait a minute. Adding up the numbers a different way, you can check that 4067 voters liked Montroll better than Kiss, while only 3477 liked Kiss better than Montroll. And 4597 voters preferred Montroll to Wright, but only 3668 preferred Wright to Montroll.
In other words, a majority of voters liked the centrist candidate Montroll better than Kiss, and a majority of voters liked Montroll better than Wright. That’s a pretty solid case for Montroll as the rightful winner—and yet Montroll was tossed in the first round. Here you see one of IRV’s weaknesses. A centrist candidate who’s liked pretty well by everyone, but is nobody’s first choice, has a very hard time winning.
To sum up:
Traditional American voting method: Wright wins
Instant-runoff method: Kiss wins
Head-to-head matchups: Montroll wins
Confused yet? It gets even worse. Suppose those 495 voters who wrote “Wright, Kiss, Montroll” had decided to vote for Kiss instead, leaving the other two candidates off their ballot. And let’s say 300 of the Wright-only voters switch to Kiss too. Now Wright has lost 795 of his first-place votes, setting him back to 2502; so he, not Montroll, gets eliminated in the first round. The election then goes down to Montroll vs. Kiss, and Montroll wins, 4067–3777.
See what just happened? We gave Kiss more votes—and instead of winning, he lost!
It’s okay to be a little dizzy at this point.
But hold on to this to steady yourself: at least we have some reasonable sense of who should have won this election. It’s Montroll, the Democrat, the guy who beats both Wright and Kiss head to head. Maybe we should toss all these Borda counts and runoffs and just elect the candidate who’s preferred by the majority.
Do you get the feeling I’m setting you up for a fall?
When to use it
Choosing among multiple candidates in an election where voters have ranked preferences.
Deciding on a group decision for a committee or board when multiple options are available.
Evaluating the fairness of alternative voting methods for a organization's election process.
FROM THEORY TO EVERYDAY LIFE
In practice
2009 Burlington Mayoral Election
In the 2009 mayoral election in Burlington, Vermont, using instant-runoff voting (IRV), the three main candidates were Kurt Wright (Republican), Andy Montroll (Democrat), and Bob Kiss (Progressive). First-choice votes: Wright 3297, Kiss 2982, Montroll 2554. Under IRV, Montroll was eliminated first despite being the Condorcet winner (preferred head-to-head over both Kiss and Wright). The final round gave Kiss 4314 votes to Wright's 4064, so Kiss won. However, 4067 voters preferred Montroll to Kiss, and 4597 preferred Montroll to Wright, showing that Montroll would have defeated both in direct matchups. IRV can fail to elect the Condorcet winner.
Monotonicity Failure Example
A hypothetical scenario based on the Burlington election: if 495 voters who ranked 'Wright, Kiss, Montroll' had instead voted only for Kiss, and 300 of Wright-only voters also switched to Kiss, Wright would lose 795 first-place votes, dropping to 2502. This would eliminate Wright first, and then Montroll would defeat Kiss 4067–3777. Paradoxically, giving Kiss more first-place votes causes him to lose the election, demonstrating IRV's failure of monotonicity.
A MOMENT TO REFLECT
Put it to work.
In what other contexts might a voting system fail to select the most broadly supported option?
Can you think of a real-world election where a similar paradox could occur?
How would you design a voting system that ensures the Condorcet winner always wins?
Often confused with
CONTINUE THE CONVERSATION
Source texts
- How Not to Be Wrong The Power of Mathematical Thinking (Ellenberg, Jordan)