The Condorcet method
Also called pairwise or head-to-head voting. The Schulze method and ranked pairs are popular versions.
A Condorcet method uses everyone's rankings to compare every pair of options. If one option beats every other option head to head, it's called the Condorcet winner, and it wins. When no option beats all the others, a tiebreaker such as the Schulze method decides.
- Nickname
- The tournament
- Proposed
- 1785, and by Ramon Llull in the 1200s
- Used for
- Electing the leader of the Debian software project, Kubernetes' steering committee, and party primaries in Sweden's Pirate Party
- Best at
- Finding the option with the broadest support, one matchup at a time
- Watch out for
- Loops, and being slow to count by hand
How it works
- Everyone ranks the options. Best first.
- Hold every matchup. For each pair of options, count how many ballots rank one above the other.
- Find the undefeated option. If one option wins all of its matchups, it's the Condorcet winner, and it wins.
- Break any loop. If no option is undefeated, use a tiebreaker, called a completion method, to pick the winner.
| Matchup | Result |
|---|---|
| 🍕 Pizza vs. 🌮 Tacos | Pizza, 5–4 |
| 🍕 Pizza vs. 🍣 Sushi | Pizza, 6–3 |
| 🍣 Sushi vs. 🌮 Tacos | Sushi, 5–4 |
Pizza wins both of its matchups, so it's the Condorcet winner, even though only two friends ranked it first. See this tournament in the story.
When there's a loop
Sometimes nobody wins every matchup. Picture a group where tacos beats pizza 6–3, pizza beats sushi 7–2, and sushi beats tacos 5–4. Like rock, paper, scissors, every option loses to something. This is called a Condorcet cycle or Condorcet paradox, and every Condorcet method needs a rule for it:
- Schulze method
- Looks for the strongest chain of wins from each option to each other. With three options, that comes down to setting aside the narrowest win in the loop. Here, that's sushi's 5–4 win, so tacos wins.
- Ranked pairs
- Locks in matchup results from the biggest win to the smallest, skipping any that would create a loop.
- Minimax
- Picks the option whose worst defeat is the smallest.
- Copeland
- Picks the option that wins the most matchups. Simple, but it often ends in a tie.
When there's a Condorcet winner, all of these pick it. They only differ when there's a loop.
A short history
- 1200s
The Catalan philosopher Ramon Llull describes choosing church officials by comparing candidates two at a time. His writings on elections were lost for centuries, and rediscovered only in modern times.
- 1785
The French mathematician the Marquis de Condorcet publishes an essay on the mathematics of majority decisions. He argues that an option a majority prefers to every other option should win, and shows that majority preferences can go in a circle.
- 1794
Condorcet dies in prison during the French Revolution, the year after being declared an enemy of the state.
- 1987
The economist Nicolaus Tideman devises ranked pairs, a Condorcet method that resists spoilers.
- 1997
Markus Schulze devises the Schulze method. Over the next decade, it's adopted by organizations including the Debian software project and, until 2013, the Wikimedia Foundation, which runs Wikipedia.
- Today
Condorcet methods are used by software projects and political parties for internal votes, but no government is known to use one for public elections.
Strengths and weaknesses
Strengths
- Broad support wins. The winner is the option that more voters prefer over each alternative, not just the one with the most first choices.
- Fewer spoilers. Adding a similar option doesn't split the vote. With Schulze or ranked pairs, adding or dropping a near-copy of an option doesn't change who wins.
- More support always helps. Ranking an option higher can only help it, never hurt it.
- Counted anywhere, added up later. Each polling place can tally its own matchup results, and the totals add up, just like a show of hands.
- A clear winner, usually. Real elections with many voters almost always have a Condorcet winner, so the tiebreaker rarely matters.
Weaknesses
- Loops need a tiebreaker. When there's a cycle, the result depends on which completion method you use, and those rules are harder to explain.
- Hard to count by hand. Ten options mean 45 matchups. Computers make this easy, but it's a lot for a room full of paper ballots.
- Ranking a backup can hurt your favorite. Your second choice can beat your first choice in their matchup because you ranked it, so some voters are tempted to leave backups off.
- Room for strategy. A group of voters can rank a strong rival last on purpose, trying to create a loop the tiebreaker resolves in their favor. This is called burying.
- A compromise can feel lukewarm. The winner might be nobody's favorite. Supporters see that as the point; critics see a winner few people are excited about.
How it does on the fairness tests
- Passes: if more than half the voters rank an option first, it wins.
- Passes: an option that beats every other one-on-one always wins.
- Passes: more support never makes an option lose (Schulze and ranked pairs).
- Passes: similar options don't split the vote (Schulze and ranked pairs).
- Passes: votes can be counted in separate places and added up.
- Fails: ranking a backup can hurt your favorite.
Questions
- What is a Condorcet winner?
- An option that beats every other option in a head-to-head matchup, using everyone's rankings. Not every election has one, but most large ones do.
- What is the Condorcet paradox?
- Majority preferences can form a loop: a majority prefers A to B, another prefers B to C, and another prefers C to A. Condorcet described it in 1785.
- Which Condorcet method is best?
- They all agree when there's a Condorcet winner. Schulze and ranked pairs are popular because they resist spoilers and never punish an option for gaining support.
- Why don't governments use it?
- It's harder to explain and to count by hand than other methods, and few places have tried it. Instant runoff voting (IRV) got there first in most places that switched away from plurality.