Unlabeled

Condorcet's paradox: elections with no head-to-head winner, 3 to 18 candidates

With 3 candidates and a huge electorate ranking them at random, 8.8% of elections have no candidate who beats every rival head to head. From 11 candidates on, that is more likely than not.

What does this axis measure?
What does this axis measure?
Condorcet's paradox: elections with no head-to-head winner, 3 to 18 candidates: 16 points, a bar chart.
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Chance of no head-to-head winner (%)
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Candidates on the ballot (count)
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The data: 16 points
Candidates on the ballot (count)Chance of no head-to-head winner (%)
38.8
417.5
525.1
631.5
736.9
841.5
945.5
1048.9
1151.9
1254.5
1356.9
1459
1560.9
1662.6
1764.2
1865.6
Source
Condorcet's paradox for a huge electorate ranking at random (Guilbaud 1952; Niemi and Weisberg 1968), calculated by Unlabeled
Licence
Original computation by Unlabeled; nothing copied from the source
Idea
Alex Stone
Read
2026-11-08
Drawn
16 points as the puzzle showed them · bar
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