Research data · Collective choice
The Condorcet atlas
Seven alternatives. Millions of domains. Two ends of the spectrum to explore.
How can a collection of individual rankings prevent every majority cycle? Explore the size distribution of maximal unitary Condorcet domains (MUCDs) on seven alternatives, then open individual examples at the two ends of the distribution.
- Archived representatives
- 171,870,480
- Small domains · sizes ≤16
- 3,353
- Large domains · sizes ≥88
- 4,679
Complete counts, up to isomorphism
Every archived example included
Every archived example included
01 · The full picture
Small domains, large domains
Each bar counts archived representatives of one size. Teal bars have every example available here: sizes ≤16 and ≥88. The middle of the distribution is much larger; its counts are included in full.
Swipe sideways to see the full chart, or use the size selector.
- 1 Choose a size
- 2 Load its collection
- 3 Inspect a domain
No domain collection has been loaded.
02 · Inside a domain
Selected domain
Each dot is a complete ranking. A line is one swap of neighbouring alternatives that stays inside the domain. Horizontal position counts inversions relative to 1 ≻ 2 ≻ 3 ≻ 4 ≻ 5 ≻ 6 ≻ 7; vertical position separates dots. Select a dot to highlight its neighbours.
The graph scrolls sideways on a small screen. Every ranking is also available in the list below.
Select a ranking from the full list
Why can’t majority preferences cycle?
Restrict the rankings to any three alternatives. At least one alternative never occupies a particular position. This is the triple condition behind the Condorcet property.
File details & verification
Exact counts for every size, with selected collection downloads
| Domain size | Representatives | Local collection |
|---|---|---|
| 1 | 0 | No domains |
| 2 | 0 | No domains |
| 3 | 0 | No domains |
| 4 | 46 | Original gzip · 0.4 KB · Browser JSON |
| 5 | 0 | No domains |
| 6 | 0 | No domains |
| 7 | 0 | No domains |
| 8 | 44 | Original gzip · 0.8 KB · Browser JSON |
| 9 | 24 | Original gzip · 0.6 KB · Browser JSON |
| 10 | 270 | Original gzip · 3.6 KB · Browser JSON |
| 11 | 188 | Original gzip · 2.6 KB · Browser JSON |
| 12 | 147 | Original gzip · 2.6 KB · Browser JSON |
| 13 | 176 | Original gzip · 3.5 KB · Browser JSON |
| 14 | 284 | Original gzip · 4.8 KB · Browser JSON |
| 15 | 548 | Original gzip · 8.2 KB · Browser JSON |
| 16 | 1,626 | Original gzip · 22.2 KB · Browser JSON |
| 17 | 2,178 | Summary only |
| 18 | 4,435 | Summary only |
| 19 | 9,994 | Summary only |
| 20 | 11,864 | Summary only |
| 21 | 7,040 | Summary only |
| 22 | 12,685 | Summary only |
| 23 | 18,570 | Summary only |
| 24 | 25,954 | Summary only |
| 25 | 44,660 | Summary only |
| 26 | 81,579 | Summary only |
| 27 | 94,158 | Summary only |
| 28 | 132,114 | Summary only |
| 29 | 159,868 | Summary only |
| 30 | 194,674 | Summary only |
| 31 | 247,692 | Summary only |
| 32 | 404,009 | Summary only |
| 33 | 356,618 | Summary only |
| 34 | 480,195 | Summary only |
| 35 | 461,900 | Summary only |
| 36 | 609,624 | Summary only |
| 37 | 678,422 | Summary only |
| 38 | 928,441 | Summary only |
| 39 | 930,304 | Summary only |
| 40 | 1,244,522 | Summary only |
| 41 | 1,273,738 | Summary only |
| 42 | 1,739,772 | Summary only |
| 43 | 1,849,074 | Summary only |
| 44 | 2,701,280 | Summary only |
| 45 | 2,644,266 | Summary only |
| 46 | 3,491,780 | Summary only |
| 47 | 3,686,966 | Summary only |
| 48 | 4,911,214 | Summary only |
| 49 | 4,790,868 | Summary only |
| 50 | 6,426,642 | Summary only |
| 51 | 6,253,444 | Summary only |
| 52 | 8,174,653 | Summary only |
| 53 | 7,497,364 | Summary only |
| 54 | 9,180,598 | Summary only |
| 55 | 8,270,608 | Summary only |
| 56 | 11,160,909 | Summary only |
| 57 | 8,540,924 | Summary only |
| 58 | 10,269,782 | Summary only |
| 59 | 7,723,932 | Summary only |
| 60 | 9,606,176 | Summary only |
| 61 | 6,518,148 | Summary only |
| 62 | 7,839,946 | Summary only |
| 63 | 5,191,166 | Summary only |
| 64 | 7,728,718 | Summary only |
| 65 | 3,436,076 | Summary only |
| 66 | 3,750,621 | Summary only |
| 67 | 2,034,070 | Summary only |
| 68 | 2,440,206 | Summary only |
| 69 | 1,152,526 | Summary only |
| 70 | 1,351,871 | Summary only |
| 71 | 671,796 | Summary only |
| 72 | 808,375 | Summary only |
| 73 | 357,970 | Summary only |
| 74 | 405,334 | Summary only |
| 75 | 186,106 | Summary only |
| 76 | 244,369 | Summary only |
| 77 | 101,268 | Summary only |
| 78 | 116,958 | Summary only |
| 79 | 48,120 | Summary only |
| 80 | 56,464 | Summary only |
| 81 | 23,402 | Summary only |
| 82 | 25,154 | Summary only |
| 83 | 11,344 | Summary only |
| 84 | 14,503 | Summary only |
| 85 | 6,108 | Summary only |
| 86 | 4,273 | Summary only |
| 87 | 2,066 | Summary only |
| 88 | 2,038 | Original gzip · 102.0 KB · Browser JSON |
| 89 | 1,248 | Original gzip · 66.0 KB · Browser JSON |
| 90 | 647 | Original gzip · 38.2 KB · Browser JSON |
| 91 | 214 | Original gzip · 12.9 KB · Browser JSON |
| 92 | 274 | Original gzip · 19.8 KB · Browser JSON |
| 93 | 106 | Original gzip · 7.8 KB · Browser JSON |
| 94 | 76 | Original gzip · 7.0 KB · Browser JSON |
| 95 | 18 | Original gzip · 2.3 KB · Browser JSON |
| 96 | 36 | Original gzip · 3.5 KB · Browser JSON |
| 97 | 16 | Original gzip · 1.8 KB · Browser JSON |
| 98 | 4 | Original gzip · 0.8 KB · Browser JSON |
| 99 | 0 | No domains |
| 100 | 2 | Original gzip · 0.6 KB · Browser JSON |
The mathematics
What the numbers mean
A domain is a set of rankings
The seven alternatives are labelled 1–7. A ranking lists all seven from best to worst. A domain of size 100 therefore allows 100 different rankings; the size does not count voters.
A Condorcet domain allows only rankings that keep pairwise majority preferences free of cycles, for every odd electorate whose members use those rankings.
Maximal is different from maximum
Maximal means that adding any ranking outside the domain destroys the Condorcet property. Even a domain with only four rankings can be maximal.
Unitary means that the natural order 1 ≻ 2 ≻ 3 ≻ 4 ≻ 5 ≻ 6 ≻ 7 is included. The archive classifies these domains up to isomorphism. For seven alternatives the maximum size is 100, with two archived representatives at that size.
A graph can have isolated nodes or several components. That does not contradict maximality: being connected by adjacent swaps is a different property from being impossible to extend as a Condorcet domain.
03 · Reuse the data
Downloads, provenance & citation
The complete size summary is hosted here, together with all 8,032 archived representatives at sizes ≤16 or ≥88. Original compressed files are preserved byte for byte. Browser files relabel the source alternatives 0–6 as 1–7, retaining every alternative and the original order of domains and rankings.
All 3,353 small domains (ZIP)All 4,679 large domains (ZIP)
- Source: Klas Markström’s Condorcet-domain archive, snapshot 13 September 2026. The full collection includes the larger middle-range downloads.
- Checks: every hosted example was checked for valid distinct rankings, inclusion of the natural order, the Condorcet triple condition, and maximality against all 7! rankings. Counts, isomorphism classification, and completeness of the census are attributed to the authors’ archive.
- Software: CDL provides tools for working with Condorcet domains and restricted permutation sets. Software paper and citation →
Research paper: Dolica Akello-Egwel, Charles Leedham-Green, Alastair Litterick, Klas Markström and Søren Riis. Condorcet domains on at most seven alternatives. Mathematical Social Sciences 133 (2025), 23–33. Publisher / DOI · Full publication record.