Research game 02 · 5–8 min
The Assembly
How much disagreement can a majority hold?
Optional advanced workshop: Build Fishburn’s five-option domain ↓
Optional advanced workshop · Build Fishburn’s five-option domain
What is happening? · The idea and the research
Pairwise majority voting can turn consistent individual rankings into a collective cycle. A Condorcet domain is a set of permitted ranking types that prevents this for every odd-sized electorate using those rankings, with repetition allowed.
The challenge is to retain many possible preferences. Restriction need not mean unanimity: a rich domain can still let voters disagree about every pair.
Why the certificate covers every odd electorate
We use Ward’s value-restriction criterion, as stated in Section 2 of the cited paper. On each triple of alternatives there must be some alternative that never occupies one of the three positions in any allowed ranking. A domain is certified only if every triple has such a law.
For three alternatives, removing the two orders with B last leaves four. The six orders split into two cyclic triples; any five contain one complete cyclic triple, so four is the maximum. For four alternatives, the app checks every triple separately. The eight-card recipe is single-peaked on the axis A–B–C–D. Nine is possible; a separate exhaustive check of all 9⁴ choices of one law per triple verifies that no four-alternative domain is larger.
The opening has exactly three voters, strict rankings, and no ties. All universal claims in this app concern an odd number of voters. Even-electorate ties would be left undecided; no tie-breaking rule is used to claim transitivity.
No historical permutation archive is imported. All rankings are generated locally with every symbol retained.
Research connection and credits
Condorcet’s voting paradox and classical domain theory are background. Charles Leedham-Green, Klas Markström and Søren Riis proved that a Condorcet domain on eight alternatives has at most 224 orders, attained uniquely up to isomorphism. The number counts orders, not voters. The small workshop does not re-prove their result.
The largest Condorcet domain on 8 alternatives, Social Choice and Welfare 62, 109–116 (2024; published online 5 September 2023).