Cyclic majorities

How can a group prefer A to B, B to C, and C to A when no member of it does?

You would not tolerate a friend who preferred tea to coffee, coffee to cocoa, and cocoa to tea. We would say he had not really got a preference at all, just three opinions that fail to fit together. Transitivity is not an extra virtue we ask of preferences; it is close to what having a preference means.

So here is the discomfort. Take three people whose preferences are each perfectly transitive, ask them to vote pairwise by simple majority, and the group produces exactly the incoherence we would not tolerate in one person. Where did the inconsistency come from, if it was in nobody?

Build the smallest example and check every claim by counting.

| | first | second | third | |---|---|---|---| | Voter 1 | A | B | C | | Voter 2 | B | C | A | | Voter 3 | C | A | B |

Each row is orderly. Now hold three referendums, counting each. A against B: Voters 1 and 3 place A above B — A wins two to one. B against C: Voters 1 and 2 place B above C — B wins two to one. C against A: Voters 2 and 3 place C above A — C wins two to one.

So the group prefers A to B, B to C, and C to A, each by a solid majority, and no individual is confused about anything. Notice what makes it work: read the table down its columns and each candidate appears exactly once in first place, once in second, once in third. The three voters disagree not about which option is best but about the whole shape of the ordering — they are three rotations of one another. That rotational structure is what defeats majority rule, because every pairwise contest is decided by a different two-thirds of the electorate.

How special is it? Count. Each of the three voters picks one of 3! = 6 rankings, so there are 6 cubed = 216 profiles. If any two voters share a ranking, that ranking commands a majority on every pair and the group ordering is simply that voter's — transitive. So all three must differ. And only two sets of three distinct rankings rotate in the way above: {ABC, BCA, CAB} and its mirror {ACB, CBA, BAC}. Each can be handed to the three labelled voters in 3! = 6 ways, giving 12 cyclic profiles out of 216 — one in eighteen, about 5.6 percent, if every ranking is equally likely.

Rare, then. But what the cycle does matters more than how often it occurs. Suppose the group decides by a sequence of pairwise votes — a committee agenda, a parliamentary amendment procedure. Someone chooses the order. Try all three.

Vote A against B first: A wins, then A meets C, and C wins. C is elected. Vote B against C first: B wins, then B meets A, and A wins. A is elected. Vote C against A first: C wins, then C meets B, and B wins. B is elected.

The pattern is exact: whichever option is held back from round one wins. So the agenda-setter — the chair, the clerk drafting the order paper — determines the outcome without casting a vote, without lying, and without anyone voting insincerely. The procedure is impeccable and the result chosen in advance.

What this adds to its neighbour. The walk-through on voting rules already shows different counting rules electing different winners from identical ballots, and states Arrow's theorem. This piece is narrower and more mechanical: the internal structure of a single cycle, and the fact that a cycle converts procedural control into decisive power. Arrow tells you no rule escapes; agenda control tells you what someone can do on a Tuesday afternoon with a rule that has not escaped.

The rescue that works. Cycles need that rotational disagreement. Suppose instead every voter places the options on one shared dimension and prefers whatever is nearer their own ideal point. Then no cycle can form, and the Condorcet winner is the option at the median voter's ideal point — Black's median voter theorem. That is the limiting case that breaks the result: where everyone agrees about the dimension and disagrees only about position, majority rule behaves. Cycles are the signature of multi-dimensional disagreement, common in bundled policy and rare in questions of "how much".

Agenda power is real but not unlimited. The three agendas above assumed sincere voting each round. In the first agenda, Voter 1 can see that if A survives round one, C takes the prize — and he prefers B to C, so he votes for B in round one against his own ranking, and B is elected. Sophisticated voting changes which option the agenda delivers. It does not restore neutrality, since the setter can anticipate the anticipation, but "the chair simply picks the winner" is a claim about sincere voting only.

The folklore. "Condorcet's paradox proves the general will does not exist" overshoots. What is proved is that pairwise majority comparison need not be transitive, that with three options and three voters under equally likely rankings it fails about 5.6 percent of the time, and that the rate rises with more options and, more slowly, with more voters — toward roughly 9 percent for three options and a large electorate, a figure I state from recall. What is not proved is that any particular assembly is cycling. Detecting a real cycle needs full rankings, which ballots rarely record, so confident claims of empirical cycles usually rest on reconstruction.

• The chaos theorems for spatial voting, where in two or more dimensions almost any outcome is reachable by some agenda.
• Tournament solutions — the top cycle, the uncovered set, Copeland — which pick a winner from a cycle by rule rather than agenda.

Every term the collection defines is gathered in the glossary.