Why voting can produce a result nobody wanted

Three people rank three options. Every ranking is coherent. A majority prefers A to B, a majority prefers B to C, and a majority prefers C to A. There is no winner, and whichever pair is voted on first determines the result.

What everyone sees

A collective decision is imagined as a large version of an individual one — preferences summed into a group preference that behaves like a preference. It does not behave like one. Transitivity is a property of individual rankings that is not inherited by majority comparison, and its absence has consequences the procedure conceals.

What is actually happening

Condorcet identified the cycling case in the eighteenth century, showing that pairwise majority comparisons over three or more options can produce a preference loop even when every individual ranking is transitive. Arrow generalised the result: no aggregation rule can simultaneously satisfy a short list of reasonable conditions — unrestricted domain, transitivity of the social ranking, independence of irrelevant alternatives, and non-dictatorship — which means every voting system available makes a trade among them rather than avoiding the problem. The practical consequence is that agenda order becomes decisive whenever a cycle exists, so control over sequencing is control over outcome, exercised through a rule that appears purely procedural.

Why it stays hidden

The cycle hides because voting procedures return a single winner regardless. A sequence of pairwise votes terminates, a name is announced, and nothing in the output indicates that a different order would have produced a different name. The record shows a decision, not the loop it was cut from.