When Majority Rule Eats Its Own Tail

Picture three voters and three candidates. Voter 1 ranks them A, B, C. Voter 2 ranks them B, C, A. Voter 3 ranks them C, A, B. Add up the head-to-head matchups. A beats B by 2 votes to 1. B beats C by 2 votes to 1. And C beats A by 2 votes to 1. The group's preference cycles. There is no consistent collective ranking, even though every individual voter ranked the candidates without contradiction.

This is the Condorcet paradox, named for the Marquis de Condorcet, who described it in his 1785 Essay on the Application of Analysis to the Probability of Majority Decisions. He was studying juries, not parliaments, but the discovery applied to any voting body forced to choose among three or more options.

Kenneth Arrow proved in 1951 that the paradox is unavoidable. His impossibility theorem showed that no ranked voting system can simultaneously satisfy a small list of reasonable fairness conditions when there are three or more candidates. Something has to give: either the system can be gamed, or it can ignore some voters' preferences, or it can produce cycles. Arrow won the Nobel Prize in economics in 1972, partly for this result.

The practical consequence is humbler than it sounds. Most real elections do not encounter the paradox in its pure form. But when they do, the choice of voting rule starts to matter more than the votes themselves. Different reasonable systems pick different winners from the same ballots — and the only thing they agree on is that someone is going to be unhappy.