Can a majority prefer A over B, B over C, and C over A?

Yes — and it's not a trick question. Imagine the voters of an imaginary Cactus County choosing between the incumbent Alex, rival Beatrice, and independent newcomer Charlie. Depending on how each voter ranks the three, a clear majority can prefer Alex to Beatrice, another majority prefer Beatrice to Charlie, and yet another majority prefer Charlie to Alex — even though every single voter's own ranking is perfectly rational and consistent. Combine the three majorities and you get a loop: Alex beats Beatrice beats Charlie beats Alex, with no candidate who can beat all the others one-on-one.

This is Condorcet's paradox, named for the French mathematician and political philosopher who worked it out in the late 18th century — though the Catalan philosopher and theologian Ramon Llull had actually found the same problem five hundred years earlier while thinking about how to elect church officials, and his writing on it sat forgotten until the 21st century. The paradox turns out to be a special case of a much bigger result, Arrow's impossibility theorem, which shows that any system for turning individual rankings into a group decision must either contradict itself, hand power to one dictator, or use more information than a simple ranking provides.

In practice, these cycles are rarer than the simple three-voter example suggests — one realistic model of large elections puts the odds well under 1%, and an analysis of 883 three-candidate contests drawn from real ranked-ballot elections of the Electoral Reform Society found a cycle in only 0.7% of them. But they do happen: in Minneapolis's 2021 Ward 2 city council election, voters' preferences among three candidates circled back on themselves with no one beating both of the others.

The paradox isn't just a curiosity for mathematicians — it's a tool. In legislatures, it's the logic behind the "poison pill amendment": deliberately engineering a circular vote to sink a bill that would otherwise pass. And it explains why, when a cycle is lurking, simply adding or removing a losing candidate can change who wins, even though no voter changed their mind.