How Voting Is Like a Klein Bottle

That may sound like the premise of a Silicon Valley billionaire’s six-hour podcast, but it’s actually a statement about a remarkably stubborn problem: How do you translate a group of individual preferences into a single choice? In 1785, the French mathematician and philosopher Nicolas de Condorcet discovered that a majority can sometimes vote itself into a logical loop. In elections with three or more candidates, each voter might rank the candidates in a perfectly clear and consistent way — yet when all the votes are tallied, every candidate loses in a head-to-head contest against one of their rivals.

Changing the rules only gets us so far: As the economist Kenneth Arrow later proved, any method for combining individual rankings into a collective order must sacrifice at least one seemingly reasonable democratic requirement. Now three researchers have shown that Condorcet’s 240-year-old paradox has a shape, and it’s suitably peculiar: a Klein bottle, a surface with no distinct inside or outside. The result gives mathematicians a geometric way to explore why seemingly reasonable rules for collective decision-making can become impossible to reconcile.

When there are only two choices, the will of the people is easy to divine: Barring a tie, the candidate preferred by a majority wins. Add a third contender, however, and things get weird. Suddenly, even agreeing on what it means to win becomes a problem.

You might simply elect whoever receives the most votes, as in a typical American election. But suppose Alice receives 35% of the vote, Bob 34%, and Carol 31%. Yet if Carol’s supporters all prefer Bob to Alice, then 65% of the electorate would rather have Bob.

By one perfectly reasonable definition of majority rule, the wrong candidate has won. Before anyone reaches for a tinfoil hat, let’s be clear: Every ballot is counted correctly. The question is what should count as winning.

One solution is to ask voters to rank the candidates, then choose whoever would defeat every rival in a head-to-head contest. But consider the following scenario: Every voter has a perfectly coherent ranking, yet together they have produced a result that runs in circles. This is the paradox that Condorcet, a passionate believer in democracy, encountered in the late 18th century.