Mathematically speaking, democracy is next to impossible.
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.
Everyone Loses to Someone
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%. Alice wins. 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:
- One-third of voters rank Alice over Bob over Carol.
- One-third rank Bob over Carol over Alice.
- One-third rank Carol over Alice over Bob.
Now compare the candidates in pairs:
- Alice beats Bob, because the first and third groups prefer Alice to Bob.
- Bob beats Carol, because the first and second groups prefer Bob to Carol.
- Carol beats Alice, because the second and third groups prefer Carol to Alice.
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. As the secretary of the French Academy of Sciences, he’d been looking for ways to make sure the group was choosing new members fairly. (More enlightened than most, Condorcet would also question the democratic arithmetic of excluding half the population, explicitly defending women’s political rights in a 1790 essay. A few years later, during the Reign of Terror, the revolution he had supported put him in a prison cell; he was found dead two days after his arrest.)
Over the centuries, mathematicians, political scientists, and reformers have devised an astonishing variety of voting systems in hopes of breaking Condorcet’s impasse. Yet every method brings its own vulnerabilities. Change the system and you may well change the winner, even when the voters and their preferences remain exactly the same.
In 1951, the economist Kenneth Arrow showed how deep the problem ran. He sought a voting system that stayed consistent while meeting just a few simple democratic requirements. Voters should be free to rank the candidates in any order. No single voter should unilaterally dictate the outcome of an election. (Read: no kings.) If everyone prefers one option to another, the group’s ranking should reflect that preference. And the ranking of two candidates should depend only on how voters compare those two: Where voters place Carol on their ballots shouldn’t affect whether the group ranks Alice above Bob.
Each requirement sounds reasonable enough on its own. But, Arrow showed, no method can satisfy them all. “Quite often the will of the people doesn’t exist,” said Iain McLean of the University of Oxford.
Arrow’s impossibility theorem, as this result is now known, extended Condorcet’s paradox beyond majority rule, establishing a fundamental limit on how individual votes can be combined into a group decision. Different voting systems can change which of Arrow’s requirements fail, but no single system can ever preserve them all. “You have to make a choice as a system designer as to which of the conditions it’s most harmless to fail,” McLean said.
In 1980, to better understand this phenomenon, the mathematician and economist Graciela Chichilnisky turned to topology, an area of math in which shapes are considered equivalent if one can be stretched into another without tearing. In a footnote, she mused that Condorcet’s paradox in particular could be thought about as a topological problem.
Now mathematicians have proved her intuition true.
The Shape of a Paradox
What, then, does a voting paradox look like?
To answer this question, the new work builds on a model that Yuliy Baryshnikov, now a professor at the University of Illinois, Urbana-Champaign, proposed in 1993. First, consider a single voter’s ballot in an election with three candidates. There are six possible ways to rank the options. Each one can be represented by a triangle:
