Condorcet Paradox

THE RABID SHEEP WRESTLES WITH PARADOX

Consider a hypothetical simplification of a Burlington election, with just three kinds of ballots:

Montroll, Kiss, Wright	1332
Kiss, Wright, Montroll	371
Wright, Montroll, Kiss	1513

A majority of voters—the Kiss-first and Wright-first groups—prefers Wright to Montroll. Another majority, the Montroll-first and Kiss-first groups, prefers Kiss to Wright. If most people like Kiss better than Wright, and most people like Wright better than Montroll, doesn’t that mean Kiss should win again? There’s just one problem: people like Montroll better than Kiss by a resounding 2845 to 371. There’s a bizarre vote triangle: Kiss beats Wright, Wright beats Montroll, Montroll beats Kiss. Every candidate would lose a one-on-one race to one of the other two candidates. So how can anyone at all rightfully take office?

Vexing circles like this are called Condorcet paradoxes, after the French Enlightenment philosopher who first discovered them in the late eighteenth century. Marie-Jean-Antoine-Nicolas de Caritat, Marquis de Condorcet, was a leading liberal thinker in the run-up to the French Revolution, eventually becoming president of the Legislative Assembly. He was an unlikely politician—shy and prone to exhaustion, with a speaking style so quiet and hurried that his proposals often went unheard in the raucous revolutionary chamber. On the other hand, he became quickly exasperated with people whose intellectual standards didn’t match his own. This combination of timidity and temper led his mentor Jacques Turgot to nickname him “ le mouton enragé,” or “the rabid sheep.”

The political virtue Condorcet did possess was a passionate, never-wavering belief in reason, and especially mathematics, as an organizing principle of human affairs. His allegiance to reason was standard stuff for the Enlightenment thinkers, but his further belief that the social and moral world could be analyzed by equations and formulas was novel. He was the first social scientist in the modern sense. (Condorcet’s term was “social mathematics.”) Condorcet, born into the aristocracy, quickly came to the view that universal laws of thought should take precedence over the whims of kings. He agreed with Rousseau’s claim that the “general will” of the people should hold sway on governments but was not, like Rousseau, content to accept this claim as a self-evident principle. For Condorcet, the rule of the majority needed a mathematical justification, and he found one in the theory of probability.

Condorcet lays out his theory in his 1785 treatise Essay on the Application of Analysis to the Probability of Majority Decisions. A simple version: suppose a seven-person jury has to decide a defendant’s guilt. Four say the defendant is guilty, and only three believe he’s innocent. Let’s say each of these citizens has a 51% chance of holding the correct view. If their judgments are independent, a 4−3 majority in the correct direction is more likely than a 4−3 majority favoring the incorrect choice.

It’s a little like the World Series. If the Phillies and the Tigers are facing off, and we agree that the Phillies are a bit better than the Tigers—say, they have a 51% chance of winning each game—then the Phillies are more likely to win the Series 4−3 than to lose by the same margin. If the World Series were best of fifteen instead of best of seven, Philadelphia’s advantage would be even greater.

Condorcet’s so-called “jury theorem” shows that a sufficiently large jury is very likely to arrive at the right outcome under the simplified assumptions just described: independent judgments and an individual chance of correctness above even odds. Shared errors or misleading common evidence can undermine that result. If the majority of people believe something, Condorcet said, that must be taken as strong evidence that it is correct. We are mathematically justified in trusting a sufficiently large majority—even when it contradicts our own preexisting beliefs. “ I must act not by what I think reasonable,” Condorcet wrote, “but by what all who, like me, have abstracted from their own opinion must regard as conforming to reason and truth.” The role of the jury is much like the role of the audience on Who Wants to Be a Millionaire? When we have the chance to query a collective, Condorcet thought, even a collective of unknown and unqualified peers, we ought to value their majority opinion above our own.

Condorcet’s wonkish approach made him a favorite of American statesmen of a scientific bent, like Thomas Jefferson (with whom he shared a fervent interest in standardizing units of measure). John Adams, by contrast, had no use for Condorcet; in the margins of Condorcet’s books he assessed the author as a “quack” and a “ mathematical charlatan.” Adams viewed Condorcet as a hopelessly wild-eyed theorist whose ideas could never work in practice, and as a bad influence on the similarly inclined Jefferson. Indeed, Condorcet’s mathematically inspired Girondin Constitution, with its intricate election rules, was never adopted, in France or anywhere else. On the positive side, Condorcet’s practice of following ideas to their logical conclusions led him to insist, almost alone among his peers, that the much-discussed Rights of Man belonged to women, too.

In 1770, the twenty-seven-year-old Condorcet and his mathematical mentor, Jean le Rond d’Alembert, a coeditor of the Encyclopédie, made an extended visit to Voltaire’s house at Ferney on the Swiss border. The mathophile Voltaire, then in his seventies and in faltering health, quickly adopted Condorcet as a favorite, seeing in the young up-and-comer his best hope of passing rationalistic Enlightenment principles to the next generation of French thinkers. It might have helped that Condorcet wrote a formal éloge (memorial appreciation) for the Royal Academy about Voltaire’s old friend La Condamine, who had made Voltaire rich with his lottery scheme. Voltaire and Condorcet quickly struck up a vigorous correspondence, Condorcet keeping the older man abreast of the latest political developments in Paris.

Some friction between the two arose from another of Condorcet’s éloges, the one for Blaise Pascal. Condorcet rightly praised Pascal as a great scientist. Without the development of probability theory, launched by Pascal and Fermat, Condorcet could not have done his own scientific work. Condorcet, like Voltaire, rejected the reasoning of Pascal’s wager, but for a different reason. Voltaire found the idea of treating metaphysical matters like a dice game to be offensively unserious. Condorcet, like R. A. Fisher after him, had a more mathematical objection: he didn’t accept the use of probabilistic language to talk about questions like God’s existence, which weren’t literally governed by chance. But Pascal’s determination to view human thought and behavior through a mathematical lens was naturally appealing to the budding “social mathematician.”

Voltaire, by contrast, viewed Pascal’s work as fundamentally driven by religious fanaticism he had no use for, and rejected Pascal’s suggestion that mathematics could speak to matters beyond the observable world as not only wrong but dangerous. Voltaire described Condorcet’s éloge as “ so beautiful that it was frightening . . . if he [Pascal] was such a great man, then the rest of us are total idiots for not being able to think like him. Condorcet will do us great harm if he publishes this book as it was sent to me.” One sees here a legitimate intellectual difference, but also a mentor’s jealous annoyance at his protégé’s flirtation with a philosophical adversary. You can almost hear Voltaire saying, “Who’s it gonna be, kid, him or me?” Condorcet managed never to make that choice (though he did bow to Voltaire and tone down his praise of Pascal in later editions). He split the difference, combining Pascal’s devotion to the broad application of mathematical principles with Voltaire’s sunny faith in reason, secularism, and progress.

When it came to voting, Condorcet was every inch the mathematician. A typical person might look at the results of Florida 2000 and say, “Huh, weird: a more left-wing candidate ended up swinging the election to the Republican.” Or they might look at Burlington 2009 and say, “Huh, weird: the centrist guy who most people basically liked got thrown out in the first round.” For a mathematician, that “Huh, weird” feeling comes as an intellectual challenge. Can you say in some precise way what makes it weird? Can you formalize what it would mean for a voting system not to be weird?

Condorcet thought he could. He wrote down an axiom —that is, a statement he took to be so self-evident as to require no justification. Here it is:

If the majority of voters prefer candidate A to candidate B, then candidate B cannot be the people’s choice.

Condorcet wrote admiringly of Borda’s work, but considered the Borda count unsatisfactory because, in Borda’s system, as with plurality voting, the addition of a third alternative can flip the election from candidate A to candidate B. That violates Condorcet’s axiom: if A would win a two-person race against B, then B can’t be the winner of a three-person race that includes A.

Condorcet intended to build a mathematical theory of voting from his axiom, just as Euclid had built an entire theory of geometry on his five axioms about the behavior of points, lines, and circles:

• There is a line joining any two points.
• Any line segment can be extended to a line segment of any desired length.
• For every line segment L, there is a circle that has L as a radius.
• All right angles are congruent to each other.
• If P is a point and L is a line not passing through P, there is exactly one line through P parallel to L.

Imagine what would happen if someone constructed a complicated geometric argument showing that Euclid’s axioms led, inexorably, to a contradiction. Does that seem completely impossible? Be warned—geometry harbors many mysteries. In 1924, Stefan Banach and Alfred Tarski showed how to take a sphere apart into six pieces, move the pieces around, and reassemble them into two spheres, each the same size as the first. How can it be? Because some natural set of axioms that our experience might lead us to believe about three-dimensional bodies, their volumes, and their motions simply can’t all be true, however intuitively correct they may seem. Of course, the Banach-Tarski pieces are shapes of infinitely complex intricacy, not things that can be realized in the crude physical world. So the obvious business model of buying a platinum sphere, breaking it into Banach-Tarski pieces, putting the pieces together into two new spheres, and repeating until you have a wagonload of precious metal is not going to work.

If there were a contradiction in Euclid’s axioms, geometers would freak out, and rightly so—because it would mean that one or more of the axioms they relied on was not, in fact, correct. We could even put it more pungently—if there’s a contradiction in Euclid’s axioms, then points, lines, and circles, as Euclid understood them, do not exist.

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That’s the disgusting situation that faced Condorcet when he discovered his paradox. For the hypothetical ballots in the table above, Condorcet’s axiom says Montroll cannot be elected, because he loses the head-to-head matchup to Wright. The same goes for Wright, who loses to Kiss, and for Kiss, who loses to Montroll. There is no such thing as the people’s choice. It just doesn’t exist.

Condorcet’s paradox presented a grave challenge to his logically grounded worldview. If there is an objectively correct ranking of candidates, it can hardly be the case that Kiss is better than Wright, who is better than Montroll, who is better than Kiss. Condorcet was forced to concede that in the presence of such examples, his axiom had to be weakened: the majority could sometimes be wrong. But the problem remained of piercing the fog of contradiction to divine the people’s actual will—for Condorcet never really doubted there was such a thing.