The Condorcet Jury Theorem: The 240-Year-Old Math Behind Better Decisions

The marquis who did math about democracy

Marie Jean Antoine Nicolas de Caritat, Marquis de Condorcet (1743–1794), was the Enlightenment’s most committed believer in the idea that human affairs could be reasoned about with the same rigor as physics. A brilliant mathematician elected to the Académie des sciences in his twenties, he was also — unusually for his century and his class — an early and vocal advocate of free public education, women’s suffrage, and the abolition of slavery. Where his contemporaries debated whether ordinary people should govern, Condorcet asked a stranger and more productive question: under what conditions would their collective judgment actually be correct?

His answer appeared in 1785 in the Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix — the “Essay on the Application of Analysis to the Probability of Majority Decisions.” Buried in its dense probability calculations is a result of startling simplicity and power, now known as the Condorcet Jury Theorem.

The story of the man ends darkly. Condorcet backed the Revolution, drafted a constitution, and then fell foul of the Terror. Outlawed in 1793, he hid in a Paris boarding house and — facing death — wrote his most optimistic work, the Sketch for a Historical Picture of the Progress of the Human Mind. Arrested in March 1794 after fleeing his refuge, he was found dead in his cell at Bourg-la-Reine two days later. The mathematician of collective wisdom was killed by a collective that had stopped listening to reasons — an irony no historian of the period fails to note.

The theorem, in plain English

Strip away the eighteenth-century notation and the theorem says this:

Take a yes/no question that has a correct answer. If each person in a group is more likely than chance to get it right, and each judges independently, then the probability that the majority gets it right grows with the size of the group — and approaches certainty as the group becomes large.

The intuition is error cancellation. Each voter is signal plus noise. Because everyone is right more often than wrong, the signal points the same way for everyone, while the noise scatters randomly. Individual mistakes are uncorrelated coin flips; the shared grain of truth accumulates. Majority voting is simply a machine for letting the noise cancel and the signal add up — the same arithmetic behind the wisdom of crowds, which would later make 787 fairgoers collectively out-judge cattle experts in Galton’s ox experiment, applied to votes instead of estimates.

The numbers: how fast majorities get smart

The theorem’s force is easiest to feel with concrete numbers. Suppose each person independently answers a binary question correctly 60% of the time — decent but far from expert. The probability that a majority of the group answers correctly:

Read the 60% column downward: three people barely improve on one; eleven people are meaningfully better; a hundred and one are right 97.9% of the time; five hundred and one are practically infallible. Even a whisker of individual competence — the 51% column — compounds relentlessly with scale: about 58% for a group of 101, roughly 74% at a thousand voters, and about 98% at ten thousand. Mediocre individual judgment, aggregated correctly, produces excellent collective judgment. That is the theorem’s optimistic face.

The dark twin: when p falls below one half

Now read the 45% column. The same mathematics that makes competent crowds nearly infallible makes incompetent crowds nearly infallibly wrong. If each individual is right only 45% of the time — systematically misled on this particular question — a majority of 101 is right just 15.6% of the time, and a majority of 501 about 1.2% of the time. Aggregation doesn’t launder bad judgment; it concentrates it. A large group that shares a systematic misconception will be more confidently wrong than any of its members.

This is the half of the theorem that rarely makes it into celebrations of crowd wisdom, and it has a sharp practical edge. Whether a group’s scale is an asset or a liability depends entirely on which side of the 50% line its members sit for the question at hand — which is why misinformation, and the media environment that shapes individual accuracy, is not a side issue for collective decision-making but the central one. Scale amplifies whatever you feed it.

The assumptions — and what breaks when they break

The theorem is an if-then statement, and everything interesting about applying it lives in the “if.” Four assumptions carry the result:

Two and a half centuries of scholarship have stress-tested each assumption. Grofman, Owen and Feld’s “Thirteen theorems in search of the truth” (Theory and Decision, 1983) showed the result survives heterogeneous competence — voters don’t all need the same accuracy; under reasonable distributional conditions, a mean accuracy above one half suffices. Krishna Ladha (1992) tackled correlation and found that correlated votes weaken the guarantee without always destroying it: the effective size of a correlated crowd is smaller than its headcount, because copied votes add noise-sharing rather than fresh information.

In practice, independence is the assumption that fails first and hardest. When early opinions are visible, later voters rationally discount their own information and follow — an information cascade — and when everyone draws on the same few sources, votes correlate even without anyone copying anyone. A million voters reading the same feed can carry roughly the informational weight of a handful of independent ones. The theorem doesn’t fail gracefully; it fails silently, while the group still feels like a large, confident majority.

Cycles: the paradox Condorcet also found

The same 1785 Essai contains a second, more unsettling discovery. With three or more options, majority preferences can cycle: a group can prefer A to B, B to C, and yet C to A, even though every individual’s preferences are perfectly consistent. This Condorcet paradox means “the will of the majority” may simply not exist as a coherent ranking — which option wins can depend on the order votes are taken.

That crack in majority rule became the foundation of an entire field. In 1951, Kenneth Arrow generalized it into his impossibility theorem: no rank-based voting system can satisfy a short list of reasonable fairness conditions simultaneously. Social choice theory — how voting systems aggregate preferences, as distinct from how juries aggregate judgments and markets aggregate beliefs — is a subject we’ll return to in its own right. For now, the practical moral: the jury theorem shines on factual questions with right answers; the moment a decision is about competing values rather than competing facts, different machinery (and different math) applies.

240 years of applications

For a piece of eighteenth-century probability, the theorem has aged remarkably well:

The machine learning application deserves a second look, because it shows the theorem working as an engineering blueprint rather than a metaphor. A random forest deliberately trains each tree on a random subsample of data and features — not to make any single tree better (it makes them worse), but to make their errors less correlated. Accepting weaker individuals to buy independence is exactly the trade the jury theorem recommends, and the ensemble’s accuracy is the payoff. Teams designing DAO governance mechanisms face the same design problem in reverse: token-weighted voting concentrates influence, and publicly visible running tallies invite herding — both of which shrink the effective jury.

What the theorem can’t do — and what has to happen before the vote

The jury theorem governs one moment in the life of a decision: the aggregation step, when individual judgments become a collective verdict. It is silent about everything upstream — how the question was framed, which options made it to the ballot, what information voters saw — and everything downstream. A perfectly aggregated vote on a badly framed question yields an excellent answer to the wrong problem.

It also, by design, discards the reasons. A vote records that forty people favored option A, never why — so the minority’s strongest objection vanishes, the majority’s reasoning can’t be audited later, and the group learns nothing transferable for the next decision. This is where structured deliberation complements aggregation rather than competing with it. In collaborative decision making done well, the group first surfaces arguments independently — preserving the independence the theorem needs — then evaluates them openly, and only then converges.

Argumentree is built around exactly that sequence. Participants contribute pro and con arguments independently and asynchronously, before anyone anchors the room — protecting the theorem’s load-bearing assumption. Each argument is then rated on explicit dimensions (helpfulness, clarity, accuracy, completeness), and ratings aggregate into consensus scores: Condorcet’s error-cancelling arithmetic, applied to the quality of reasons rather than just the count of votes. And unlike a ballot box, the full argument tree and audit trail preserve why the group decided — so the decision can be explained, revisited, and learned from. The marquis supplied the math for trusting a majority; two hundred and forty years on, the remaining work is making sure the majority’s judgment is independent, informed, and on the record.