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Philosophy for Kids

Can a Bigger Group Always Make Better Decisions?

A Jury of Twelve—or a Thousand?

If twelve jurors are good, would a thousand be even better? Not always.

Imagine you are in a courtroom. The jury has twelve members who must decide whether the defendant is guilty. You might think adding more jurors would make the verdict more reliable. But what if adding people actually made the group more likely to be wrong? This puzzle is at the heart of jury theorems—mathematical ideas that ask how group size affects the quality of decisions. The decisions could be legal verdicts, parliamentary votes, or even your classroom choosing a movie. As long as there is a correct or better answer (like who really committed the crime), we can ask: does a bigger group make a smarter choice?

The first person to tackle this question with math was the French philosopher and mathematician Marquis de Condorcet (1743–1794). He started something that philosophers, political scientists, and mathematicians still debate today.

Condorcet’s Optimistic Math

Condorcet imagined voters as biased coins, each slightly tilted toward the truth.

Condorcet imagined a very simple situation: a group votes on one yes-or-no question, like “guilty or innocent?” The group uses majority rule—whichever side gets more than half the votes wins. He made two strong assumptions about the voters:

  • Independence: Each voter’s judgment is not influenced by anyone else’s. Knowing how one person voted tells you nothing about how another voted.
  • Competence: Each voter has a better-than-even chance of being correct (more than 50 %), and all voters have the same chance.

Given these, Condorcet proved a striking result. As you add voters, the probability that the majority is correct increases. And if the group could grow forever, the probability would approach 100 %—the group would be almost certainly right. The crowd gets wiser and wiser, eventually becoming nearly perfect.

You can picture this with a biased coin. Each toss has, say, a 55 % chance of landing “correct.” With many tosses, the majority of tosses will be correct almost all the time. Condorcet’s theorem seemed to show that large democracies would make excellent decisions—as long as voters were independent and minimally competent.

When everyone watches the same news, their opinions can become linked—even if they vote in secret.

But real voters are not independent. Imagine every juror watches the same misleading news report before the trial. If the report wrongly suggests the defendant is guilty, many jurors will believe that—together. Their votes are now correlated. If one juror makes a mistake, others are more likely to make the same mistake. This is the problem of common causes: factors that influence many people at once, like shared evidence, the same education, or even a noisy courtroom. Causal independence (voting in secret) doesn’t guarantee probabilistic independence, because votes can still be connected through these shared influences.

In real life, common causes usually create positive correlation: if someone votes correctly, it raises the odds that others do too, because they’ve all been exposed to helpful evidence. If someone errs, others likely err as well. This breaks Condorcet’s independence assumption, and his optimistic conclusion becomes shaky.

Fixing Independence—at a Price

If we could “unlock” the hidden facts behind everyone’s vote, we might restore independence.

Some philosophers and mathematicians tried to save the idea by conditionalising on the hidden facts. They said: okay, votes aren’t independent overall, but they might be independent once you know the common causes. For example, if you know exactly what evidence all jurors saw, then knowing Jane’s vote tells you nothing extra about John’s vote—you already know the evidence that guided them both.

This leads to the Conditional Jury Theorem. It uses two new assumptions:

  • Conditional Independence: Given any particular value of the “facts” (like the true state of guilt or innocence, or the shared evidence), voters’ judgments are independent.
  • Conditional Competence: Given any value of those facts, each voter still has a better-than-even chance of being correct.

Under these assumptions, the theorem again says that majority correctness increases with group size and eventually reaches 100 %. But there is a serious tension. To make conditional independence plausible, you need to pack a lot of information into the “facts” variable—ideally all common causes. But if you include everything, then in some fact scenarios (like a case with deliberately misleading evidence), voters might be wrong more often than right, violating conditional competence. If you keep the facts minimal to avoid that, you lose independence. You can’t easily have both.

The Competence-Sensitive Solution: Wise, Not Infallible

The group’s accuracy depends on how often the facts are helpful rather than misleading.

A more realistic approach drops the demand that voters are always competent. Instead, it says voters tend to be competent: their chance of being correct is more often above 50 % than below it by the same amount. For instance, they might be right 51 % of the time more often than they are right only 49 % of the time. This is called Tendency to Competence.

The Competence-Sensitive Jury Theorem combines conditional independence with tendency to competence. The result? Majority correctness still increases as the group gets larger. But it does not race to 100 %. Instead, it approaches a limit: the probability that the underlying facts are truth-conducive (plus half the chance they are neutral). Large groups can still be wrong if the facts are misleading, but they do better than small groups up to that point.

This theorem captures the Increasing-Reliability Hypothesis—bigger groups are more reliable—without claiming they become infallible. It matches our intuition that a huge crowd can’t overcome fundamentally bad evidence. It also helps explain why real democracies can be wise but not perfect.

What This Means for Your World

When your class votes, the size of the group matters—but so does the quality of the information you share.

These theorems matter far beyond courtrooms. Think about how your student council makes decisions, or how you and your friends pick a movie. If everyone streams the same trailer and discusses it together, you are sharing common evidence. Deliberation can make you more competent (you learn from others) but it also reduces independence (you start thinking alike). Finding the right balance is key.

Jury theorems also fuel a big debate about democracy. Should we trust large elections to make wise choices, or rely on small groups of experts? Condorcet’s original work suggested that if voters are independent and slightly competent, huge electorates are almost always right. But modern refinements show that independence is fragile, and that the quality of information matters enormously. Diversity—in backgrounds, perspectives, and information—helps because it makes views less correlated. Without it, even a million voters can march in the wrong direction together.

So next time you’re part of a group decision, ask: Are we truly independent thinkers, or are we all watching the same “trailer”? And how competent are we when facing tricky facts? The math doesn’t give final answers, but it sharpens the questions we all need to ask.

Think about it

  1. Imagine your class votes on the best movie of the year after everyone watches the same five trailers. Does that shared experience make the vote more or less reliable? Why?
  2. If you could add 500 randomly chosen students to your school’s governing board, would the board’s decisions become smarter? What risks might come with a much larger group?
  3. Should a country use a large popular vote instead of a small panel of experts to decide a complex scientific policy, like how to fight climate change? What are the strongest arguments for each side?

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