Larger groups make better collective decisions—if voters meet one key condition
In 1785, the Marquis de Condorcet mathematically proved that if each individual voter has a greater than 50% chance of choosing correctly, majority voting approaches a 100% chance of being right as group size grows. However, if individual accuracy drops below 50%, adding more voters guarantees a wrong decision.
The 1785 Proof of Collective Intelligence
In 1785, the French philosopher and mathematician Nicolas de Condorcet published a treatise examining how probability could be applied to collective decision-making. Writing during the Enlightenment, Condorcet sought a mathematical framework to evaluate whether juries and democratic assemblies were genuinely reliable instruments for arriving at the truth. His work culminated in what is now known as Condorcet's jury theorem, a foundational result in social choice theory and formal political philosophy.
The theorem models a group of decision-makers faced with a binary choice, such as a jury determining whether a defendant is guilty or innocent, or an assembly voting on a yes-or-no policy question. Crucially, the model assumes that one of the two alternatives is objectively correct. Condorcet set out to calculate how the likelihood of the group choosing the correct option changes as the size of the voting body increases, comparing the collective outcome under simple majority rule to the reliability of an individual voter.
The Mechanism Behind the Math
The theorem relies on three core conditions: there are two options, one of which is objectively right; each voter has an independent probability of voting correctly; and every voter shares the same baseline competence, represented as a probability value. If each individual voter has a probability greater than 50 percent of choosing the correct alternative, the collective probability that a majority of the group chooses correctly will always be higher than any single member's individual accuracy.
As more voters are added to the group, this collective accuracy climbs steadily. In the mathematical limit, as the size of the assembly approaches infinity, the probability that the majority reaches the correct verdict approaches 100 percent. The intuitive mechanism behind this is identical to the law of large numbers in coin tossing: if a coin is slightly biased toward landing on heads, flipping it a handful of times might yield more tails by chance, but flipping it thousands of times virtually guarantees a majority of heads. When voters are better than random guessing, their individual errors cancel out in the aggregate, while their slight tendency toward truth reinforces itself.