Collective-decision theorem
Condorcet Jury Theorem
If voters independently choose between two alternatives and each is more likely than not to be correct, majority accuracy approaches one as the group grows.
P(majority correct) = sum from j>n/2 of C(n,j) p^j (1-p)^(n-j)
The convergence requires p greater than one half and sufficiently independent errors. Common information, incentives, discussion, and unequal expertise can overturn it.
Independent private signals appear first; correlated influence then passes over the same jury to show why effective sample size can be smaller than headcount.
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The animation runs automatically, pauses on the conclusion, and then repeats. The main control changes the scenario rather than scrubbing the timeline.
- CHANGE
- Jury size
- WATCH
- majority accuracy
- MEANING
- Independent private signals appear first; correlated influence then passes over the same jury to show why effective sample size can be smaller than headcount.
Many weak independent signals can make one strong majority.
A binomial majority distribution is paired with a dependency web that exposes the theorem's most important assumption.
What it actually says
The theorem formalizes one route to collective intelligence: aggregating many better-than-random, conditionally independent judgments. With p below one half, a larger majority becomes more reliably wrong.
Real institutions must therefore improve information quality and diversity, not merely add voters. Deliberation can share useful evidence but can also correlate errors.
"A useful law compresses a pattern. It does not erase the conditions that make the pattern true."
How the idea developed
The modern form emerged through observation, argument, and later refinement. The timeline separates the first insight from the version now used in textbooks and practice.[1]
Condorcet publishes his essay on majority decisions.
Social choice theory generalizes competence and dependence.
Forecasting and ensemble methods study diversity and calibration.
How the pattern works
The relation becomes useful only when its mechanism, measurement process, and operating range are visible.
Each vote contains a small signal above chance.
Independent mistakes tend not to align.
The binomial mass moves above the threshold.
The convergence requires p greater than one half and sufficiently independent errors. Common information, incentives, discussion, and unequal expertise can overturn it.
Where it earns its keep
Applications are strongest when the law changes a decision, measurement, model, or experiment rather than merely providing an analogy.
Aggregate calibrated judgments
ApplicationWeighting and independence may improve a panel.
Measure shared sources.
Protect information diversity
ApplicationLarger groups help only when signals add information.
Avoid coerced consensus.
Where it stops working
Binary truth, equal competence, sincere voting, independence, and common objectives are strong assumptions; many political choices are value conflicts rather than factual classification.
"Democracy is mathematically infallible"
Better: The theorem is conditional and narrowly framed."More people always improve accuracy"
Better: Correlated or below-chance judgments can do the opposite.Sources and further reading
Original publications and serious secondary scholarship are prioritized over summaries.
- Condorcet - Essai sur l'application de l'analyseDigitized 1785 primary text.https://gallica.bnf.fr/ark:/12148/bpt6k417181
- Nitzan and Paroush - Collective Decision MakingFormal collective-competence treatment.https://doi.org/10.1017/CBO9780511522161
- Grofman, Owen, and Feld - Thirteen Theorems in Search of the TruthReview and extensions.https://doi.org/10.1007/BF00138312