Condorcet's 1785 jury theorem: majority-vote accuracy rises with group size only if voters decide independently

condorcetvoting-theoryepistemicsindependencejury-theoremhistory-of-mathematics

The Marquis de Condorcet (1743–1794) proved, in his 1785 Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix, what is now called the Condorcet Jury Theorem: "if each member of a voting group is more likely than not to make a correct decision, the probability that the highest vote of the group is the correct decision increases as the number of members of the group increases." Larger juries, committees, or panels converge toward certainty — but only under one precondition the theorem requires as a premise and cannot itself guarantee: the voters must decide independently of one another. Christian List's Stanford Encyclopedia of Philosophy entry on social choice theory states the assumption formally — "The votes of different individuals V1, V2, …, Vn are independent of each other, conditional on each value x ∈ {-1,1} of X" — and spells out the cost of losing it: where voters are perfectly correlated, or merely "mimic a small number of opinion leaders," "the collective judgment is no more reliable than the judgment among a small number of independent individuals." Correlated voters do not average out their mistakes; they compound them.

The theorem is the accuracy-theoretic ancestor of a family of modern "weighted majority voting" schemes that reason about combining judgments to get closer to a correct answer — distinct from mechanisms that use "weighted voting" for consistency (Gifford's 1979 quorum-replication protocol) or for power (Banzhaf's 1968 voting-power index); see observation-weighted-voting-power-gap-recurs-across-cs-law-regulation for how those two non-accuracy lineages relate to each other. A 2025 retrieval-augmented-generation system, RA-RAG, fuses retrieved answers by "weighted majority voting" without citing Condorcet, but is reasoning about the same problem the theorem formalized 240 years earlier: does combining judgments make the combined judgment more likely correct. Whether that independence precondition actually holds for a 2024 LLM ensemble is tested directly in claim-lefort-2024-llm-ensembling-marginal-gains-non-independent-errors. The theorem's dependence on independence is also the mathematical mirror of Talmudic law's disqualification of a unanimous capital verdict: one treats voter independence as the source of a majority's accuracy, the other treats total agreement as proof independence broke down.