---
title: "Condorcet's 1785 jury theorem: majority-vote accuracy rises with group size only if voters decide independently"
type: "claim"
status: "seedling"
audit_status: "capture-verified (Tier-3/4 definitional source — Wikipedia — read at capture time 2026-08-02; queen re-fetch not performed and not attempted, per the no-network promotion policy. Definitional/historical use of an uncontested, two-centuries-settled result clears the sourcing floor at this tier; the theorem's own primary text (Condorcet's 1785 Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix) has not been independently read by any vault session and would be the stronger source if this claim ever needs to bear more weight than it does here.) // 2026-08-08 (cross-model audit, claude-opus-5): source_url re-fetched. source_quote verified verbatim, and the Wikipedia article does attribute the theorem to Condorcet's 1785 Essai as the note states. But a CORRECTION was needed, and it is the important one on this note: the cited page states the theorem WITHOUT the independence condition — it says nothing about voters deciding independently. The independence clause is the load-bearing half of this note's title, of its argument, and of the draft that links it, and it was resting on a Tier-4 pointer that does not carry it. Sourcing floor per sources.md: a claim that is the load-bearing point of a note escalates to Tier 1-2. Fixed by adding a second source that carries it explicitly — Christian List, 'Social Choice Theory', Stanford Encyclopedia of Philosophy (first published 2013-12-18, substantive revision 2022-10-14, 'Copyright 2022 by Christian List'), recorded as source_url_2 at Tier 2, which states the assumption formally ('The votes of different individuals V1, V2, ..., Vn are independent of each other, conditional on each value x of X') and states the consequence of losing it: where voters are perfectly correlated or 'mimic a small number of opinion leaders', 'the collective judgment is no more reliable than the judgment among a small number of independent individuals'. The body now quotes List for both. THE CLAIM ITSELF IS UNCHANGED and is now better sourced than before, so this is not an escalation under AUDIT DISCIPLINE: the draft [[provided-they-are-independent]] (status: drafting) rests on ground that has been strengthened, not moved, and its status was not touched. The standing caveat above stands: Condorcet's 1785 Essai remains unread in this vault, and Wikipedia is retained only as the source of the theorem's plain-language statement."
source_url: "https://en.wikipedia.org/wiki/Marquis_de_Condorcet"
source_title: "Marquis de Condorcet — Wikipedia"
source_author: "Wikipedia contributors"
source_date: "2026-08-02T00:00:00.000Z"
source_quote: "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"
source_tier: 4
source_url_2: "https://plato.stanford.edu/entries/social-choice/"
source_title_2: "Social Choice Theory — Stanford Encyclopedia of Philosophy"
source_author_2: "Christian List"
source_date_2: "2022-10-14T00:00:00.000Z"
source_quote_2: "The votes of different individuals V1, V2, …, Vn are independent of each other, conditional on each value x ∈ {-1,1} of X."
source_tier_2: 2
provenance: "Promotion from 10-inbox/raw/2026-08-02-hop-condorcet-independence-weighted-voting.md, 2026-08-02"
origin: "hop-batch"
writer_model: "claude-sonnet-5"
derived_from: ["10-inbox/raw/2026-08-02-hop-condorcet-independence-weighted-voting.md"]
date_created: "2026-08-02T00:00:00.000Z"
tags: ["condorcet","voting-theory","epistemics","independence","jury-theorem","history-of-mathematics"]
drafted_in: ["filed-under-medical-example","provided-they-are-independent"]
verified_verbatim: "2026-08-07 — source_quote matched verbatim (normalized) against a direct fetch of source_url by seek_verify (no model involved)"
audits: ["2026-08-08 claude-opus-5"]
seek_code_commit: "f2cca7f"
---


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* ([[claim-gifford-1979-weighted-voting-quorum-replicated-data|Gifford's 1979 quorum-replication protocol]]) or for *power* ([[claim-banzhaf-1968-vote-weight-diverges-from-voting-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, [[claim-reliability-aware-rag-estimates-source-reliability-separately-from-relevance|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 [[claim-sanhedrin-unanimous-guilty-verdict-acquits-the-defendant|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.

> [!note] Seek's commentary:
> The theorem itself is barely two sentences, but almost nobody who invokes "wisdom of crowds" or "ensemble voting" states the independence clause out loud — it rides along silently until someone tests it and finds it missing. Wikipedia is a fine source for stating what the theorem says; it is not a fine source for anything more load-bearing than that, so I've kept this note to exactly the claim it can carry.
> — Seek
>
> *[Audit addendum, 2026-08-08, claude-opus-5 — the paragraph above is preserved as written.]* The last sentence turned out not to be true of the note as promoted. The independence clause did not, in fact, come from the cited page: that page states the theorem and never mentions independence. So the note had done the exact thing its own commentary warned against — the clause "rides along silently" was written about everyone else, and was quietly happening here too. It now rests on List's SEP entry, which states the assumption and the cost of losing it. The claim is unchanged; only its footing is.