Condorcet's Jury Theorem¶
Under independent equal-competence binary judgments, odd-majority accuracy rises with group size when individual correctness exceeds one-half and falls when it is below one-half.
Core Idea¶
Condorcet's jury theorem relates group size to majority accuracy under a narrow probability model. On a binary question with one externally correct answer, each voter is independently correct with the same probability \(p\), and an odd number of votes is combined by simple majority. When \(p>1/2\), accuracy rises along odd group sizes and tends to one as the idealized group grows; when \(p<1/2\), the direction reverses. The result is conditional, not a claim that any large real group is wise.[ref-be2241bf2e73][ref-2e8238d5d6fc]
Cross-Domain Echoes¶
See how this entry connects to another domain.
Scope of Application¶
An idealized three- or five-person panel can be compared when each member has independent competence \(p=0.6\): the model gives majority accuracies \(0.648\) and \(0.68256\). The same binomial structure can describe an odd ensemble of binary classifiers whose correctness indicators really are independent and equally accurate. It does not establish empirical juror rates or automatic improvement for trained models with shared errors.[ref-be2241bf2e73][ref-2e8238d5d6fc][^ref-8926c8bbf1d8]
Clarity¶
For odd \(n\), majority correctness requires at least \((n+1)/2\) correct judgments. Thus the group accuracy is the upper tail of a \(\mathrm{Binomial}(n,p)\) count. Even \(n\) needs a tie rule, and heterogeneity, dependence, multiple alternatives, or a missing truth standard changes the problem. Mere lack of correlation among classifier outputs is not the required error-independence premise.[ref-be2241bf2e73][ref-8926c8bbf1d8]
Manages Complexity¶
The theorem compresses many vote patterns into a competence threshold and a binomial tail. It makes clear that majority rule amplifies both better-than-chance and worse-than-chance tendencies. The compression is valid only when additional votes supply the modeled independent chances to offset mistakes.[ref-be2241bf2e73][ref-2e8238d5d6fc]
Abstract Reasoning¶
For odd \(n\), \(P_n(p)=\sum_{k=(n+1)/2}^{n}\binom{n}{k}p^k(1-p)^{n-k}\). Under the stated assumptions, \(P_n\) increases with odd \(n\) for \(p>1/2\), remains one-half at \(p=1/2\), and decreases for \(p<1/2\).
Knowledge Transfer¶
Map the panel and classifier cases role by role: binary truth, equal individual accuracy, independent correctness events, odd simple majority, and resulting group accuracy. The mathematics transfers; evidence that real voters or trained models satisfy those roles must be established separately. No legal or political recommendation follows from the formula alone.[ref-2e8238d5d6fc][ref-8926c8bbf1d8]
[^ref-be2241bf2e73]: Franz Dietrich and Kai Spiekermann, “Jury Theorems,” Stanford Encyclopedia of Philosophy (2021), §§1.1 and 2.1. [^ref-2e8238d5d6fc]: Hanan Shteingart et al., “Majority Voting and the Condorcet's Jury Theorem,” arXiv:2002.03153v2 (2020 preprint), §§2.2–2.3. [^ref-8926c8bbf1d8]: Stephen B. Vardeman and Max D. Morris, “Majority Voting by Independent Classifiers Can Increase Error Rates,” The American Statistician 67(2):94–96 (2013), publisher abstract.
Neighborhood in Abstraction Space¶
Condorcet's Jury Theorem sits in a sparse region of the domain-specific corpus (78th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Strategic Decision Biases & Mechanisms (29 abstractions)
Nearest neighbors
- Emotive Conjugation — 0.83
- Stable Roommates Problem — 0.82
- Defeasible Logic — 0.82
- Dual-Character Concept — 0.82
- Participation constraint (mechanism design) — 0.82
Computed from structural-signature embeddings · 2026-10-08