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 is a conditional law of error aggregation. Suppose an odd number of decision makers each choose between two alternatives, one externally correct and one incorrect. If their correctness events are independent and have the same probability \(p\), a simple majority is more reliable as the odd group grows when \(p>1/2\), and its accuracy approaches one in the idealized unlimited-size limit. If \(p<1/2\), the direction reverses: a majority of increasingly many below-chance decision makers becomes increasingly likely to be wrong. At \(p=1/2\), symmetric random judgments leave majority accuracy at one-half.[1][2]
The theorem's abstraction is not “large groups are wise.” It is the reusable binary truth / independent equal-quality judgments / odd simple majority / threshold-dependent group accuracy structure. The same probability model can describe a hypothetical juror panel or a carefully specified binary classifier ensemble. It cannot be transferred to actual juries, elections, or trained models without checking the competence, independence, truth-standard, and voting-rule assumptions.[1][3]
Structural Signature¶
Sig role-phrases:
- Binary correctness-apt decision: exactly two available answers with one externally correct answer. A preference contest without a truth standard is outside this theorem.[1]
- Independent correctness indicators: each judgment has probability \(p\) of being right, and the right/wrong events are mutually independent under the model.[1]
- Equal competence: the same \(p\) applies to every member; heterogeneous competence belongs to a generalized theorem.[1]
- Odd simple majority: \(n=2m+1\) judgments are aggregated by the alternative receiving at least \(m+1\) votes, so no tie rule is needed.[1][2]
- Accuracy threshold: the binomial-tail probability of a correct majority moves upward for \(p>1/2\), downward for \(p<1/2\), and remains at one-half at the boundary.[1][2]
Independence is about the correctness events, not a claim that human beings or classifiers have no social or computational relationship of any sort. Such relationships matter when they couple errors.
What It Is Not¶
The theorem is not a universal defense of majoritarian politics or of increasing a jury's size. Real decisions may lack one accepted correct answer, voters may influence one another, and competence may differ. It is not the claim that merely uncorrelated classifier outputs suffice: a published classifier-fusion analysis demonstrates that this loose inference can fail. Nor is it a rule for even \(n\) without specifying what happens in a tie. It is a theorem under a declared probability model, not a measurement of an actual population's competence.[1][3]
Scope of Application¶
In a formal jury model, a panel votes on a binary, truth-apt question; every juror is modeled as independently correct with the same \(p\). The theorem then compares the probability of a correct majority for odd panel sizes. This does not establish what any real juror's \(p\) is, whether deliberation preserves independence, or which legal jury size is desirable.[1]
In a formal machine-learning model, independent binary prediction errors with the same accuracy can be combined by unweighted majority. Shteingart and colleagues develop the monotonic and limit derivations in a classifier formulation. That is a genuine second instantiation of the Bernoulli-majority law, not proof that ordinary trained models—often exposed to shared data and features—satisfy it. The classifier literature explicitly warns against treating weakly related outputs as if they supplied independent errors.[2][3]
Clarity¶
For odd \(n\), let \(X_i=1\) when member \(i\) is correct and \(0\) otherwise, with independent identically distributed \(X_i\sim\mathrm{Bernoulli}(p)\). The majority is right when \(\sum_i X_i\ge (n+1)/2\). Thus its accuracy is
This expression states what must be counted: all outcomes with more correct than incorrect judgments. It is not a general formula for correlated votes. At \(p=0.6\), \(P_1=0.6\), \(P_3=0.648\), and \(P_5=0.68256\); these are exact values of the model, not empirical accuracy estimates for actual juries or classifiers.[1][2]
Manages Complexity¶
The theorem compresses a potentially large set of joint voting outcomes into a single binomial tail when the assumptions hold. It also highlights the otherwise easily missed threshold: adding more votes amplifies a small advantage above one-half, but also amplifies a small disadvantage below it. The same compression is dangerous when shared evidence, copying, deliberation, or common training data couples decisions; then \(n\) bodies do not supply \(n\) independent chances to correct one another.[1][3]
Abstract Reasoning¶
Under the stated assumptions the count of correct judgments is \(\mathrm{Binomial}(n,p)\). For \(p>1/2\), increasing odd \(n\) to \(n+2\) adds two independent judgments and raises the upper-tail majority probability; as \(n\) grows through odd values, the law of large numbers places the correct-vote share near \(p>1/2\), so \(P_n(p)\to1\). By swapping “correct” and “incorrect,” the \(p<1/2\) case has \(P_n(p)\to0\). These are model conclusions, not a theorem that more people always add independent information.[1][2]
An even group can split equally. A tie-breaking rule can produce a different sequence of accuracies, so “adding one voter always helps” is not the classical odd-size monotonic statement. Likewise, differing \(p_i\) or correlation changes the distribution of the correct count and requires a different theorem.[1]
Knowledge Transfer¶
The juror and classifier settings share a role map: truth = externally correct binary alternative; units = individual judgments; competence = common \(p\); dependence condition = independent right/wrong events; aggregator = unweighted odd majority; output = probability of correct group decision. The machine-learning application substitutes algorithmic predictions for human votes without changing the mathematics. What does not transfer is evidence that any actual set of models has independent errors or that its collective decision has legal legitimacy.[2][3]
Cross-Domain Echoes¶
See how this entry connects to another domain.
Examples¶
Idealized odd jury. Let each member of an imagined panel be independently correct with \(p=0.6\) on one binary factual question. Mapped back: truth = one correct answer; agents = three or five equal-competence jurors; rule = simple majority; result = model accuracies \(0.648\) and \(0.68256\), respectively. No claim is made about observed human accuracy or a recommended court procedure.[1]
Idealized classifier ensemble. Let an odd number of binary classifiers each match the fixed reference label with probability \(0.6\), with mutually independent correctness indicators. Mapped back: truth = reference label; agents = classifiers; rule = unweighted majority; result = the same binomial-tail law. If the classifiers share important errors, this mapping's independence role fails and the Condorcet conclusion cannot simply be imported.[2][3]
Structural Tensions¶
Group size versus independent evidence. More independent, better-than-chance judgments can cancel errors; more correlated judgments may merely repeat the same error. Diagnostic: Are the correctness indicators independent in the model being used, or only superficially different outputs?[1][3]
Competence threshold versus amplification. Majority aggregation raises reliability above \(p=1/2\) but drives accuracy downward below it. Diagnostic: Which side of one-half does the assumed individual correctness lie on, and how was that assumption justified?[1][2]
Structural–Framed Character¶
Evaluative weight. The theorem gives a conditional probability comparison, not a moral verdict that majority rule is legitimate. Collective welfare, deliberation and fairness require premises beyond voter correctness.[1]
Human-practice bound. Formal Bernoulli votes and a declared tie rule define the model; applying it to jurors or classifiers requires human choices about truth labels, competence and dependence. Institutional origin. The historical jury framing names one use, but the mathematics does not require a court; an exact binary classifier model can instantiate the same probability structure.[1][2]
Vocabulary travel. Majority, independence and conditional accuracy transfer literally wherever binary decisions and a truth standard are specified. “Wisdom of crowds” outside those assumptions is only an interpretive slogan. Import versus recognition. Recognize a new instance by testing common (p), conditional independence, panel size and tie handling; merely assembling many opinions imports the name without the theorem.[2][3]
Its character: mixed-structural—a probabilistic aggregation result whose real-world application is strongly model-framed.
Structural Core vs. Domain Accent¶
Portable skeleton. Live Aggregation supplies the many-to-one majority operation used in the proof, but the theorem is not itself an aggregation operator; accordingly no strict DAG parent is staged. A general prime for “guarantee under independent evidence aggregation” would be a future-prime candidate, not an admitted node.[2]
Domain-bound mechanism. For binary truth, independent identically competent decisions with common (p), and odd-majority or specified tie handling, the binomial majority tail yields the familiar (p=½) threshold. Juror and classifier settings differ in how truth, competence and dependence are established, not in the formal conditional calculation.[1][2]
Why not prime. A broad aggregation principle travels, but this named result is a theorem with exact probability hypotheses. Legal legitimacy, deliberation, calibration and model diversity do not follow from its formula. Outside a model satisfying those hypotheses, “the crowd becomes wiser” is analogy, not an instantiation of Condorcet's theorem.[3]
Instantiates / Related Primes¶
Live prime Aggregation names the general collapse of multiple items into a summary, which the majority rule performs, but the named theorem is not itself a kind of aggregation operation. Live Ensemble Learning is an application neighbor, not a parent of the juror theorem. No canonical edge is applied.
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
Not to Be Confused With¶
The Condorcet winner criterion concerns pairwise preference comparisons and is not this binary correctness theorem. General jury theorems may relax equal competence or independence, but their conclusions require their own premises. The wisdom of crowds is a broader slogan; it does not specify \(p\), error dependence, binary truth, or tie handling. Majority voting by trained models is an implemented ensemble rule, not proof that its errors meet the theorem's assumptions.[1][3]
References¶
[1] Franz Dietrich and Kai Spiekermann, “Jury Theorems,” Stanford Encyclopedia of Philosophy (2021), §§1.1 and 2.1, formal Condorcet assumptions, odd-size monotonicity, limit, and caveats. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t
[2] Hanan Shteingart, Eran Marom, Igor Itkin, Gil Shabat, Michael Kolomenkin, Moshe Salhov, and Liran Katzir, “Majority Voting and the Condorcet's Jury Theorem,” arXiv:2002.03153v2 (2020 preprint), §§2.2–2.3, Theorems 2–3 and classifier framing. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m
[3] 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. It challenges the loose application of the theorem to classifiers without the appropriate error-independence assumptions. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j