Decision Science, Governance & Ensemble ML
Condorcet Jury Theorem Majority Accuracy Formula
Calculate majority-vote accuracy for 3-, 5-, and 7-member independent committees/ensembles and verify Marquis de Condorcet’s (1785) p < 0.50 crowd-degradation trap.
Quantify Wisdom of Crowds (and ensemble classifier boosting!) via Condorcet’s (1785) Jury Theorem!
Published in 1785 by the Marquis de Condorcet and foundational today in both board governance and random-forest ensemble machine learning, the Condorcet Jury Theorem evaluates an odd N-member panel voting independently on a binary truth where each member is right with probability p in (0, 1) (and wrong with q = 1 - p): 3-Voter Majority Accuracy is P_3(p) = p^3 + 3*p^2*q = 3*p^2 - 2*p^3; 5-Voter Majority Accuracy is P_5(p) = p^5 + 5*p^4*q + 10*p^3*q^2; and 7-Voter Majority Accuracy is P_7(p) = p^7 + 7*p^6*q + 21*p^5*q^2 + 35*p^4*q^3! Whenever p > 0.50, adding independent voters pushes P_N(p) -> 1.00; whenever p < 0.50 (systematic bias!), adding voters makes the majority STRICTLY WORSE than a single individual (P_N(p) -> 0.00)!
Verify by hand at Individual_Accuracy_p = 0.70 (B2): a solo expert is right 70.00% of the time; a 3-person majority rises to 3*(0.49) - 2*(0.343) = 0.7840 (78.40%); a 5-person majority reaches 0.8369 (83.69%); and a 7-person majority reaches 0.8740 (87.40%, +17.40 percentage points over a solo expert!). Now test systematically biased reviewers p = 0.40: P_3(0.40) = 0.3520 < 0.4000 and P_7(0.40) = 0.2898!
| Panel / Ensemble Size | Exact Binomial Majority Formula (q = 1 - p) | Accuracy at p=0.70 vs p=0.40 |
|---|---|---|
| Solo Decision-Maker (N = 1) | p | 0.7000 | 0.4000 |
| 3-Voter Majority (N = 3) | 3*p^2 - 2*p^3 | 0.7840 | 0.3520 (Worse!) |
| 5-Voter Majority (N = 5) | p^5 + 5*p^4*q + 10*p^3*q^2 | 0.8369 | 0.3174 (Worse!) |
| 7-Voter Majority (N = 7) | p^7 + 7*p^6*q + 21*p^5*q^2 + 35*p^4*q^3 | 0.8740 | 0.2898 (Worse!) |
Exact 5-Member Committee / Ensemble Majority Accuracy Formula
Validate 0 < p < 1 and keep q = 1 - p inside LET so binomial terms stay crisp and self-documenting.
=LET(p_ac, B2, q_er, 1 - p_ac, IF(OR(p_ac <= 0, p_ac >= 1), "Require 0 < p < 1", ROUND((p_ac^5) + 5 * (p_ac^4) * q_er + 10 * (p_ac^3) * (q_er^2), 4)))Exact 7-Member Committee Majority Gain Over Solo Expert (P_7(p) - p) Formula
Positive when p > 0.50, exact 0.0000 at coin-toss p = 0.50, and strictly negative when systematic bias pushes p < 0.50!
=LET(p_ac, B2, q_er, 1 - p_ac, IF(OR(p_ac <= 0, p_ac >= 1), "Require 0 < p < 1", LET(p_7, (p_ac^7) + 7 * (p_ac^6) * q_er + 21 * (p_ac^5) * (q_er^2) + 35 * (p_ac^4) * (q_er^3), ROUND(p_7 - p_ac, 4))))Try the example
Use the original values or change them to test your understanding.