Theory
Equal agg: x̄ = (1/n)∑xᵢ
Var(x̄) = σ²/n → 0 as n→∞
Confidence weighting:
x* = ∑wᵢxᵢ / ∑wᵢ
wᵢ = 1/σᵢ² (optimal)
Social learning update:
xᵢ(t+1) = (1-α)xᵢ + α·x̄(t)
Herding: α→1 kills diversity
The wisdom of crowds requires independence. Social influence introduces correlation — the effective sample size drops below n. Beyond a critical herding level, the crowd becomes worse than a single individual. Joshua Becker's work quantifies this tradeoff empirically.