Square-root law for group signal detection and optimal group size
Each agent independently measures a weak signal S with Gaussian noise ε ~ N(0,σ²).
The collective estimate averages N independent measurements: SNR_group = S·√N / σ.
This √N improvement is the statistical basis of wisdom-of-crowds phenomena (Galton 1907).
When communication carries a cost c per agent, the effective net benefit peaks at an
optimal group size N* ≈ (S/σ·c)² — too large a group pays more than it gains.
Right panel: SNR vs group size, with cost-adjusted net benefit curve (gold).