Behavioral Finance & Information Economics
Informational Cascade & Bayesian Herding Formula
Calculate Bikhchandani, Hirshleifer & Welch’s (1992) Bayesian herding posterior after two sequential actions, early cascade probability 1 - 2*q*(1 - q), and false-cascade risk (1 - q)^2.
Prove why rational Bayesians ignore their own private signal once two predecessors agree!
Sushil Bikhchandani, David Hirshleifer, and Ivo Welch (1992) showed that rational herding does NOT require irrational crowd psychology! Suppose state V in {0, 1} has 50/50 prior and sequential decision-makers i = 1, 2, 3, ... observe predecessor actions PLUS a binary private signal s_i in {L, H} with accuracy q in (0.5, 1). If Agents 1 and 2 both adopt (revealing signals H, H), what happens when Agent 3 observes a CONTRADICTORY private signal L? Agent 3’s single L cancels one H, leaving net signal count (+1 H) and exact Bayes posterior P(V = 1 | H, H, L) = (q^2 * (1 - q)) / (q^2 * (1 - q) + (1 - q)^2 * q) = q > 0.50! Thus Agent 3 rationally adopts REGARDLESS of s_3, her action conveys ZERO new information to Agent 4, and an Informational Cascade locks in forever!
Check by hand with Signal_Accuracy_q = 0.75 (B2) and Prior_Prob_p0 = 0.50 (B3): after two Adopts + private L, Agent 3’s posterior is still 0.7500 > 0.50; Total Cascade Probability starting at Agent 3 is q^2 + (1 - q)^2 = 1 - 2*q*(1 - q) = 0.5625 + 0.0625 = 0.6250 (62.50%), Correct-Cascade Probability is q^2 = 0.5625, and False-Cascade Probability (everyone herding onto the WRONG choice forever!) is (1 - q)^2 = 0.0625 (6.25%)!
| BHW Cascade Metric (p_0 = 0.50, q = 0.75) | Formula | Verified Output |
|---|---|---|
| Agent 3 Posterior After (Adopt, Adopt, Private L) | (p_0 * q) / (p_0 * q + (1 - p_0) * (1 - q)) | 0.7500 (> 0.50 -> Herds!) |
| Probability Cascade Starts at Agent 3 | q^2 + (1 - q)^2 = 1 - 2*q*(1 - q) | 0.6250 (62.50%) |
| Correct Cascade Probability (True State) | q^2 | 0.5625 (56.25%) |
| False Cascade Probability (Wrong Herd!) | (1 - q)^2 | 0.0625 (6.25%) |
General Bayes Posterior After Net Public-Plus-Private Signal Count k_net Formula
Let B3 hold Prior_p0 in (0, 1) and B4 hold Net_High_Minus_Low_Signals k_net (e.g., +2 public H - 1 private L = +1): exact Bayes odds scale prior odds p_0 / (1 - p_0) by (q / (1 - q))^k_net!
=LET(q_ac, B2, p_0, B3, k_nt, B4, IF(OR(q_ac <= 0.5, q_ac >= 1, p_0 <= 0, p_0 >= 1), "Require 0.5 < q < 1 and 0 < p_0 < 1", LET(lr_k, (q_ac / (1 - q_ac))^k_nt, p_num, p_0 * lr_k, ROUND(p_num / (p_num + (1 - p_0)), 4))))Agent-3 Informational Cascade & False-Herd Probability Formula
Notice that as signal noise rises toward q = 0.51, False-Cascade risk (1 - q)^2 surges toward 24.01% — explaining fragile fads in early product reviews and IPO order books!
=LET(q_ac, B2, IF(OR(q_ac <= 0.5, q_ac >= 1), "Require 0.5 < q < 1 and 0 < p_0 < 1", ROUND((1 - q_ac)^2, 4)))Try the example
Use the original values or change them to test your understanding.