Following the crowd can be rational

A viewer scrolls into a live stream. Two thousand people are watching and the comments are moving fast; she stays. Another stream selling the same product has eleven viewers; she leaves. She may know nothing else about either seller. Is she being foolish?

In 1992, Sushil Bikhchandani, David Hirshleifer and Ivo Welch, and independently Abhijit Banerjee, showed that following others can be perfectly rational, and that this rationality can still lead whole populations astray (Bikhchandani et al. 1992; Banerjee 1992).

The model

The state of the world is either good (the product is worth buying) or bad. Each person receives a private signal that points the right way with probability $p > 1/2$, observes the choices, but not the signals, of everyone before them, and then decides. The first person follows her own signal. The second can infer the first’s signal from her choice and combines it with his own. But once the choices so far lean two ways more to one side than the other, a new person’s single signal can no longer outweigh the evidence of the choices she sees. From that moment she ignores her own signal and copies the crowd, and so does everyone after her, because her choice reveals nothing new. This is an information cascade.

The arithmetic is short. Signals arrive in pairs that either agree, pushing the count two steps to one side and starting a cascade, or disagree, cancelling out. The probability that a cascade, once it starts, is the correct one is

$$P(\text{correct cascade}) = \frac{p^2}{p^2 + (1-p)^2} .$$

import random

def run(p, n, rng):
    """Bikhchandani-Hirshleifer-Welch: the true state is 'good'. Each viewer gets a private signal that is
    right with probability p, sees all earlier choices, and buys if the evidence favours 'good'.
    Ties follow the own signal. Once earlier buys outnumber skips by 2 (or the reverse), every later viewer
    ignores their own signal: a cascade."""
    # buys minus skips among the choices that still carried a signal
    d = 0
    for _ in range(n):
        if d >= 2:  return "buy cascade"
        if d <= -2: return "skip cascade"
        signal_good = rng.random() < p
        d += 1 if signal_good else -1
    return "no cascade yet"

rng = random.Random(3)
print("signal accuracy   buy cascade   skip cascade (wrong)   none after 20")
for p in (0.55, 0.6, 0.7, 0.8):
    res = [run(p, 20, rng) for _ in range(100000)]
    f = lambda k: res.count(k) / len(res)
    print(f"{p:15.2f} {f('buy cascade'):13.1%} {f('skip cascade'):22.1%} {f('no cascade yet'):15.2%}")
# signal accuracy   buy cascade   skip cascade (wrong)   none after 20
#            0.55         59.7%                  40.1%           0.18%
#            0.60         68.9%                  30.9%           0.14%
#            0.70         84.5%                  15.5%           0.04%
#            0.80         94.0%                   6.0%           0.00%

With signals that are right 60 per cent of the time, almost every sequence locks into a cascade within the first few viewers, and about 31 per cent lock into the wrong one, as the formula predicts ($0.36/0.52 = 0.69$ correct). Three features of the model match what creators and sellers see. Cascades form fast, after very few people. They can be wrong. And they are fragile: since they rest on the choices of the first few, a small piece of public information, such as a credible review or a visible return rate, can overturn them.

Thresholds and the phase transition

The cascade model explains why the first few viewers matter. A second model explains why a room can go from almost empty to packed with no change in what is being sold.

Mark Granovetter proposed that each person has a threshold: the share of others who must already be taking part before they will join (Granovetter 1978). A few people with very low thresholds join anyway. If enough others have thresholds just above them, they join next, and so on. Whether this chain reaction runs to the whole population or stops after a handful depends on the whole distribution of thresholds, and it can switch abruptly.

from statistics import NormalDist

def final_share(mu, sigma=0.12, seed=0.02):
    """Granovetter threshold model: each viewer joins once the share already joined reaches their threshold.
    Thresholds ~ Normal(mu, sigma); a seed share joins anyway. Iterate r -> seed + (1 - seed) * F(r)."""
    F = NormalDist(mu, sigma).cdf
    r = seed
    for _ in range(2000):
        r = seed + (1 - seed) * F(r)
    return r

print("mean threshold   final share joined")
for mu in (0.20, 0.24, 0.25, 0.26, 0.265, 0.27, 0.28, 0.30, 0.35):
    print(f"{mu:14.3f} {final_share(mu):18.1%}")
# mean threshold   final share joined
#          0.200             100.0%
#          0.240             100.0%
#          0.250             100.0%
#          0.260             100.0%
#          0.265             100.0%
#          0.270               5.8%
#          0.280               4.4%
#          0.300               3.3%
#          0.350               2.3%

A change in the average threshold of half a percentage point, from 0.265 to 0.270, moves the outcome from almost everyone to under 6 per cent. Nothing about the product changed. Duncan Watts showed the same kind of discontinuity for cascades on random networks: global cascades occur only within a window of network connectivity, and when they occur they are rare and large (Watts 2002).

What this means for creators and sellers

The first minutes of a live stream, and the first few hundred impressions of a video, carry far more weight than later ones, because they decide which side of a cascade or threshold the room falls on. That is why sellers try to start streams with an audience already present and why platforms test new content on small audiences first.