The crowded room appears to know something the empty room does not. A traveler looking for dinner has little experience of either kitchen, but can see where other people have chosen to eat. It is reasonable to treat their choices as evidence. They may have eaten there before, spoken to someone who has or noticed something the traveler missed. Yet the next arrival can reason in precisely the same way, taking the traveler's choice as one more informed endorsement. A line begins to lengthen. The question is how much independent knowledge the length of that line contains, and at what point the act of following conceals the very information for which the follower came.
A decision reveals less than the information on which it rests. Someone who enters a restaurant may have received a favorable report, or may have received an unfavorable one that the crowd seemed to outweigh. The action is visible; the report remains private. Bikhchandani, Hirshleifer and Welch made this loss of detail central to their account of informational cascades. Their decision makers observe the actions of their predecessors and combine those actions with signals of their own. Eventually an observed history can become persuasive enough that a person takes the same action whatever his signal says. His choice then carries no further information about that signal. Later people inherit the decision while losing access to the reason he might have had to doubt it.
A simple sequence shows how individually sensible choices can stop the sequence from learning. Suppose an opportunity is equally likely to be good or bad. Adopting earns one in the good state and loses one in the bad state; declining earns zero. Each person receives a private favorable or unfavorable signal that identifies the true state two-thirds of the time. Signals are independent conditional on that state, and everyone knows the structure. People decide in order, see earlier actions and maximize their expected payoff. When exactly indifferent, they follow their own signal. The first person therefore adopts on a favorable signal and declines on an unfavorable one. Until social evidence overrides the private report, the choice lets the next person infer something that report contained.
Two favorable signals at the beginning can be enough to silence the evidence that follows. The first adoption reveals a favorable signal. The second person, if privately unfavorable, would be indifferent and would decline under the stated rule; a second adoption therefore reveals a second favorable signal. Together they raise the probability that the opportunity is good to four-fifths. Now let the third person's signal be unfavorable. Two favorable reports and one unfavorable report leave a two-thirds probability of the good state, so adopting remains preferable. A favorable private signal would also lead to adoption. The third person thus adopts in either case, and the fourth learns nothing about the third person's private evidence by seeing him join.
The growing crowd can preserve the same uncertainty beneath a growing appearance of agreement. Once everyone knows that the third person would adopt on either private signal, his adoption adds no new evidence about the state. The fourth faces the same public odds and the same possible private signals, and again adopts either way. So does the fifth. Under these assumptions, a hundred subsequent adoptions leave the public probability at four-fifths. They do not turn two early reports into a hundred and two independent confirmations. Each follower understands the limitation and still finds adoption best. Rational inference can sustain the crowd without converting its unanimity into certainty.
The crowd can also begin in error with positive probability. Conditional on the opportunity being bad, each of the first two people receives a misleading favorable signal with probability one-third. The chance that both do so is one-ninth. On that path their actions start a mistaken adoption cascade. This is one route into error, rather than the total probability of all possible wrong cascades. Other early histories remain informative for longer. If the first two signals disagree, their evidence cancels and the third person's private signal matters again. The result turns on the sequence of information revealed through choices, not on an axiom that the first mover is always right or that followers are always mistaken.
Direct access to the underlying reports changes what the same number of people can teach. Four independent favorable signals would produce odds of sixteen to one, a probability of sixteen-seventeenths that the opportunity is good. Four observed adoptions in the cascade reveal only the first two favorable signals and preserve odds of four to one. If private reports continue to enter a common record, later unfavorable reports can also overturn the early evidence. After two favorable and three unfavorable signals, the odds become one to two and declining is preferable. The information suppressed by the visible consensus can therefore contain exactly the reason that would reverse it.
Dependence between reports must be counted even when everyone speaks. Ten people repeating a single review do not supply ten independent observations of the kitchen. The same can be true when several analysts use one dataset or when successive decisions inherit a common preliminary judgment. An assessment that counts the repetitions as separate evidence will become more confident than its information warrants. This is a different error from the rational cascade just described: the cascade's followers correctly understand why later actions are uninformative. Both problems, however, make the origin of a claim matter. Asking who saw what, and whether later observations add anything, can be more revealing than counting the people who agree.
A fresh public signal can break a cascade whose visible size suggests permanence. Retain the public odds of four to one, and suppose an independent public test now reports against adoption with ninety-percent accuracy. That result multiplies the odds by one-ninth, leaving four to nine. Even a favorable private signal only raises them to eight to nine, so the next person declines. A crowd built on a small stock of evidence can change when that stock changes. The mechanism explains a possibility of fragility without proving that every sudden reversal is informational. A change in payoffs, a changed rule or a new common shock can also move many people at once.
More public information need not improve every stage of social learning in the same way. Before a cascade begins, a shared announcement changes both what people know directly and what their subsequent choices reveal to others. Bikhchandani and his coauthors show that an early disclosure can reduce the information conveyed by those choices enough to harm some later decision makers in expectation. After a cascade has begun, the particular loss of fresh information through subsequent actions has already occurred. The comparison depends on timing and on the information structure. A rule to publish everything as early as possible needs more reasoning than the observation that information can be valuable.
Several other motives can produce the same outward conformity. People may benefit directly from using the same system, fear punishment for departing from a convention or conceal their preferences because they believe everyone else supports the norm. The earlier Collected Trap and the separate Utility Cascade examine such failures of coordination and public expression. An informational cascade isolates a different motive: each person wants the better option and treats predecessors as evidence about which option that is. In an actual crowd these forces can coexist. Observing uniform behavior alone cannot identify which one sustains it, and a remedy addressed to the wrong force may leave the relevant obstacle untouched.
An institution can preserve independent judgment by preserving the evidence before the crowd absorbs it. Separate preliminary assessments, a record of reasons and a way to report contrary observations can keep later decision makers from seeing only a final vote. The benefit depends on truthful reporting and on the independence that the procedure actually secures. It also has a cost: gathering an additional observation takes time and resources, and a quick decision may sometimes matter more than a more informed one. The model supplies a reason to examine these arrangements. It does not prescribe a single voting order, disclosure rule or duty to dissent for every committee and market.
The value of a dissent lies in the information it adds, rather than in its distance from the majority. An uninformed contrary opinion supplies no automatic correction, and deliberately ignoring the observations of others can discard knowledge worth having. In the example, the third person with an unfavorable signal does better in expectation by adopting. What the group loses is the continued circulation of that person's signal, not the moral courage to choose a personally inferior action. A useful institution tries to let the observation survive even when it does not change its owner's immediate decision. That is how the group can retain the benefits of social learning without treating each act of deference as fresh proof.
A crowd can grow while the evidence beneath it stands still.
The cliff in the original question is a possible destination of reasonable decisions made from an impoverished record. Nothing in the example requires stupidity, dishonesty or a taste for obedience. Each person can read the public evidence correctly while helping to stop new evidence from becoming public. What keeps the crowd from walking on indefinitely is whatever lets contrary knowledge enter and matter: disclosed reasons, informative tests, different experiences or changed circumstances. These are resources a market may use and institutions it may need. The last plate asks whether the missing knowledge can itself be bought, and what becomes of a bargain when showing the goods also gives them away. The argument remains open.