Why Starting Beliefs Can Decide Whether a Crowd Learns

Rational choices, collective mistakes

In Herding with Multiple States: When the Prior Matters, Xiaoyu Chen studies people who act one after another. Each person observes every predecessor’s action, receives a private signal about an unknown state of the world, and chooses the action with the highest expected payoff.

Everyone reasons rationally within the model. Nevertheless, observing choices rather than the private information behind them can prevent society from learning enough to make correct decisions.

The paper’s central finding concerns the prior: the initial probability assigned to each possible state. A full-support prior gives every state a positive chance, ruling nothing out. Chen shows that, with more than two states, changing these starting probabilities can change whether society learns—even when preferences and signal distributions stay fixed.

This contradicts a conjecture in the 2024 working-paper version of research by Kartik, Lee, Liu, and Rappoport. In the classic binary-state setting with finitely many actions, the learning-versus-herding distinction does not depend on which full-support prior society starts with.

Three possibilities, two choices

The key counterexample has three states and two actions. One action is best in states 1 and 3; the other is best in state 2. Although the extreme states call for the same decision, they generate different private signals.

Some public beliefs lie in a cascade region: people choose the first action regardless of their private signal. Their actions then convey no additional information, and the group can remain mistaken when state 2 is true.

But the existence of this region does not mean every starting belief leads toward it. Chen proves that learning occurs for generic full-support priors outside the region. Here, “generic” has a precise mathematical meaning: the exceptions form a set of Lebesgue measure zero that is also meager, a topological notion of smallness. It does not mean that exceptions are impossible.

Symmetry matters. For certain starting beliefs assigning equal probabilities to states 1 and 3, an uninterrupted sequence of the first action has positive probability and leads toward an incorrect cascade. For generic starting beliefs outside the cascade region, society instead learns the true state almost surely.

That is a striking distinction: learning which action is correct can require distinguishing states that offer exactly the same payoffs.

How the result is established

This is mathematical research, not an experiment or analysis of observed crowds. Chen tracks the public belief—the probability distribution implied by the action history before the next private signal arrives.

Public beliefs have a martingale property: their expected future value, given current public information, equals their current value. The proofs use convergence results to study where these beliefs can end up, then examine explicit updating rules to determine which limits are reachable from particular priors.

The distinction between a cascade’s existence and its accessibility is crucial. A substantial region of cascade beliefs need not attract the belief paths starting outside it.

The paper also perturbs the counterexample’s signal distributions. In one direction, prior-dependent learning persists; in the other, incorrect herding becomes possible from every full-support prior. Thus, the dynamics of information updating matter, not merely the shape of the cascade region.

Can private evidence overturn a mistake?

Chen introduces escapability, which asks whether private signals can provide arbitrarily strong evidence that a given action is suboptimal, relative to states where it is strictly optimal. A weaker, pairwise version asks the corresponding question one state against another.

If escapability fails, herding occurs for some initial prior. If pairwise escapability fails, herding occurs for every full-support prior. “Herding” here means a persistent payoff shortfall with positive probability; in the general model, it need not mean everyone eventually chooses one identical action.

These are necessary conditions for learning, not universal guarantees. They become a complete characterization under additional assumptions: finitely many actions, a unique best action in each state, ordered states and signals, and two structural restrictions. Each action must be best over an interval of states, while higher signals must increasingly favor higher states in the likelihood-ratio sense.

What the model does—and does not—show

The findings establish mechanisms within a specified model, not measured causal effects in markets or other real groups. Agents share payoffs, observe all previous actions, and receive signals that are independent conditional on the true state. Learning is assessed in the infinite-sequence limit, not over a practical deadline.

The general classification also remains incomplete: when pairwise escapability holds but escapability fails, the paper does not determine which priors produce learning in every environment.

Released as an arXiv preprint, the study offers a mathematical warning rather than a tested intervention: with multiple possible states, starting beliefs can shape whether rational observation ultimately corrects a collective mistake.