This paper explores how Bayes-rational individuals learn sequentially from the discrete actions of others. Unlike earlier informational herding papers, we admit heterogeneous preferences. Not only may type-specific ‘herds’ eventually arise, but a new robust possibility emerges: confounded learning. Beliefs may converge to a limit point where history offers no decisive lessons for anyone, and each type's actions forever nontrivially split between two actions. To verify that our identified limit outcomes do arise, we exploit the Markov-martingale character of beliefs. Learning dynamics are stochastically stable near a fixed point in many Bayesian learning models like this one.

In an influential paper, David S. Scharfstein and Jeremy C. Stein (1990) modeled sequential investment by agents concerned about their reputation as good forecasters. Consider an agent who acts after observing the behavior of another ex ante identical agent. Scharfstein and Stein argue that reputational herding requires that better agents have more correlated signals conditionally on the state of the world. They claim that without correlation the second agent would have no incentive to attempt to manipulate the market inference about ability by imitating the behavior of the first agent. In this Note we show that in their model, correlation is not necessary for herding, other than in degenerate cases. Our clarification exploits a parallel with statistical herding, introduced by Abhijit V. Banerjee (1992) and Sushil Bikhchandani et al. (1992) (henceforth, BHW). BHW feature investors who maximize expected profits in a common-value environment and have access to conditionally independent private signals of bounded precision, while still observing the behavior of others. Eventually, the evidence accumulated from observing earlier decisions is sufficiently strong to swamp the private information of a single decision maker. Thereafter, everyone rationally copies the prevailing behavior. We notice that payoffs have a common-value nature in both the statistical and the reputational model. The observed behavior of other agents possibly affects the probability belief attached to different states of the world as well as the payoff conditional on each state. Herding arises from the interaction of these two channels affecting the expected payoff, be it physical or reputational. Positive differential conditional correlation of signals in the reputational model is tantamount to the introduction of positive payoff externalities in the statistical model. This reinforces the tendency to herd already present with independence. The fact that differential conditional correlation is not needed for herding is a clear strength of the reputational herding model. It is not necessary to assume common unpredictable components of returns at the individual level in order to rationalize the empirical findings that individual prediction errors of security analysts are correlated. After setting up Scharfstein and Stein’s model in Section I, we summarize their findings in Section II and provide a unified definition of herd behavior in Section III. Section IV contains our critique of their line of argument and clarifies the role of differential conditional correlation. In Section V we propose alternative robust scenarios where herding would indeed be driven by correlation. Section VI concludes.