We formulate and solve the welfare optimization in a herding model, in which individuals care about later people, discounting their welfare. We characterize the constrained efficient outcome by solving a social planner’s problem as a Bayesian optimal experimentation problem. The paper then makes key contributions to both of these literatures.

For herding, we find that herding is socially efficient for all discount factors less than one, but should occur less readily, since cascade sets shrink. In other words, we prove that herds and inefficient herds owe not to the selfishness of agents, but to the problem of signaling private information through finitely many actions.

In our earlier (2000) paper, we showed that “cascades” (sets of public beliefs where actions reflect no private information) require the following bizarre possibility: You see someone take an action in a herding model, and this leads you to a LOWER posterior public belief when the prior is HIGHER. This non-monotone map from prior to posterior public belief is clearly a pathology in some sense. Here we derive simple a sufficient log-concavity condition on the distribution of log likelihood ratios that formalizes this fragility. This condition is generally met, but the failure of this property with multinomial signals led to the incredible focus on cascades.

We argue more sharply that efficiency entails contrarian behaviour — i.e. individuals should optimally lean against taking the myopically more popular actions, under a robust new information log-concavity condition.

Finally, we derive a simple mechanism— transfers depending on the current and next actions— that decentralizes this outcome. With two actions, it amounts to rewarding individuals if their successor mimics their action.