On the role of responsiveness in rational herds

Models of rational herding typically involve a finite action space. An intuition for herding is that this coarseness of the action space relative to the space of potential beliefs is responsible for herding, and were the action space sufficiently rich, learning would be complete. That intuition is false: simple examples illustrate that learning may be incomplete even if the action space is isomorphic to the space of beliefs. What then distinguishes models with ”coarse” versus ”rich” action spaces? This paper develops the language of responsiveness to formalize this distinction. Responsiveness assesses the sensitivity of optimal actions with respect to their rationalizing beliefs. If the optimal action always changes with beliefs, then complete learning is guaranteed regardless of the information structure. By contrast, if the action that is optimal at certainty remains optimal near-certainty, then complete learning is guaranteed if and only if information can induce unbounded likelihood ratios. The lens of responsiveness unifies results across coarse and rich action spaces.