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In order to study updating rules, we consider the problem of a malevolent
principal screening an imperfectly Bayesian agent. We uncover a fundamental
dichotomy between underreaction and overreaction to information.
If, for some signal realization, the agent’s updating rule produces a posterior
that lies outside the convex hull of the Bayesian posteriors induced by the
signal structure, then she can be exploited by the principal without bound:
her ex ante expected loss can be made arbitrarily large. In contrast, an agent
who underreacts–whose posterior after each signal realization lies between
the prior and the corresponding Bayesian posterior–cannot be exploited at all.
