Asymptotic Behavior of Bayesian Learners with Misspecified Models
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We consider an agent who represents uncertainty about her environment via a possibly misspecified model. Each period, the agent takes an action, observes a consequence, and uses Bayes' rule to update her belief about the environment. This framework has become increasingly popular in economics to study behavior driven by incorrect or biased beliefs. Current literature has either characterized asymptotic behavior in general settings under the assumption that the agent's action converges (which sometimes does not) or has established convergence of the action in specific applications. By noting that the key element to predict the agent's behavior is the frequency of her past actions, we are able to characterize asymptotic behavior in general settings in terms of the solutions of a generalization of a differential equation that describes the evolution of the frequency of actions. Among other results, we provide a new interpretation of mixing in terms of convergence of the frequency of actions, and we also show that convergent frequencies of actions are not necessarily captured by previous Nash-like equilibrium concepts.
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