Asymptotic Behavior of Bayesian Learners with Misspecified Models

TL;DR

This paper analyzes how Bayesian learners with potentially incorrect models behave over time, providing a general framework to predict their asymptotic action frequencies using differential equations, with practical implications for economics.

Contribution

It introduces a unified approach to characterize the long-term behavior of misspecified Bayesian learners through differential equations, extending previous specific assumptions.

Findings

Behavior is determined by the frequency of past actions.

Asymptotic behavior can be described by solutions to a generalized differential equation.

Provides practical tools for economic applications involving misspecified learning.

Abstract

We consider an agent who represents uncertainty about the 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 characterized asymptotic behavior under fairly specific assumptions. By first showing 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. We then present a series of implications that can be readily applied to economic applications, thus providing off-the-shelf tools that can be used to characterize behavior under misspecified learning.