[Paper Review] Asymptotic Behavior of Bayesian Learners with Misspecified Models

Ignacio Esponda, Demián Pouzo+1 more|arXiv (Cornell University)|Apr 18, 2019

Decision-Making and Behavioral EconomicsDecision Sciences42 references19 citations

TL;DR

This paper develops a general framework for analyzing the long-run behavior of Bayesian agents who use misspecified models to learn about their environment. By linking action frequency dynamics to a generalized differential equation, it characterizes almost-sure convergence to attracting equilibria, providing a unified tool for predicting persistent biases in economic learning under model misspecification.

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.

Motivation & Objective

• To understand the long-term behavior of Bayesian agents who hold misspecified models of reality.
• To identify conditions under which learning leads to persistent biases despite repeated feedback.
• To generalize existing convergence results beyond specific, restrictive model assumptions.
• To provide a systematic method for predicting asymptotic behavior in economic applications with misspecified learning.
• To characterize the role of action frequency in determining long-run belief and behavior dynamics.

Proposed method

• The authors model the agent’s learning process as a Bayesian updating mechanism under a misspecified statistical model.
• They show that the asymptotic behavior of beliefs and actions is determined by the frequency of past actions, not the actions themselves.
• A generalized differential equation is derived to describe the evolution of action frequencies over time.
• The framework uses the concept of attracting sets in the space of action frequencies to predict long-run convergence.
• Key tools include weak identification, upper hemi-continuity of best-response correspondence, and martingale properties of Bayesian updating.
• The analysis distinguishes between strict equilibria, repelling equilibria, and boundary cases (e.g., constant beliefs over intervals), using topological and dynamic stability concepts.

Experimental results

Research questions

• RQ1 Under what conditions does a Bayesian agent with a misspecified model converge to a stable long-run behavior?
• RQ2 How does the frequency of past actions influence the asymptotic evolution of beliefs and behavior?
• RQ3 Why do persistent biases emerge in learning even with repeated feedback, and what determines their direction?
• RQ4 What characterizes the set of long-run equilibria in misspecified learning models, and which are stable?
• RQ5 How can the convergence of beliefs and actions be predicted in general economic settings with model misspecification?

Key findings

• The sequence of beliefs almost surely converges to a Dirac measure centered on a parameter value that corresponds to an attracting equilibrium.
• Action frequencies converge almost surely to a pure action that is optimal under the limiting belief, provided the equilibrium is attracting.
• Endpoints of vertical segments in the best-response correspondence are ruled out as limits due to non-strictness of the associated equilibria.
• Equilibria corresponding to interior points of horizontal segments or constant belief regions (cases 3 and 4) are attracting and thus almost surely realized.
• The limiting belief is consistent with the long-run action frequency, and the value function is continuous at the limit due to upper hemi-continuity of the best-response correspondence.
• The set of long-run action frequencies is contained within the set of myopic best responses under a common belief, implying stability of the limiting behavior.