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Learning and Equilibrium under Model Misspecification

Ignacio Esponda, Demian Pouzo

TL;DR

The chapter develops a unified framework for misspecified learning in settings where agents optimize within incorrect models and data are endogenous to actions. It blends Bayesian posterior analysis under misspecification with an equilibrium concept, Berk–Nash, to characterize long-run behavior in both single-agent and strategic contexts. Key results include posterior concentration around KL-projections in exogenous data, dynamic learning with convergence to KL-minimizers (or non-convergence when minimizers are tied), and various routes to Berk–Nash steady states in finite-action environments. The analysis extends to forward-looking agents and games, with convergence results grounded in weak identification and perturbation-based or empirical-frequency interpretations of mixed strategies. Overall, the framework clarifies when misspecified beliefs yield stable equilibria, how data-generation and action selection interact, and the implications for welfare and strategic behavior under bounded rationality and model misspecification.

Abstract

This chapter develops a unified framework for studying misspecified learning situations in which agents optimize and update beliefs within an incorrect model of their environment. We review the statistical foundations of learning from misspecified models and extend these insights to environments with endogenous, action-dependent data, including both single agent and strategic settings.

Learning and Equilibrium under Model Misspecification

TL;DR

The chapter develops a unified framework for misspecified learning in settings where agents optimize within incorrect models and data are endogenous to actions. It blends Bayesian posterior analysis under misspecification with an equilibrium concept, Berk–Nash, to characterize long-run behavior in both single-agent and strategic contexts. Key results include posterior concentration around KL-projections in exogenous data, dynamic learning with convergence to KL-minimizers (or non-convergence when minimizers are tied), and various routes to Berk–Nash steady states in finite-action environments. The analysis extends to forward-looking agents and games, with convergence results grounded in weak identification and perturbation-based or empirical-frequency interpretations of mixed strategies. Overall, the framework clarifies when misspecified beliefs yield stable equilibria, how data-generation and action selection interact, and the implications for welfare and strategic behavior under bounded rationality and model misspecification.

Abstract

This chapter develops a unified framework for studying misspecified learning situations in which agents optimize and update beliefs within an incorrect model of their environment. We review the statistical foundations of learning from misspecified models and extend these insights to environments with endogenous, action-dependent data, including both single agent and strategic settings.
Paper Structure (36 sections, 15 theorems, 60 equations, 1 figure)

This paper contains 36 sections, 15 theorems, 60 equations, 1 figure.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Five strands in the development of misspecified learning in economics
  4. Learning from exogenous data
  5. Posterior Concentration
  6. Doob's Martingale Convergence Theorem.
  7. Kullback--Leibler (KL) divergence.
  8. Asymptotic Concentration of Beliefs.
  9. Discussion.
  10. Implications of Proposition \ref{['pro:Berk']}.
  11. Biographical Notes
  12. Endogenous learning: single agent
  13. Setup and equilibrium
  14. Assumptions (single agent).
  15. Optimality correspondence.
  16. ...and 21 more sections

Key Result

Proposition 2.1

For all closed set $C \subset \Theta \setminus \Theta^{m}$, there exists a constant $L<\infty$ such that The symbol $o_{as}(1)$ denotes a positive random variable that $\mathbf P$-a.s. converges to zero. where $\rho_{C} : = \min_{\theta \in C} \mathrm{KL}(Q \Vert Q_\theta) - \min_{\theta \in \Theta} \mathrm{KL}(Q \Vert Q_\theta) > 0$.

Figures (1)

  • Figure 1: Bernoulli KL curves for two truths, $\theta^\star=0.5$ and $\theta^\star=0.75$.

Theorems & Definitions (41)

  • Proposition 2.1
  • proof : Sketch of the Proof of Proposition \ref{['pro:Berk']}
  • Corollary 2.1
  • Example 2.1: Coin Flips (Bernoulli model)
  • Definition 3.1
  • Example 3.1
  • Example 3.2: Trade with adverse selection
  • Lemma 3.1: Posterior concentration near KL minimizers
  • Proposition 3.1
  • proof : Sketch of the proof of Proposition \ref{['prop:SA-converges-to-BN']}
  • ...and 31 more