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Berk-Nash Rationalizability

Ignacio Esponda, Demian Pouzo

TL;DR

This paper develops Berk–Nash rationalizability, a learning-based solution concept for complete-information games under misspecified models. It shows that, when players update beliefs from past actions using data-consistent forecasts, any action played infinitely often must be Berk–Nash rationalizable, yielding convergence-free predictions and strong refinement over correlated rationalizability under correct specification. The framework generalizes to multiplayer games, provides an iterative method to compute the BN rationalizable set, and connects BN rationalizability to classical notions like BP under appropriate conditions, while offering extensions to distributions, mixed strategies, and player-specific histories. Practically, BN rationalizability gives robust long-run predictions without requiring convergence of actions or beliefs, and it sharpens equilibrium predictions when the underlying model is misspecified or partially identified.

Abstract

We study learning in complete-information games, allowing the players' models of their environment to be misspecified. We introduce Berk--Nash rationalizability: the largest self-justified set of actions -- meaning each action in the set is optimal under some belief that is a best fit to outcomes generated by joint play within the set. We show that, in a model where players learn from past actions, every action played (or approached) infinitely often lies in this set. When players have a correct model of their environment, Berk--Nash rationalizability refines (correlated) rationalizability and coincides with it in two-player games. The concept delivers predictions on long-run behavior regardless of whether actions converge or not, thereby providing a practical alternative to proving convergence or solving complex stochastic learning dynamics. For example, if the rationalizable set is a singleton, actions converge almost surely.

Berk-Nash Rationalizability

TL;DR

This paper develops Berk–Nash rationalizability, a learning-based solution concept for complete-information games under misspecified models. It shows that, when players update beliefs from past actions using data-consistent forecasts, any action played infinitely often must be Berk–Nash rationalizable, yielding convergence-free predictions and strong refinement over correlated rationalizability under correct specification. The framework generalizes to multiplayer games, provides an iterative method to compute the BN rationalizable set, and connects BN rationalizability to classical notions like BP under appropriate conditions, while offering extensions to distributions, mixed strategies, and player-specific histories. Practically, BN rationalizability gives robust long-run predictions without requiring convergence of actions or beliefs, and it sharpens equilibrium predictions when the underlying model is misspecified or partially identified.

Abstract

We study learning in complete-information games, allowing the players' models of their environment to be misspecified. We introduce Berk--Nash rationalizability: the largest self-justified set of actions -- meaning each action in the set is optimal under some belief that is a best fit to outcomes generated by joint play within the set. We show that, in a model where players learn from past actions, every action played (or approached) infinitely often lies in this set. When players have a correct model of their environment, Berk--Nash rationalizability refines (correlated) rationalizability and coincides with it in two-player games. The concept delivers predictions on long-run behavior regardless of whether actions converge or not, thereby providing a practical alternative to proving convergence or solving complex stochastic learning dynamics. For example, if the rationalizable set is a singleton, actions converge almost surely.

Paper Structure

This paper contains 41 sections, 15 theorems, 96 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Single-agent problem
  3. Primitives.
  4. Optimal action correspondence
  5. Kullback--Leibler divergence.
  6. Solution concept.
  7. Characterization.
  8. Limit actions are rationalizable.
  9. Games
  10. Setup
  11. Solution concept
  12. Optimal action correspondence.
  13. Kullback--Leibler divergence.
  14. Berk--Nash rationalizability.
  15. Existence and characterization.
  16. ...and 26 more sections

Key Result

Theorem 1

Almost surely, every limit action of a Bayesian agent is Berk--Nash rationalizable.

Figures (1)

  • Figure 1: Returns to effort example. The optimal action $a^{\text{opt}}$ is where the marginal cost curve intersects the true return $\theta^*$. Berk--Nash equilibrium actions lie at intersections of the marginal cost curve and the KL-minimizing curve $\theta^m(\delta_\cdot)$. In the overconfident case, there are 3 Berk--Nash equilibria, $a_S$, $a_M$, and $a_L$, and the Berk--Nash rationalizable set is $[a_S, a_L]$; in the underconfident case, there is a unique Berk--Nash equilibrium, and the Berk--Nash rationalizable set is the 2-cycle interval $[a_{\min}^\infty, a_{\max}^\infty]$.

Theorems & Definitions (38)

  • Example : Returns to effort
  • Example : continued
  • Example : continued
  • Definition 1
  • Example : continued
  • Theorem 1
  • Example : continued
  • Definition 2
  • Theorem 2: Existence and characterization of rationalizable set
  • proof
  • ...and 28 more