Berk-Nash Rationalizability

TL;DR

The paper introduces Berk-Nash rationalizability, a concept describing the set of actions consistent with players' misspecified models in learning games, providing insights into long-term behavior without requiring convergence.

Contribution

It defines Berk-Nash rationalizability, extending rationalizability to misspecified models, and shows its relevance in predicting long-term outcomes in learning in games.

Findings

Actions played infinitely often lie in Berk-Nash rationalizable set.

When models are correct, Berk-Nash rationalizability matches rationalizability.

Predicts long-run behavior without requiring convergence.

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.