Limit Points of Endogenous Misspecified Learning

Abstract

We study how an agent learns from endogenous data when their prior belief is misspecified. We show that only uniform Berk–Nash equilibria can be long‐run outcomes, and that all uniformly strict Berk–Nash equilibria have an arbitrarily high probability of being the long‐run outcome for some initial beliefs. When the agent believes the outcome distribution is exogenous, every uniformly strict Berk–Nash equilibrium has positive probability of being the long‐run outcome for any initial belief. We generalize these results to settings where the agent observes a signal before acting.

Robust Model Misspecification and Paradigm Shifts

“…The proof of Corollary 2 is similar to Theorem 2 in Fudenberg et al (2021) and Theorem 1 in Frick et al (2020). Suppose we have a uniformly quasi-strict SCE σ with a supporting belief π ∈ ∆Ω θ .…”

“…Analogous to Corollary 2, the following corollary provides a sufficient condition for Theorem 3. As an implication of Theorem 2 in Fudenberg et al (2021), given any uniformly strict BN-E, there exists a prior belief such that, with positive probability, a dogmatic modeler's action process converges to this equilibrium. This establishes the p-absorbing condition.…”

“…Further, Esponda et al (2019) find general conditions for a single agent's action frequency to converge to the Berk-Nash equilibrium using tools from stochastic approximation. Fudenberg et al (2021) establish that a uniformly strict Berk-Nash equilibrium is uniformly stable in the sense that starting from any prior that is sufficiently concentrated on the Kullback-Leibler minimizers, the dogmatic modeler's action converges to the equilibrium with arbitrarily high probability. This paper contributes to the literature by allowing for model switching and proposing various robustness notions for misspecified models to persist.…”

“…This paper builds on the literature of learning with subjective models, including Berk (1966), Easley and Kiefer (1988), Esponda and Pouzo (2016), Bohren and Hauser (2021), Esponda, Pouzo, and Yamamoto (2019), Fudenberg, Lanzani, and Strack (2021) and Frick, Iijima, and Ishii (2020), all of which study asymptotic learning outcomes of dogmatic modelers in relatively general environments. Esponda and Pouzo (2016) first propose the concept of Berk-Nash equilibrium.…”

“…I assume the dogmatic modeler and the switcher have the same discount factor throughout. 9 For example, this is weaker than the stability notion proposed byFudenberg et al (2021). They define that a pure BN-E a * under θ is stable if for every κ ∈ (0, 1), there exists a belief π ∈ ∆Ω θ and some > 0 such that for any prior π θ 0 ∈ B (π), the dogmatic modeler's action sequence a t converges to a * with probability larger than κ.…”