Belief Convergence under Misspecified Learning: A Martingale Approach

Abstract

We present an approach to analyze learning outcomes in a broad class of misspecified environments, spanning both single-agent and social learning. We introduce a novel “prediction accuracy” order over subjective models, and observe that this makes it possible to partially restore standard martingale convergence arguments that apply under correctly specified learning. Based on this, we derive general conditions to determine when beliefs in a given environment converge to some long-run belief either locally or globally (i.e., from some or all initial beliefs). We show that these conditions can be applied, first, to unify and generalize various convergence results in previously studied settings. Second, they enable us to analyze environments where learning is “slow”, such as costly information acquisition and sequential social learning. In such environments, we illustrate that even if agents learn the truth when they are correctly specified, vanishingly small amounts of misspecification can generate extreme failures of learning.

Mislearning from censored data: The gambler's fallacy and other correlational mistakes in optimal‐stopping problems

“…Agents' stopping decisions determine how many signals they observe about the fundamentals. Other recent papers (Esponda, Pouzo, and Yamamoto (2021), Fudenberg, Lanzani, and Strack (2021), Frick, Iijima, and Ishii (2021a), Heidhues, Kőszegi, and Strack (2021)) prove general theorems about the convergence of misspecified learning in different settings. Though not the primary contribution of this work, the convergence result in Proposition 7 deals with a setting that is not covered by these papers: a multi‐dimensional inference problem with a continuum of states, signals, and actions.…”

Which misspecifications persist?

“…Esponda, Pouzo, and Yamamoto (2021) use stochastic approximation to establish when the agent's action frequency converges. Frick, Iijima, and Ishii (2023) provide conditions for local and global convergence of the agent's beliefs without explicitly modeling the agent's actions. Fudenberg, Lanzani, and Strack (2021) introduce uniform Berk–Nash equilibria and uniformly strict Berk–Nash equilibria.…”

Pathwise concentration bounds for Bayesian beliefs

“…11 Subsequent papers include Fudenberg, Romanyuk, and Strack (2017), Molavi (2019), Bohren and Hauser (2021), Fudenberg and Lanzani (2023), He and Libgober (2021), Esponda, Pouzo, and Yamamoto (2021), Heidhues, Kőszegi, and Strack (2021), Levy, Moreno de Barreda, and Razin (2021), He (2022), and Frick, Iijima, and Ishii (2023). Before this, Arrow and Green (1973) gave the first general framework for this problem, and Nyarko (1991) pointed out that the combination of misspecification and endogenous observations can lead to cycles.…”