CFDP 2235R3

Belief Convergence under Misspecified Learning: A Martingale Approach


Publication Date: May 2020

Revision Date: April 2021December 2021April 2022

Pages: 55


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.

Keywords: Misspecified learning, Stability, Robustness, Berk-Nash equilibrium

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