Complexity and limited ability have profound effect on how we learn and make decisions under uncertainty. Using the theory of finite automaton to model belief formation, this paper studies the characteristics of optimal learning behavior in small and big worlds, where the complexity of the environment is low and high, respectively, relative to the cognitive ability of the decision maker. Optimal behavior is well approximated by the Bayesian benchmark in very small world but is more different as the world gets bigger. In addition, in big worlds, the optimal learning behavior could exhibit a wide range of well-documented non-Bayesian learning behavior, including the use of heuristics, correlation neglect, persistent over-confidence, inattentive learning, and other behaviors of model simplification or misspecification. These results establish a clear and testable relationship among the prominence of non-Bayesian learning behavior, complexity, and cognitive ability.

翻译：复杂性与有限能力深刻影响着我们在不确定性条件下的学习与决策方式。本文利用有限自动机理论对信念形成过程进行建模，研究了在"小世界"与"大世界"（分别对应环境复杂度相对于决策者认知能力较低和较高的情况）中，最优学习行为的特征。在极小世界中，最优行为可被贝叶斯基准良好近似；但随着世界规模的增大，二者差异愈发显著。此外，在大世界中，最优学习行为可能呈现出一系列文献充分记载的非贝叶斯学习特征，包括启发式策略使用、相关性忽视、持续性过度自信、注意力缺失学习以及其他模型简化或设定错误行为。这些结果在非贝叶斯学习行为的突出程度、环境复杂度与认知能力之间建立了清晰且可检验的关系。