Abstract Complexity and limited ability affect how we learn and make decisions under uncertainty. Using finite automata 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. The optimal long-run average payoff and distribution of actions are well approximated by the Bayesian benchmark in very small worlds, but the gap widens as the worlds get bigger. In big worlds, it can be optimal for the decision maker to ignore some states entirely, concentrating limited cognitive resources on a subset of states. This ``optimal ignorance'' is reminiscent of well-documented non-Bayesian learning patterns such as the use of heuristics, correlation neglect, and inattentive learning. The results establish a relationship between complexity, cognitive ability, and the scope for non-Bayesian learning behavior.
