Reasoning about Bounded Reasoning

Interactive decision-making relies on strategic reasoning. Two prominent frameworks capture this idea. One follows a structural perspective, exemplified by level-$k$ and Cognitive Hierarchy models, which represent reasoning as an algorithmic process. The other adopts an epistemic perspective, formalizing reasoning through beliefs and higher-order beliefs. We connect these approaches by "lifting" static complete-information games into incomplete-information settings where payoff types reflect players' levels. Within this unified framework, reasoning is represented through mathematically explicit and transparent belief restrictions. We analyze three instances: downward rationalizability, a robust benchmark concept; and two refinements, L-rationalizability and CH-rationalizability, which provide epistemic foundations -- albeit with a nuance -- for the classic level-$k$ and Cognitive Hierarchy models, respectively. Our results clarify how reasoning depth relates to behavioral predictions, distinguish cognitive limits from belief restrictions, and connect bounded reasoning to robustness principles from mechanism design. The framework thus offers a transparent and tractable bridge between structural and epistemic approaches to reasoning in games.