Reasoning about Bounded Reasoning

Interactive decision-making relies on strategic reasoning. Two prominent frameworks are (1) models of bounded reasoning, exemplified by level-$k$ models, which keep reasoning implicit, and (2) epistemic game theory, which makes reasoning explicit. We connect these approaches by "lifting" static complete-information games into incomplete-information settings where payoff types reflect players' reasoning depths as in level-$k$ models. We introduce downward rationalizability, defined via minimal belief restrictions capturing the basic idea common to level-$k$ models, to provide robust and well-founded predictions in games where bounded reasoning matters. We then refine these belief restrictions to analyze the foundations of two seminal models of bounded reasoning: the classic level-$k$ model and the cognitive hierarchy model. Our findings shed light on the distinction between hard cognitive bounds on reasoning and beliefs about co-players' types. Furthermore, they offer insights into robustness issues relevant for market design. Thus, our approach unifies key level-$k$ models building on clear foundations of strategic reasoning stemming from epistemic game theory.