Level-$k$ Reasoning, Cognitive Hierarchy, and Rationalizability

We employ a unified framework to provide an epistemic-theoretical foundation for Camerer, Ho, and Chong's (2003) cognitive hierarchy (CH) solution and its dynamic extension, using the directed rationalizability concept introduced in Battigalli and Siniscalchi (2003). We interpret level-$k$ as an information type instead of specification of strategic sophistication, and define restriction $\Delta^\kappa$ on the beliefs of information types; based on it, we show that in the behavioral consequence of rationality, common belief in rationality and transparency of $\Delta^\kappa$, called $\Delta^\kappa$-rationalizability, the strategic sophistication of each information type is endogenously determined. We show that in static games, the CH solution generically coincides with $\Delta^\kappa$-rationalizability; this result also connects CH with Bayesian equilibrium. By extending $\Delta^\kappa$ to dynamic games, we show that Lin and Palfrey's (2024) dynamic cognitive hierarchy (DCH) solution, an extension of CH in dynamic games, generically coincides with the behavioral consequence of rationality, common strong belief in rationality, and transparency of (dynamic) $\Delta^\kappa$. The same framework can also be used to analyze many variations of CH in the literature.