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

TL;DR

This paper provides a unified epistemic foundation for cognitive hierarchy models, connecting static and dynamic game solutions with rationalizability and Bayesian equilibrium through a new framework.

Contribution

It introduces a novel interpretation of level-$k$ as an information type and links CH solutions to rationalizability and equilibrium concepts within a unified framework.

Findings

CH solution coincides with $ ext{Δ}^ ext{k}$-rationalizability in static games.

Dynamic CH (DCH) aligns with rationality and common belief in rationality in dynamic games.

Framework applies to various CH models in the literature.

Abstract

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.