In Chapter 41, "Epistemic Conditions for Nash Equilibrium," Robert J. Aumann and Adam Brandenburger explore the epistemic conditions necessary for a strategy profile to be a Nash equilibrium. They question the common wisdom that Nash equilibrium is based on common knowledge of the game's structure, players' rationality, and the strategies actually played. The authors aim to clarify these issues within a formal framework, focusing on the epistemic conditions involving what players know or believe about each other, such as payoff functions, strategy choices, decision procedures, and beliefs. Surprisingly, they find that common knowledge of payoff functions and rationality is never necessary, and common knowledge of the strategies being played is also irrelevant. Instead, the key condition is common knowledge of the beliefs that players hold about the strategies of others, but this is only relevant when there are at least three players. The chapter includes sections on the main results, interactive belief systems, illustrations, formal statements and proofs, counterexamples, and a discussion of the conceptual aspects and related literature. The authors provide a detailed framework and examples to support their findings, emphasizing the importance of understanding the specific epistemic conditions required for Nash equilibrium.In Chapter 41, "Epistemic Conditions for Nash Equilibrium," Robert J. Aumann and Adam Brandenburger explore the epistemic conditions necessary for a strategy profile to be a Nash equilibrium. They question the common wisdom that Nash equilibrium is based on common knowledge of the game's structure, players' rationality, and the strategies actually played. The authors aim to clarify these issues within a formal framework, focusing on the epistemic conditions involving what players know or believe about each other, such as payoff functions, strategy choices, decision procedures, and beliefs. Surprisingly, they find that common knowledge of payoff functions and rationality is never necessary, and common knowledge of the strategies being played is also irrelevant. Instead, the key condition is common knowledge of the beliefs that players hold about the strategies of others, but this is only relevant when there are at least three players. The chapter includes sections on the main results, interactive belief systems, illustrations, formal statements and proofs, counterexamples, and a discussion of the conceptual aspects and related literature. The authors provide a detailed framework and examples to support their findings, emphasizing the importance of understanding the specific epistemic conditions required for Nash equilibrium.