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Epistemic Conditions for Nash Equilibrium

Robert Aumann
Adam Brandenburger

Abstract

Sufficient conditions for Nash equilibrium in an $n$-person game are given in terms of what the players know and believe--about the game, and about each other's rationality, actions, knowledge, and beliefs. Mixed strategies are treated not as conscious randomizations, but as conjectures, on the part of other players, as to what a player will do. Common knowledge plays a smaller role in characterizing Nash equilibrium than had been supposed. When $n = 2$, mutual knowledge of the payoff functions, of rationality, and of the conjectures implies that the conjectures form a Nash equilibrium. When $n \geq 3$ and there is a common prior, mutual knowledge of the payoff functions and of rationality, and common knowledge of the conjectures, imply that the conjectures form a Nash equilibrium. Examples show the results to be tight.