 Brandenburger and Dekel (1988) show that in 2-agent games, if the beliefs of the agents are common knowledge, condition (3.iii) characterizes a Nash equilibrium-in-beliefs. As they note, condition (3.iii) characterizes a Nash equilibrium in beliefs for the \(n\)-agent case if the probability distributions are consistent and satisfy probabilistic independence. Proposition 3.7 extends Brandenburger and Dekel’s result to the endogenous correlated equilibrium concept by relaxing the consistency and probabilistic independence assumptions.