 Hence, the endogenous correlated equilibrium \(\boldsymbol{\mu}^*\) restricts the set of strategies that the agents might follow, as do the Bayes concordant beliefs of rationalizability. However, the endogenous correlated equilibrium concept is a proper refinement of rationalizability, because the latter does not presuppose that condition (3.iii) holds with respect to the beliefs one’s opponents actually have. If exactly one pure strategy combination \(\boldsymbol{s}^*\) satisfies (3.iv) given \(\boldsymbol{\mu}^*\), then \(\boldsymbol{\mu}^*\) is a strict equilibrium, and in this case one can predict with certainty what the agents will do given common knowledge of the game, rationality and their beliefs. Note that Definition 3.5 says nothing about whether or not the agents regard their opponents’ strategy combinations as probabilistically independent. Also, this definition does not require that the agents’ probabilities are consistent, in the sense that agents’ probabilities for a mutual opponent’s acts agree. A simple refinement of the endogenous correlated equilibrium concept characterizes the Nash equilibrium concept.