Rational behavior under correlated uncertainty

In complete information games, Dekel and Fudenberg (1990) and Börgers (1994) have proposed the solution concept S^∞W (one round of elimination of weakly dominated strategies followed by iterated elimination of strongly dominated strategies), motivating it by a characterization in terms of "approximate common certainty" of admissibility. We examine the validity of this characterization of S^∞W in an incomplete information setting. We argue that in Bayesian games with a nontrivial state space, the characterization is very sensitive to the way in which uncertainty in the form of approximate common certainty of admissibility is taken to interact with the uncertainty already captured by players' beliefs about the states of nature: We show that S^∞W corresponds to approximate common certainty of admissibility when this is not allowed to coincide with any changes to players' beliefs about states. If approximate common certainty of admissibility is accompanied by vanishingly small perturbations to beliefs, then S^∞W is a (generally strict) subset of the predicted behavior, which we characterize in terms of a generalization of Hu's (2007) perfect p-rationalizable set.