Center for the Study of Rationality, Discussion Paper #610

**Abstract: **We extend Aumann's theorem (Aumann, 1987) in deriving correlated equilibria as a consequence of

common priors and common knowledge of rationality by explicitly allowing for non-rational

behavior. We replace the assumption of common knowledge of rationality with a substantially weaker

notion, p-belief of rationality, where agents believe the other agents are rational with

probabilities p or more. We show that behavior in this case constitutes a constrained correlated

equilibrium of a doubled game satisfying certain p-belief constraints and characterize the

topological structure of the resulting set of p-rational outcomes. We establish continuity in the

parameters p and show that, for p sufficiently close to one, the p-rational outcomes are close to

the correlated equilibria and, with high probability, supported on strategies that survive the

iterated elimination of strictly dominated strategies. Finally, we extend Aumann and Dreze's

theorem (Aumann and Dreze, 2008) on rational expectations of interim types to the broader

p-rational belief systems, and also discuss the case of non-common priors.

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