Approximate Knowledge of Rationality and Correlated Equilibria

Fabrizio Germano and Peio Zuazo-Garin
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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