Rational behavior with payoff uncertainty

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The iterated deletion of weakly dominated strategies has been advanced as a necessary requirement for “rational” play. However, this requirement relies on the assumption that the players have no doubts about their opponents' payoffs. We show that once such doubts are introduced, all that can be justified by an appeal to rationality is one round of deletion of weakly dominated strategies, followed by iterated deletion of strategies that are strongly dominated. This extends the Fudenberg, Kreps, and Levine ( J. Econ. Theory 12 (1988), 354–380) study of the robustness of Nash equilibrium refinements to the robustness of solution concepts based only on rationality. Our results also clarify the relationship between various notions of what it means for payoff uncertainty to be “small.”