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Abstract Recently, the use of Knightian (uncertain) strategies in normal form games has received increasing attention. The use of uncertain acts in games leads to new (Ellsberg) equilibria. We provide a foundation of the new equilibrium concept in the spirit of Harsanyi by proving an extension of the Purification Theorem for $$2\times 2$$2×2 normal form games. Our result implies that Ellsberg equilibria are limits of equilibria in slightly perturbed games with private information. In such equilibria, players use pure or maxmin strategies only.