Strategic choice data fromfive classic 2×2 laboratory games is augmented by eliciting subjects’ preferences over outcomes, first- and second-order beliefs over strategies, and beliefs about opponents’ rationality. Using a theorem by Aumann & Brandenburger (Econometrica v.63(5) pp.1161–1180), the measured epistemic variables identify why subjects fail to play Nash equilibria in certain games. In several games, subjects are unable to guess accurately their opponent’s preferences; thus, they fail to agree on the game being played. In such a situation, a complete-information equilibrium concept cannot apply; Bayesian equilibrium must be used. The elicited data are also used to examine the underlying assumptions of several alternative behavioral models, such as the Level-k model and models of other-regarding preferences.