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Description: This paper theoretically and experimentally studies the dynamic cognitive hierarchy (DCH) solution, a novel hierarchical reasoning model for general extensive games, in multi-stage games of incomplete information. Using the dirty-faces game, I illustrate how the DCH solution can differ dramatically between strategically equivalent games. I test this prediction in a laboratory experiment using two strategically equivalent versions of the dirty-faces game. The parameters are calibrated to maximize the informativeness of the experiment.The experimental results reveal significant differences in behavior between the two versions, and more importantly, the observed differences align with DCH.This result suggests playing a dynamic game in a different, but strategically equivalent, version can lead to distortions in behavior in a predictable way.