A two-step guessing game

We propose a two-step guessing game to measure the depth of thinking. We apply this method to the P beauty contest game. Using our method, we find that 81% of subjects do not make choice following best response reasoning while the classical method would suggest only 12%. The result suggests that the classical method has the fundamental problem that it cannot distinguish if a submitted number is due to best response reasoning or not. It also suggests that traditional level k analysis falsely attributes some sophistication to random players, and that the degree of false attribution is large. Our procedure provides an alternative way to identify whether the individual has best response reasoning which is essential for any positive level of depth of thinking and differentiates between the depth of thinking and random choice, and hence provides a very different conclusion, which is suggestive of limitations of the classical method.