Cognitive hierarchies in adaptive play

Inspired by the behavior in repeated guessing game experiments, we study adaptive play by populations containing individuals that reason with different levels of cognition. Individuals play a higher order best response to samples from the empirical data on the history of play, where the order of best response is determined by their exogenously given level of cognition. As in Young's model of adaptive play, (unperturbed) play still converges to a minimal curb set. Random perturbations of the best response dynamic identifies the stochastically stable states. In Young's model of adaptive play with simple best-responses, the set of stochastically stable states are sensitive to the sample size that individuals from a population can draw. In generic games with higher order best-responders in both populations, the sample size is rendered irrelevant in determination of the stochastically stable set. Perhaps counter-intuitively, higher cognition may actually be bad for both the individual with higher cognition and his parent population.