Cognitive hierarchies for games in extensive form

In the cognitive hierarchy (CH) framework, players in a game have heterogeneous levels of strategic sophistication. Each player believes that other players in the game are less sophisticated, and these beliefs correspond to the truncated distribution of a “true” distribution of levels. We develop the dynamic cognitive hierarchy (DCH) solution by extending the CH framework to games in extensive form. Initial beliefs are updated as the history of play provides information about players' levels of sophistication. We establish some general properties of DCH and fully characterize the DCH solution for a wide class of centipede games. DCH predicts a strategy-reduction effect: there will be earlier taking if the centipede game is played as an alternating-move sequential game rather than as a simultaneous move game in its reduced normal form. Experimental evidence reported in García-Pola et al. (2020a) supports this prediction. In all three centipede games for which the DCH strategy-reduction effect is predicted, termination occurs earlier when played sequentially rather than simultaneously with reduced strategies. In a fourth centipede game, where this effect is not predicted, it is not observed.