We study structural properties of a turn-based game called the Marble Drop Game, which is an experimental paradigm designed to investigate higher-order social reasoning. We show that the cognitive complexity of game trials, measured with respect to reaction time, can be predicted by looking at the structural properties of the game instances. In order to do this, we define complexity measures of finite dynamic two-player games based on the number of alternations between the game players and on the pay-off structure. Our predictions of reaction times and reasoning strategies, based on the theoretical analysis of complexity of Marble Drop game instances, are compared to subjects' actual reaction times. This research illustrates how formal methods of logic and computer science can be used to identify the inherent complexity of cognitive tasks. Such analyses can be located between Marr's computational and algorithmic levels.