We develop and analyze an axiomatic model of strategic thinking in games, and demonstrate that it accommodates well-known empirical puzzles that are inconsistent with standard game theory. We model the reasoning process as a Turing machine, thereby capturing the stepwise nature of the procedure. Each `cognitive state' of the machine is a complete description of the agent's state of mind at a stage of the reasoning process, and includes his current understanding of the game and his attitude towards thinking more. We take axioms over the reasoning about games within each cognitive state. These axioms restrict the scope of the reasoning process to understanding the opponent's behavior. They require that the value of reasoning be purely instrumental to improving the agent's choice, and relate the suitability of the reasoning process to the game at hand. We then derive representation theorems which take the form of a cost-benefit analysis. Players behave as if they weigh the instrumental value of thinking deeper against the cost. To compare behavior across different games, we introduce a notion of cognitive equivalence between games. Games within the same cognitive equivalence class induce the same cost of reasoning function and only differ in the incentives to reason, where incentives are related to the difference in payoffs across actions. We further enrich our model and allow the agent to account for the opponent's own cost-benefit reasoning procedure. Lastly, we apply our model to Goeree and Holt's (2001) well-known `little treasures' games. We show that the model is consistent with all the `treasures' that fall within the domain of our theory. We perform a single-parameter calibration exercise using stringent restrictions on the model's degrees of freedom, and show that the model closely matches the experimental data.