This paper suggests a novel way to import the approach of axiomatic theories of individual choice into strategic settings and demonstrates the benefits of this approach. We propose both a tractable behavioral model as well as axioms applied to the behavior of the collection of players, focusing first on Prisoners' Dilemma games. A representation theorem establishes these axioms as the precise behavioral content of the model, and that the model's parameters are (essentially) uniquely identified from behavior. The behavioral model features magical thinking: players behave as if their expectations about their opponents' behavior vary with their own choices. The model provides a unified view of documented behavior in a range of often-studied games, such as the Prisoners' Dilemma, the Battle of the Sexes, Hawk-Dove, and the Stag Hunt, and also generates novel predictions across games.