Psychological game theory, first introduced by Geanakoplos et al. (1989), has proven to be a potent framework for modeling belief-dependent motivations and emotional mechanisms such as surprise, anxiety, anger, guilt, and intention-based reciprocity. At the same time, general psychological games may be thought implausibly complicated in that utility potentially depends on the complete belief hierarchy. In particular, this means that the “higher-order beliefs” considered in the standard framework are much more complicated objects than the layman’s intuitive concept of “what others expect I will do”, “what others think I expect they will do”, and so on. This complexity gap is factually recognized in experimental applications, where games with simple utility functions that capture the intuitive understanding of “higher-order beliefs” have been used exclusively. Nevertheless, a formal definition of such “expectation-based games“ has been lacking. Here, we provide the  first such definition and systematically examine the properties of expectation-based games. For general expectation-based games, we develop an iterative procedure that determines all rationalizable choices. In addition, we exploit the product structure of the utility-relevant state space to provide a natural generalization of the expected-utility representation that is standardly presumed in traditional games. The resulting class of additive games not only allows us to recover a matrix representation of utility functions, but also yields a characterizing algorithm for rationalizability that inherits the linearity properties of traditional iterated elimination of strictly dominated choices.