  We develop methods to formally describe and compare games, in order to probe
questions of game structure and design, and as a stepping stone to predicting
player behavior from design patterns. We define a grammar-like formalism to
describe finite discrete games without hidden information, allowing for
randomness, and mixed sequential and simultaneous play. We make minimal
assumptions about the form or content of game rules or user interface. The
associated game trees resemble hybrid extensive- and strategic-form games, in
the game theory sense. By transforming and comparing game trees, we develop
equivalence relations on the space of game systems, which equate games that
give players the same meaningful agency. We bring these together to suggest a
method to measure distance between games, insensitive to cosmetic variations in
the game logic descriptions.
