Gamorphisms

Gamorphisms are structure-preserving mappings between games, in the sense of category theory. A gamorphism from a game G to a game H is a function that assigns to each position (or state) of G a position of H, while preserving the essential game structure. Concretely, the mapping preserves the initial position, the legality of moves, and the terminal outcomes: if p can move to p' in G, then f(p) can move to f(p') in H; if p is terminal with outcome o in G, then f(p) is terminal with the same outcome o in H. In addition, gamorphisms are expected to preserve the strategic structure, so images of strategies in G yield valid strategies in H under the mapping.

Examples include projection maps from a composite game G1 + G2 to G1, where moves in G1 are mapped identically and the G2 component is ignored in a way that preserves move legality. The identity map on any game is a gamorphism, and the composition of gamorphisms is again a gamorphism, illustrating a basic categorical closure.

The concept supports a category whose objects are games and whose arrows are gamorphisms. When games support a binary operation such as a disjunctive sum, gamorphisms can be compatible with that operation, potentially giving a monoidal structure in suitable settings. Applications appear in formal verification, game semantics, and the study of simulation and bisimulation relations, where gamorphisms facilitate transfer of strategies and the reduction of complex games to simpler forms.

Origin and usage: the term gamorphism is encountered in certain discussions of game theory and theoretical computer science to describe this kind of structure-preserving map; it is not universally standardized and is sometimes referred to as a game homomorphism or a strategy-preserving map in the literature.