 Ehrenfeucht–Fraïssé game

In each round the Spoiler selects an element from either structure and the Duplicator responds in the other. The paired selected elements must preserve all atomic information and equality, giving a partial isomorphism of structures. In the infinite game the Duplicator wins by avoiding failure at every finite stage. In a timed game the Spoiler first announces the number of rounds, allowing a different response strategy for each horizon.

The Ehrenfeucht–Fraïssé (EF) game is a game-theoretic method used in model theory, a branch of mathematical logic. It serves as a tool for comparing structures in terms of their properties and behaviors. The game helps establish whether two mathematical structures (often models of a particular language or theory) satisfy the same first-order properties, which is important for understanding their equivalence in a logical sense. ### Structure of the Game 1.