Ehrenfeucht-Fraïssé theorem

For a finite relational first-order language, with finitely many constants allowed, the Duplicator wins the $n$-round game exactly when the structures are rank-n equivalent. Game-winning implies rank equivalence in arbitrary languages; the converse needs a finiteness condition and can fail even for an infinite unary vocabulary. Over a finite relational vocabulary, winning every announced finite horizon is equivalent to elementary equivalence.

For countably many unary predicates $P_i$, let $M$ contain infinitely many copies of each finite membership pattern and let $N$ add an element satisfying all predicates. Every finite vocabulary reduct has infinitely many elements in every Boolean cell in both structures, so $M\equiv N$. The Spoiler nevertheless wins the one-round full-language game by selecting the new element, whose complete atomic pattern has no counterpart in $M$.