The following theorem is Theorem 2.4.6 of Marker’s model theory book.
Theorem. Let $\mathscr{L}$ be a finite a finite language without function symbols and let $\mathcal{M}$ and $\mathcal{N}$ be $\mathscr{L}$-structures. Then $\mathcal{M}\equiv \mathcal{N}$ off and only if the second player has a winning strategy in $G_n(\mathcal{M},\mathcal{N})$ for all $n$.
Question. Can we extend this theorem for arbitrary finite language?