 One can prove that if \(A\) and \(B\) are \(m\)-equivalent for every natural number \(m\), then they are elementarily equivalent. In fact, if Eloise has a winning strategy \(\tau\) in the Hintikka game G\((\phi)\) on \(A\), where the nesting of quantifier scopes of \(\phi\) has at most m levels and Duplicator has a winning strategy \(\varrho\) in the game \(\EF_m (A, B)\), the two strategies \(\tau\) and \(\varrho\) can be composed into a winning strategy of Eloise in G\((\phi)\) on \(B\). On the other hand a winning strategy for Spoiler in \(\EF_m (A, B)\) can be converted into a first-order sentence that is true in exactly one of \(A\) and \(B\), and in which the nesting of quantifier scopes has at most \(m\) levels. So we have our necessary and sufficient condition for elementary equivalence, and a bit more besides.