One of the fundamental results in Ehrenfeucht games is the equivalence
$\mathcal{I}\sim_{m}^{\sigma}\mathcal{J}\Leftrightarrow\mathcal{I}\equiv_{m}\mathcal{J}$
with the $\sigma$-structures $\mathcal{I}$ and $\mathcal{J}$ participating (1) to the equivalence relation $\sim_m^{\sigma}$ if and only if Duplicator wins the Ehrenfeucht game of $m$ rounds and (2) to the equivalence relation $\equiv_m$ if and only if they satisfy the same formulas with quantifier rank up to $m$. It is proven that the relation $\sim_m^{\sigma}$ partitions the set of $\sigma$-structures to a finite number of equivalence classes $E_1,E_2,\dots,E_r$ and we can pick for each one of those classes $E_i$, a formula $\psi_i$ such that $\mathcal{I}\in E_i\Leftrightarrow\mathcal{I}\models\psi_i$. My question is this: these equivalence classes are the same (and equal in number) with the ones produced by the other equivalence relation $\equiv_n$ (that also partitions the set of $\sigma$-strucures in equivalence classes)? In this case, the characteristic formula $\psi_i$ also characterizes and the equivalence classes associated with $\equiv_m$? I mean, if we write the conditions (a) $\mathcal{I}\in E_i\Leftrightarrow\mathcal{I}\models\psi_i$ and (b) $\mathcal{I},\mathcal{J}\in E_i\Leftrightarrow(\mathcal{I}\models\psi_i\Leftrightarrow\mathcal{J}\models\psi_i)$, then (a) and (b) uses the same $\psi_i$?