Inexpressibility of Second order

In finite model theory, Ehrenfeucht-Fraïssé games gives us tools to prove inexpressibility results for FOL. Pebble games do the same for infinitary logic with finitely many variables.

Do we have such a framework to deal with Second order logics ? At least with $$\Sigma_1^1$$ and $$\Pi_1^1$$ ?