Elliott's classification of AF-algebras through the $K_0$-functor allows to hit these C*-algebras (or rather the associated dimension groups) with classical (infinitary) model theoretic techniques, because the invariants concerned are after all just discrete first order structures. This raises the question how the model theoretic behaviour of these invariants relates to the metric model theory of the corresponding C*-algebras. We will explain that games of Ehrenfeucht-Fraïssé type offer a convenient framework to address such questions. In particular this is due to the fact that these EF-like games turn out to combine surprisingly well with Elliott's original intertwining argument. We show how this allows to construct families of AF-algebras with members of arbitrarily high Scott rank. This talk is based on joint work with Vaccaro, Velickovic and Vignati.