Let 2 < n < ω. Then CAn denotes the class of cylindric algebras of dimension n, RCAn denotes the class of representable CAns, CRCAn denotes the class of completely representable CAns, and NrnCAω(⊆ CAn) denotes the class of n-neat reducts of CAωs. The elementary closure of the class CRCAns (Kn) and the nonelementary class At(NrnCAω) are characterized using two-player zero-sum games, where At is the operator of forming atom structures. It is shown that Kn is not finitely axiomatizable and that it coincides with the class of atomic algebras in the elementary closure of ScNrnCAω where Sc is the operation of forming complete subalgebras. For any class L such that AtNrnCAω ⊆ L ⊆ AtKn, it is proved that SPCmL = RCAn, where Cm is the dual operator to At; that of forming complex algebras. It is also shown that any class K between CRCAn ∩ SdNrnCAω and ScNrnCAn+3 is not first order definable, where Sd is the operation of forming dense subalgebras, and that for any 2 < n < m, any l ≥ n + 3 any any class K such that At(NrnCAm ∩ CRCAn) ⊆ K ⊆ AtScNrnCAl, K is not not first order definable either. [ABSTRACT FROM AUTHOR]