For a Tychonoff space X, we denote by (C(X), τ k , τ p) the bitopological space of all real-valued continuous functions on X, where τ k is the compact-open topology and τ p is the topology of pointwise convergence. In the papers [6, 7, 13] variations of selective separability and tightness in (C(X), τ k , τ p) were investigated. In this paper we continue to study the selective properties and the corresponding topological games in the space (C(X), τ k , τ p),