Topological games of bounded selections

We present a new variation of the classical selection principles ${\mathsf{S}}_{\mathrm{k}}(\mathcal{A},\mathcal{B})$ ($k\in \mathbb{N}$) and ${\mathsf{S}}_{\mathrm{f}\mathrm{i}\mathrm{n}}(\mathcal{A},\mathcal{B})$ that formally lies between these two properties. As in the case of the classical selection principles, we also obtain a new variation of topological games and discuss how new topological properties may emerge in the specific cases of covering and tightness.