  We present a new variation of the classical selection principles
$\mathsf{S}_\mathrm{k}(\mathcal A, \mathcal B)$ ($k\in\mathbb N$) and
$\mathsf{S}_\mathrm{fin}(\mathcal A, \mathcal B)$ 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.
