We will initially discuss games played on subsets of the Cantor space, for which the existence or nonexistence of winning strategies for certain players can provide a characterization of perfectness or scatteredness. We will also give an old characterization of the type of trees in 2^{<\omega} through which outer models can add branches. Finally, we will make some observations about the nature of some generalizations of these topics to the 2^{\kappa} spaces.