Hurewicz and Rothberger respectively introduced prototypes of the selection properties $S_{fin}(A, B)$ and $S_1(A, B)$. In the series of papers titled “Combinatorics of open covers” (see the bibliography) we learned that for various topologically significant families $A$ and $B$ these selection properties are intimately related to game theory and Ramsey theory. The similarity in techniques used there to explore these relationships suggests that there should be a general, unified way to obtain these results. In this paper we pursue one possibility by considering the selection principle $S_{fin}(A, B)$ for distributive lattices. The selection principle $S_1(A, B)$ for distributive lattices will be treated in [3]. We use two examples throughout to illustrate the generality of the methods developed here.