The Rothberger and Menger selection principles, and the topological games that characterize, have been well-studied by others over the decades. Likewise, much work has been done on weakened versions of the two properties and the corresponding topological games.

While it is not difficult to show that for a fixed finite k, the selection principle S_k generally "collapses" to S_1, questions about the corresponding topological game can be much more interesting, as tools that were used to prove strong results such as Galvin's Theorem do not work without additional assumptions.

After a review of some classic selection principles and their corresponding games, we'll investigate what changes when two selections are made in each inning.