  We prove that in any Hausdorff space, the Rothberger game is equivalent to
the $k$-Rothberger game, i.e. the game in which player II chooses $k$ open sets
in each move. This result follows from a more general theorem in which we show
these games are equivalent to a game we call the restricted Menger game. In
this game I knows immediately in advance of playing each open cover how many
open sets II will choose from that open cover. This result illuminates the
relationship between the Rothberger and Menger games in Hausdorff spaces. The
equivalence of these games answers a question posed by Aurichi, Bella, and
Dias, at least in the context of Hausdorff spaces.
