  By employing computer simulations for a model binary mixture, we show that a
reentrant glass transition upon adding a second component only occurs if the
ratio $\alpha$ of the short-time mobilities between the glass-forming component
and the additive is sufficiently small. For $\alpha \approx 1$, there is no
reentrant glass, even if the size asymmetry between the two components is
large, in accordance with two-component mode coupling theory. For $\alpha \ll
1$, on the other hand, the reentrant glass is observed and reproduced only by
an effective one-component mode coupling theory.
