This is joint work with Richard Ketchersid.
Schipperus introduced the countable-finite game in the early 1990s. It is
an infinite game played between two players relative to a set S. In the
presence of choice, it is obvious that player II has a winning strategy
for all S, and it is natural to ask whether choice can be dispensed with.
AD+ is a technical strengthening of AD introduced by Hugh Woodin. It is
open whether AD+ actually follows from AD. All known models of AD come
from certain canonical models produced by the derived model construction.
In these canonical models, we show that every set either embeds the reals
or else is well-orderable.
From this we deduce that, except for the case when S is countable, the
countable-finite game on S is undetermined in these models.