  Schmidt's game, and other similar intersection games have played an important
role in recent years in applications to number theory, dynamics, and
Diophantine approximation theory. These games are real games, that is, games in
which the players make moves from a complete separable metric space. The
determinacy of these games trivially follows from the axiom of determinacy for
real games, $\mathsf{AD}_\mathbb{R}$, which is a much stronger axiom than that
asserting all integer games are determined, $\mathsf{AD}$. One of our main
results is a general theorem which under the hypothesis $\mathsf{AD}$ implies
the determinacy of intersection games which have a property allowing strategies
to be simplified. In particular, we show that Schmidt's $(\alpha,\beta,\rho)$
game on $\mathbb{R}$ is determined from $\mathsf{AD}$ alone, but on
$\mathbb{R}^n$ for $n \geq 3$ we show that $\mathsf{AD}$ does not imply the
determinacy of this game. We also prove several other results specifically
related to the determinacy of Schmidt's game. These results highlight the
obstacles in obtaining the determinacy of Schmidt's game from $\mathsf{AD}$.
