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Schmidt's game, and other similar intersection games have played an importantrole in recent years in applications to number theory, dynamics, andDiophantine approximation theory. These games are real games, that is, games inwhich the players make moves from a complete separable metric space. Thedeterminacy of these games trivially follows from the axiom of determinacy forreal games, $\mathsf{AD}_\mathbb{R}$, which is a much stronger axiom than thatasserting all integer games are determined, $\mathsf{AD}$. One of our mainresults is a general theorem which under the hypothesis $\mathsf{AD}$ impliesthe determinacy of intersection games which have a property allowing strategiesto 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 thedeterminacy of this game. We also prove several other results specificallyrelated to the determinacy of Schmidt's game. These results highlight theobstacles in obtaining the determinacy of Schmidt's game from $\mathsf{AD}$.

Determinacy of Schmidt's Game and Other Intersection Games