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Abstract 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 then give an application of simple strategies and prove that the winning player in Schmidt’s $$(\alpha , \beta , \rho )$$ game on $$\mathbb {R}$$ has a winning positional strategy, without appealing to the axiom of choice. 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 $$\mathsf {AD}$$ . 

We introduce the notion of [Formula: see text]-determinacy for [Formula: see text] a pointclass and [Formula: see text] an equivalence relation on a Polish space [Formula: see text]. A case of particular interest is the case when [Formula: see text] is the (left) shift-action of [Formula: see text] on [Formula: see text] where [Formula: see text] or [Formula: see text]. We show that for all shift actions by countable groups [Formula: see text], and any “reasonable” pointclass [Formula: see text], that [Formula: see text]-determinacy implies [Formula: see text]-determinacy. We also prove a corresponding result when [Formula: see text] is a subshift of finite type of the shift map on [Formula: see text].