For an integer $\gamma>1$, in the systems $T_1: [0,1]\to [0,1], x\mapsto \gamma x \text{ mod } 1\ \text{and}\ T_2:[0,1]\to [0,1], x\mapsto ({1}/{x}) \text{ mod } 1,$ given a sequence of functions $(f_n:[0,1]\to [0,1])_{n=1}^\infty $, consider the set $\mathfrak{BA}((f_n)_n):=\{x\in X:\, \liminf_{n\to\infty}{\mathrm{dist}}(T_i^n x,f_n(x))>0\}, i\in \{1,2\},$ which generalizes the non-recurrent set (by taking $f_n$ to be identity functions) and the non-shrinking-target set (by taking the $f_n$s to be constant functions, which generalizes the set of badly approximable numbers). It has been shown that if $(f_n)_n$ is a sequence of the same constant functions, then $\mathfrak {BA}((f_n)_n)$ is winning in the sense of McMullen’s game, which implies it is dense and has full Hausdorff dimension everywhere. In this paper, we strengthen this result by extending $(f_n)_n$ to be any equicontinuous sequence.

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 $\mathsf {AD}$.