I think my question applies to most games, but for the sake of concreteness, I shall consider one specific game in this question. We consider the game posed by Ilias Kastanas in his paper On the Ramsey property for sets of reals: We first fix some $X \subseteq [\omega]^\omega$.

1. Player I start by choosing an infinite subset $A_0 \subseteq \omega$.
2. Player II responds by choosing $x_0 \in A_0$, and then some $B_0 \subseteq A_0$ with $x_0 < \min(B_0)$.
3. Player I responds by choosing some $A_1 \subseteq B_0$.
4. Player II responds by choosing $x_1 \in A_1$, and then some $B_1 \subseteq A_1$ with $x_1 < \min(B_1)$.
5. Etc.

Player I wins iff $\{x_0,x_1,\dots\} \in X$.

Assume that Player I has a winning strategy $\sigma$. If $(A_0,x_0,B_0,\dots,A_n,x_n,B_n)$ is a partial play, then clearly the only things that determine what Player I should respond with are the finite sequence $(x_0,\dots,x_n)$ and $B_n$, Player II's last played set. However, it appears to me that we cannot assume that $\sigma$ satisfies this property. Let's say that $\sigma$ is uniform if $\sigma(A_0,x_0,B_0,\dots,A_n,x_n,B_n)$ only depends on $x_0,x_1,\dots,x_n,B_n$.

If Player I has a winning strategy, does Player I necessarily have a uniform strategy?

I'm also interested in how much the axiom of choice plays a part in the above statement.