It is a well-known fact that given a first-order sentence $\psi$ in prenex normal form $\forall x_1 \exists y_1 \forall x_2 \exists y_2 \dots \forall x_n \exists y_n \theta(x_1,\dots,x_n,y_1,\dots,y_n)$ (i.e., $\theta$ is quantifier free or at least $\Delta^0_0$ in the appropriate sense), the truth of $\psi$ in a given structure $M$ can be understood in terms of a game in which two players ($\forall$ and $\exists$) alternatively choose elements of $M$ corresponding to the bound variables of $\psi$, with $\forall$ choosing for the variables $x_i$ and $\exists$ choosing for the variables $y_i$. $\exists$ wins if and only if at the end the resulting variable assignment makes $\theta$ true in $M$. The characterization is of course that $M \models \psi$ if and only if $\exists$ has a winning strategy for this game.
Such a strategy can be understood as a series of Skolem functions $f_1(x_1),f_2(x_1,x_2),\dots,f_n(x_1,x_2,\dots,x_n)$ on $M$ with the property that (after expanding $M$ with the $f_i$'s), $M \models \forall x_1\dots x_n \theta(x_1,\dots,x_n,f_1(x_1),\dots,f_n(x_1,\dots,x_n))$. In the specific case of arithmetic (i.e., the structure $(\mathbb{N},0,+,\cdot)$), these functions are closely related to the idea of realizability. If $\psi$ is provable in Heyting arithmetic, for instance, then one can extract computable functions $f_1,f_2,\dots,f_n$ giving a winning strategy for $\exists$.
Kreisel showed that an analogous weaker statement is true of Peano arithmetic. A proof of $\psi$ in Peano arithmetic gives $\exists$ a computable procedure for beating any particular strategy for $\forall$, provided that $\exists$ is given $\forall$'s strategy in advance. More formally, given a proof of $\psi$ in Peano arithmetic, we can computably extract Turing indices $e_1,\dots,e_n$ such that for any functions $g_1 : \mathbb{N}^0 \to \mathbb{N}$, $g_2 : \mathbb{N}^1 \to \mathbb{N}$, $\dots$, $g_n : \mathbb{N}^{n-1} \to \mathbb{N}$, we have that $ \mathbb{N} \models \theta(a_0,\dots,a_n,b_0,\dots,b_n), $ where $$a_i = g_i(b_1,\dots,b_{i-1})$$ and $$b_i = \Phi_{e_i}^{g_1\dots g_n}(a_1,\dots,a_{i})$$ for each $i \in [1,n]$. ($\Phi_e^X(n)$ is of course the $e$th Turing functional applied to $n$ given $X$ as an oracle. Assume that we're using some fixed computable tupling function.) This fact is often called Kreisel's 'no-counterexample interpretation' of Peano arithmetic.
Let's say that $\bar{e} = (e_1,\dots,e_n)$ is a weak classical realizer of $\psi$ if the above condition holds. (I don't know if this concept has an established name.) For the sake of this question, we'll write $\bar{e} \Vdash \psi$ to mean that $\bar{e}$ is a weak classical realizer of $\psi$. (Note that $(\exists \bar{e})\bar{e}\Vdash \psi$ is a lightface $\Pi^1_1$ condition on $\psi$.) I believe that the commonly studied notions of classical realizers for arithmetic (e.g., realizability in the sense of Krivine) are a priori stronger than this in that while they allow a certain amount of probing of $\forall$'s stragey, they do not necessarily allow arbitrary computation from it.
We can now cosmetically restate Kreisel's result as saying that if $\mathsf{PA} \vdash \psi$, then $(\exists \bar{e})\bar{e} \Vdash \psi$. It's also not too hard to show that if $(\exists \bar{e})\bar{e} \Vdash \psi$, then $\mathbb{N} \models \psi$. (No amount of prep work can beat an unbeatable strategy.)
I'm curious about the extent to which the conditions $\mathsf{PA} \vdash \psi$, $(\exists \bar{e})\bar{e} \Vdash \psi$, and $\mathbb{N} \models \psi$ differ. One immediate observation is that if $\psi$ is a true $\Pi^0_1$-sentence, then $(\exists \bar{e})\bar{e} \Vdash \psi$ (somewhat trivially), which means that there are many examples of $\mathsf{PA}$-unprovable sentences with weak classical realizers (a Gödel sentence for instance). This leads to a somewhat more nuanced question.
Question 1. Is there a sentence $\psi$ such that $(\exists \bar{e})\bar{e} \Vdash \psi$, yet $\mathsf{PA} \cup \{\chi: \mathbb{N}\models\chi,~\chi\text{ a }\Pi^0_1\text{ sentence}\} \not \vdash \psi$?
In the other direction, there is basically no way that $\mathbb{N} \models \psi$ implies $(\exists \bar{e})\bar{e} \Vdash \psi$, but finding an explicit example of this seems difficult. The theory of $\mathbb{N}$ is lightface $\Delta^1_1$, which is simpler than the apparent complexity of the set of sentences with weak classical realizers (lightface $\Pi^1_1$).
Question 2. Is there a sentence $\psi$ such that $\mathbb{N} \models \psi$, yet $\neg (\exists \bar{e}) \bar{e} \Vdash \psi$?
