Let $G_{fin}(\mathcal A,\mathcal B)$ be a selection game taking place over $\omega$ rounds. In round $n<\omega$, P1 chooses some $A_n\in\mathcal A$, then P2 chooses some finite subcollection $B_n\in[A_n]^{<\aleph_0}$. P2 wins provided $\bigcup\{B_n:n<\omega\}\in\mathcal B$.
Then the Menger game is $G_{fin}(\mathcal O,\mathcal O)$, where $\mathcal O$ is the collection of all open covers for some topological space. And the $\Omega$-Menger game is $G_{fin}(\Omega,\Omega)$, where $\Omega$ is the collection of all open $\omega$-covers for some topological space, where an $\omega$-cover has the property that every finite subset (rather than just singleton) of the space is contained in a single open set from the collection.
Considering the possible full and limited-information strategies for each player, how are these games related? For example, is there a space for which one player has a winning strategy in one game, but the other player has a winning strategy for the other game?
Best Answer
In general, if the second player can win the $\Omega$-Menger game, then they can win the Menger game. You take a given open cover and close it under finite unions. This is how to play the auxiliary game. When the second player makes finite selections, there are corresponding finite selections from the given open covers. If the second player produces an $\omega$-cover, then the corresponding choices form an open cover. This covers any level of strategy definable (where here I also mean P2 being able to beat any strategy type for P1 transfers in the same way; e.g. if P1 doesn't have a winning strategy in the $\Omega$-Menger game, then P1 also doesn't have a winning strategy in the Menger game).
Full-information strategies for P2
As in the paper linked in your hint (Theorem 35), strategically Menger is equivalent to strategically $\Omega$-Menger.
Full-information and pre-determined strategies for P1
According to https://mathoverflow.net/a/443001/57800, it is shown in https://arxiv.org/pdf/1806.10385.pdf that, consistently, a certain class of spaces are Menger if and only if they are Menger in all finite powers. By results of https://www.sciencedirect.com/science/article/pii/S0166864196000752, being Menger in every finite power is equivalent to being $\Omega$-Menger. However, this doesn't seem to answer the question whether being Menger is (consistently) equivalent to being $\Omega$-Menger for all topological spaces. I gather that level of generality is still open. Maybe there is a ZFC example of a Menger space which is not $\Omega$-Menger.
Edit: I forgot to mention that, in the Combinatorics of Open Covers II linked above, a special Luzin set is constructed which is Rothberger (hence Menger) but not $\Omega$-Menger. So it is consistent that the two properties are not equivalent.
The fact that the properties of Menger and $\Omega$-Menger are equivalent to P1 not having a winning strategy in the respective game is well known. I think those results are due to Hurewicz and Scheepers, respectively, but I could be misremembering. At any rate, this collapses the strategic information for P1 (full-information and pre-determined).
Constant strategies for P1
At the level of constant strategies for P1, by techniques from https://www.sciencedirect.com/science/article/abs/pii/S0166864120300031, one can show that P1 not having a constant winning strategy is equivalent to the Lindelof-like principle ${\mathcal A \choose \mathcal B}$; that is, every element of $\mathcal A$ contains a subset which is in $\mathcal B$. The Sorgenfrey line is a well-known example of a Lindelof space which is not an $\epsilon$-space; that is, a space which is Lindelof in all finite powers (a result of https://www.sciencedirect.com/science/article/pii/0166864182900657). These properties of being Lindelof and an $\varepsilon$-space correspond to P1 not having a constant winning strategy in the Menger and $\Omega$-Menger game, respectively.