This post is an addition to the argument posted by bof last night. In bof's post, it is proved that posets having a special kind of dense subset cannot provide a counterexample for Monroe's question. It turns out that exactly this property of posets has been studied before, as it is relevant to similar questions arising from the topological version of the Banach-Mazur game. Furthermore, a large class of posets are known to have this property assuming CH.
Let's say that a poset $P$ has property bof if there is a dense $D \subseteq P$ with "countable upper cones", by which we mean that $\{ d \in D : p \leq d\}$ is countable for every $p \in P$.
Let's say that a poset $P$ has the descending ccc if for every $X \subseteq P$ there is some countable $Y \subseteq X$ such that $X$ and $Y$ both have the exact same set of lower bounds in $P$. Note that every ccc poset has the descending ccc. However, there are many other posets that have this property as well. (For example, the poset $[A]^\omega$ of countable subsets of some set $A$, when ordered by inclusion, has the descending ccc. Also, the poset of subsets of Baire space $\omega^\omega$ that are homeomorphic to the Cantor space, when ordered by inclusion, has the descending ccc. Both these examples are far from being ccc.)
Theorem: Suppose $P$ is a separative poset with the descending ccc. Assuming the Continuum Hypothesis: $(1)$ If $|P| < \aleph_\omega$ then $P$ has property bof. $(2)$ More generally, if $\square_\kappa$ holds for every singular $\kappa \leq |P|$, then $P$ has property bof.
This theorem was proved recently in the following paper:
Brian, Will; Dow, Alan; Milovich, David; Yengulalp, Lynne, Telgársky’s conjecture may fail, Israel Journal of Mathematics 242 (2021), pp. 325-358. ZBL07370902.
In this paper, we look at a conjecture of Telgarsky concerning the topological Banach-Mazur game. Generalizing the terminology of Monroe's question, let's say that a 1-tactic is a strategy for II that depends only on the previous move of I. But more generally, a $k$-tactic is a strategy for II that depends only on the previous $k$ moves of I.
As mentioned in a post by Joel, Debs described a space $X$ in which II has a winning $2$-tactic, but no winning tactic, in the topological Banach-Mazur game on $X$. Shortly after this discovery of Debs', Telgarsky conjectured that for every $k$, there is a space $X_k$ for which II has a winning $k$-tactic but no winning $(k+1)$-tactic. Notice that, by taking the disjoint union of all the $X_k$, this gives a space in which II has a winning strategy but no winning $k$-tactic for any $k$. Roughly, the conjecture is that there should be "complicated" spaces in which II can win the game, but not with any simple strategy.
Our paper shows that this conjecture is false under the assumption of GCH+$\square_\kappa$ for every singular $\kappa$. The main combinatorial principle is one that we denote $\nabla$:
$\nabla$ is the assertion that for every separative poset $P$ with the descending $\kappa$-cc, there is a dense $D \subseteq P$ such that every upper cone of $D$ has size $<\!\kappa$.
Note that property bof is simply $\nabla$ restricted to ccc ($=\, \aleph_1$-cc) posets. The reason this principle resolves Telgarsky's conjecture is essentially the argument that Joel gives in his second post about this question. If $\nabla$ holds, then II is able to code up the history of the game into each consecutive pair of I's moves, and thereby convert an arbitrary winning strategy into a winning $2$-tactic.
Let me point out that, unfortunately, the assumption of CH in this theorem cannot be removed. If $\mathfrak{b} > \aleph_1$, then the Hechler poset fails to satisfy $\nabla$. The assumption of $\square$ at limits is also necessary: it is consistent (relative to large cardinals) that the poset $([\omega_\omega]^\omega,\subseteq)$ fails to satisfy $\nabla$. (This is sometimes referred to as having no sparse cofinal family in $\omega_\omega$.)
It is interesting to me that the same principle used to resolve this conjecture for the topological Banach-Mazur game seems so relevant to Monroe's question for the poset Banach-Mazur game.