This question was the motivation behind an earlier question of mine; having thought about it some more, that question seems nontrivial and the connection is actually pretty tenuous anyways, so I've decided to ask this directly:
For $\mathcal{K}\subseteq\omega_1$, we let $G_{CD}(\mathcal{K})$ be the game defined as follows (which is a special case of a broad class of games defined by Montalban): players $1$ and $2$ alternate building objects by finite extensions (with passing allowed) - in player $1$'s case, a single linear order $L^1$, and in player $2$'s case an array of linear orders $L^2_i$ ($i\in\omega$). Player $2$ wins iff $(i)$ each $L_i^2$ is in $\mathcal{K}$, and $(ii)$ if $L^1\in\mathcal{K}$ then $L^1\cong L^2_i$ for some $i\in\omega$. Put another way, player $1$ is trying to either build an element of $\mathcal{K}$ which player $2$ doesn't build, or trick player $2$ into building something not in $\mathcal{K}$.
If $\mathcal{K}$ is unbounded then $G_{CD}(\mathcal{K})$ can't be a win for player $2$ by $\Sigma^1_1$-bounding, and assuming CH it's easy to build an unbounded $\mathcal{K}$ such that $1$ doesn't have a winning strategy in $G_{CD}(\mathcal{K})$ either. However, in the absence of CH things seem much harder: intuitively, there are continuum-many strategies we need to defeat but only $\omega_1$-many "choices" involved in creating $\mathcal{K}$.
My question is:
Does ZFC alone prove that there is some $\mathcal{K}\subseteq\omega_1$ such that $G_{CD}(\mathcal{K})$ is undetermined?
"Obviously" the answer should be yes, but I don't see it immediately.
