*(Previously asked at MSE.)*

Let the **determinacy number**, $\mathfrak{g}$ (for "game"), be the smallest cardinal such that for every (two-player, perfect-information, length-$\omega$) game on $\omega$ at least one of the following holds:

There is a set $\Sigma$ of strategies for player $1$ such that $\vert\Sigma\vert\le\mathfrak{g}$ and, for every strategy ${\bf t}$ for player $2$, there is some ${\bf s}\in\Sigma$ such that ${\bf s}\otimes{\bf t}$ is a win for player $1$.

There is a set $\Sigma$ of strategies for player $2$ such that $\vert\Sigma\vert\le\mathfrak{g}$ and, for every strategy ${\bf s}$ for player $1$, there is some ${\bf t}\in\Sigma$ such that ${\bf s}\otimes{\bf t}$ is a win for player $2$.

In $\mathsf{ZF+AD}$ we have $\mathfrak{g}=1$, and in $\mathsf{ZFC}$ we have $\aleph_1\le \mathfrak{g}\le 2^{\aleph_0}$. A bit less trivially, $\mathsf{ZFC+MA}$ implies $\mathfrak{g}=2^{\aleph_0}$. I'm curious whether a low determinacy number is consistent with choice:

Is $\mathsf{ZFC}$ + $\mathfrak{g}<2^{\aleph_0}$ consistent?

I strongly suspect the answer is negative, but I don't see how to prove it; the possibility of $\kappa<2^{\aleph_0}<2^{\kappa}$ breaks every construction of a "hard-to-cover" game I can think of.