The present question is a follow-up to this one. Assume GCH holds in $V$ and suppose $G \subseteq Add(\omega_1,\omega_2)$ is generic over $V$. For any $g \subseteq Add(\omega_1,\alpha)$ in $V[G]$ for $\alpha< \omega_2$ that is generic over $V$, there is a filter $h \subseteq Add(\omega_1,1)$ such that $g \times h$ is $Add(\omega_1,\alpha+1)$-generic.

Now in $V[G]$ define a game between two players who take turns playing mutually generic finite sequences of $Add(\omega_1,1)$-generics over $V$. At stage $n$, the player whose turn it is must play $g_n$ such that $g_0 \times ... \times g_n$ is generic. Player II wants to build an $Add(\omega_1,\omega)$-generic over $V$, and player I wants this to fail. By Joel's answer in the linked question, there is a play of the game in which player I wins. But does one of the players have a winning strategy?