Consider a $9 \times 9$ matrix that consists of $9$ block matrices of $3 \times 3$. Let each $3 \times 3$ block be a game of tic-tac-toe. For each game, label the $9$ cells of the game from $1$ to $9$ with order from left to right, from above to down, call this a cell number. Label the $9$ games of the big matrix $1$ to $9$ with the same order, call this a game number.
The rule is the following:
$1$. Player $1$ starts with any game number and any cell number.
$2$. Player $2$ can make a move in the game whose game number is the cell number where player $1$ made the last move
$3$. It continues like this, where player $1$ then plays in the game whose game number is the cell number where player $2$ made the last move.
$4$. Special case, when a player is supposed to play in game $X$, but game $X$ is already won (may not be full)/lost (may not be full)/drawn (is full), then he may choose to play in any game he wants.
$5$. Winning: whenever a player has three winning games such that the three games line up either horizontally, vertically or across the diagonals, he wins.
It is easy to see why we call it tic-tac-toe $\times$ tic-tac-toe.
We know tic-tac-toe has a non-losing strategy. Does tic-tac-toe $\times$ tic-tac-toe have a non-losing strategy? If so what is it? In general what is a good strategy?
PS: This is a fun game. Originally what was a 'good move' now sends your opponent to a 'good game position', so it is more complicated.