I was trying to start analyzing QDB using the 2x2 game, and I don't see how the 2nd player can do better than force a draw if the first player begins by dividing the grid in half. Second player has a few distinct responses. This response is equivalent to any two edge lines in any two different cells.
First player responds by playing in edge cells in the remaining two squares.
No matter what the second player does now, his first move is the third edge of a box, so he gives up that box, and when that one is complete, the other box on that side can be taken. So first player gets 2 boxes. First player's final move then gives up the other two. This response is the two remaining middle lines. No matter what the second player does now, his first move is the third edge of a box, so he gives up that box.