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.

1. B1|,B2| A3_,B3_
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.
2. A1_,B1_ A1|
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. Result: draw.

1. B1|,B2| A2_,B2_
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. Likewise, each other "free" move gives up
a box, so first player claims a box and gives one up, second player claims
that box and gives another up, etc. Result: draw.

1. B1|,B2| A2_,B3_
This response is equivalent to any one middle line and one edge line on the
other side (not part of the same box). First player responds by playing the
other middle line immediately.
2. B2_ ...
This is the third side of a box, so his turn ends immediately. Second player
claims the box, but his next move gives up another box, and we see the
alternation of boxes again.

1. B1|,B2| A2_,A3_
This response is any two lines in the same cell.
Obviously, this gives up that cell. First player takes it and then plays the
middle line on the other side.
2. A2|,B2_ ...
Any move by second player gives up another box, then first player gives up a
box to second, then second player gives up the last box to first. Result:
First player wins 3-1.

Even considering analyzing the 3x3 game has my brain spinning, so I will not
attempt that now.

Joseph DeVincentis