Another Criss-Cross Grid Game

Here is another one of my games. It isn't as fun, probably, as some of
the other games I have posted. But maybe someone will find it
enjoyable anyway.
Start with an n-by-n grid (n-by-n lines, (n-1)-by-(n-1) row/columns)
on graph-paper. (n should be at least 5, maybe in the range of 10 or
more.)
There are two players who switch roles after each round, an offensive
player and a defensive player.
Say the grid is n lines wide((n-1) columns) and n lines high ((n-1)
rows). The defensive player starts the round by writing the integers 1
through (2n) in any order to the left of the left-most vertical line
(each integer lined up with a different horizontal line of the grid)
and above the top-most horizontal line (each integer lined up with a
different vertical line of the grid).
Whether a particular integer is written either along the left side of
the grid or along the top of the grid is up to the defensive player.
Example: 6 lines -by- 6 lines:
. 2 6 11 7 1 8
9 -------------
4 | + + + + +
10| + + + + +
12| + + + + +
3 | + + + + +
5 | + + + + +
(Note: In case ascii art doesn't look correct, the pluses are the
intersections of the grid, and are supposed to each be directly below
an integer of the top row of numbers and directly to the right of the
left column of numbers.)
The two players each take turns drawing a straight line-segment (with
a straightedge) from the intersection of the grid last drawn-to by the
opposing player to an intersection determined by the order of the move
within the round.
If the move is move m -- the defensive player moves on even-numbered
moves, and the offensive player moves on odd-numbered moves -- then
the player can move to any intersection in the same row/column lined
up with the m along the grid's edges. In other words, if the value m
is written along the top of the grid, then the player on the mth move
can move to any one of the n intersections in the same COLUMN as the
value m. And if the value m is written along the left side of the
grid, then the player on the mth move can move to any one of the n
intersections in the same ROW as the value m.
Players cannot draw line-segments along already drawn line-segments.
(ie Line-segments can only intersect at most at one point.)
Line-segments cannot cross intersections that are already the
endpoints of other line-segments.
After 2n total moves (n moves for each player), the round is over.
The offensive player gets a point for every time a line-segment (drawn
by either player) crosses another segment.
If k line-segments intersect at a common point, the offensive player
gets k(k-1)/2 points for that intersection.
This is the same as counting the number of line-segments intersected
by a line-segment AS the line-segment is being drawn. (If your line
segment is crossing an intersection with (k-1) line-segments already
intersecting there, then add (k-1) to the offensive player's score. Or
wait until after the game is over to enumerate the crossings, and give
the offensive player k(k-1)/2 points for those k line-segments
intersecting at a common point.)
So, the defensive player moves so as to try to keep the lines from
crossing. The offensive player moves to try to get as many crossings
as possible.
After a round is complete the players switch who is the offensive
player and who is the defensive player. The highest score wins, after
playing an even number of rounds, of course.
What is a good strategy for this game?
Thanks,
Leroy Quet