Slice And Fill

Another game from Leroy Quet Inc.:
Start with an n-by-n grid either on graph paper or drawn carefully. I
suggest a relatively small n as far as my games usually go, about 4 to
11.
This game is for two players, each player with a pen/pencil of a
different color than his/her opponent's.
On each move, a player must first take one action (see below), and may
take a second action as well.
First, on her/his move, a player MUST fill an empty section of the
grid entirely, where the section is defined by* the lines of the grid
and/or by straight line-segments drawn by players (see below -- a
section may be a square, or it may be a polygon which is a subset of a
square).
*[By "where the section is defined by...", I mean "where the section is
BORDERED by" the lines of the grid
and/or by straight line-segments drawn by players.
There are no internal line-segments within any given "section" (at
least until lines are added later in the game, crossing the section
and subdividing the section into plural sections). ]
Unless this is the first move by a player or in case of some
other circumstances (see below), then the player must fill in a
section that is adjacent to the last section filled in by the same
player. (By "adjacent", the new section must be a previously unfilled
section that touches the player's last filled-in section along a line,
and not just touching at a point.)
Second, a player MAY on his/her move, after filling in a section,
connect any two vertices of the grid with a straight line-segment
(drawn carefully) (The vertices at the endpoints of any line-segment
must be contained within the n-by-n grid, possibly being on the grid's
border. Line segments may be diagonal, of course.) The line segment
may not cross another line segment previously drawn by either player
as part of the game. But line segments can cross grid-lines and cross
previously filled-in sections.
If a player fills in a section that is adjacent (touching along a
line, not just at a point) to a section previously filled in by the
player's opponent, a "point" is given to the player. (I put "point" in
quotes, because the goal of the game is to get as FEW points as
possible.)
Also, if a player has just filled in a section that is adjacent to a
section previously filled in by the player's opponent, then the player
may fill in any empty section of the grid on her/his next move,
starting a new path of adjoining sections.
The previous paragraph tells one situation where a player can fill any
section of the grid, not necessarily a section next to the section
previously filled in by that player. The other situation is when the
player cannot move because all adjacent sections to the player's last
filled-in section are already filled in.
Play continues until all sections are filled in.
The winner has the FEWEST number of points.
Note: Officially, the line-segments and grids act as if they were
drawn perfectly -- the line-segments pass through the appropriate grid-
vertices, given the slope of the lines.
Any strategies for this game? Any way to ensure a win for one player
or the other?
(If so, I need to fix the rules.)

Thanks,
Leroy Quet