Horizontal/Vertical (Sometimes Diagonal) Grid Game

Start with an n-by-n grid drawn on paper. (I suggest an n of at least
8, if not much larger. But not too large if you don't have a long time
to play.)
The players (2 in number) take turns filling in the squares of the
grid, one square per move.
Player 1 fills in the first square anywhere in the grid.
Play continues like this:
Player 1 fills in any EMPTY square that is either left of or right of
the last square filled in by player 2.
Player 2 fills in any EMPTY square that is either above or below the
last square filled in by player 1.
Either player may fill in an empty square that is diagonally touching
the last square filled in by the other player IF BOTH players agree
that such a move is acceptable at that time.
The players alternatingly fill in a string of squares this way until
one player cannot move. (If a player can move, the player must move.)
Then the player that could not move fills in any empty square with his/
her initials or unique symbol (chosen before play). (The other player
then moves from that position as before in the next move.)
(Note: by "empty" square, I mean a square that neither has been filled
in nor has any initials or symbols in it.)
Play continues until all squares (including isolated single squares)
are filled in or have symbols/initials in them.
The player with the MOST symbols/initials is the winner.
So it is good to NOT be able to move as much as possible during play.
Note: In the first move of the game, player 1 does NOT write his/her
symbol/initials into the first square. Only after a player cannot move
does that player write their symbol/initials in a square.
Clarification: If a player is in a situation where he cannot move
either (up/down)(left/right), but he/she can move diagonally, then
that player must move diagonally IF the other player agrees to this
move. Most often the other player will agree, since this denies the
first player a point. But sometimes for strategic reasons the other
player may deny the first player the right to move diagonally.
What would be a good strategy for this game?
Thanks,
Leroy Quet