Polygons In Permutation Grid

Here is a game for 2 players.

Start with an n-by-n grid (n-lines by n-lines, or n-1 squares by n-1 squares). I suggest that n be at least 10.

First, players take turns placing a total of n dots at intersections of the grid.
Each dot is placed at an intersection of any two lines that do not have any other dots on either of them.
So, after n dots are placed on the grid, the dots represent a permutation of (1,2,3,...n).

Reading the dots from top to bottom, let the dot on the mth horizontal line be p(m).
Reading the dots from left to right, let the dot on the mth vertical line be q(m).

Draw a straight line-segment from p(m) to p(m+1) for all m where 1<= m <= n-1.
Draw a straight line-segment from q(m) to q(m+1) for all m where 1<= m <= n-1.

Player 1 gets a point for every triangle that is formed by the line-segments.
Player 2 gets a point for every non-triangle (4 or more sides) that is formed by the line-segments.
Only polygons completely bounded by parts of line-segments (not counting the grid's lines) score any points.

For a triangle or non-triangle to score a point, the polygon must not be subdivided by any line-segments (but may be subdivided by grid-lines).


I suspect that there is a bias either towards player 1 or player 2. So, play an even number of rounds with the same-sized grids, each player playing player 1 and player 2 an equal number of times, and add up each player's score to get the players' grand total scores.

Highest grand total score wins.

Thanks,
Leroy Quet