Zigzag Edge Game

A game for two players:

Start with a grid of (n-1)-by-(n-1) squares, or n-by-n lines. I suggest that n be at least 12.

The players take turns placing dots on the grid, one dot each move at any intersection of the grid that does not yet have a dot on either of the two intersecting (horizontal and vertical) grid-lines.

After n dots are placed -- and the dots represent a permutation of (1,2,3,...n) -- the strategy portion of the game is over.

Either of the players then draws (n-1) straight line-segments (called "permutation lines"), starting at the leftmost dot, a line-segment drawn between each dot and the dot on the vertical grid-line immediately to the right. (So, you get a zigzag, in lots of cases.)

Then, either player draws (n-1) straight line-segments (also called permutation lines), starting at the topmost dot, a line-segment drawn between each dot and the dot on the horizontal grid-line immediately below.

Now, the two intersecting zigzags are inclosed within a polygon (not talking about the convex hull), the perimeter of which is made up of permutation lines and/or parts of permutation lines.
Call each straight line-segment along this perimeter an "edge line". Two permutation lines that meet at one vertex AND have the same slope are considered to both be part of one edge-line. Two edge lines may come together somewhere other than at a dot (in which case an edge line takes up only part of a permutation line).

Player 1 gets as a score the number of edge lines from the leftmost dot to the rightmost dot along the TOP of the bounding polygon's perimeter.

Player 2 gets as a score the number of edge lines from the leftmost dot to the rightmost dot along the BOTTOM of the bounding polygon's perimeter.

Largest score wins.

Note: The bounding polygon may narrow to a single point. Four edge-lines are considered to meet at such a point, even though only two permutation lines cross there.


What is a strategy for this game if you are player 1, or if you are player 2?


Example game:
Let us say we have a grid of 9-by-9 lines.
Label the grid's vertical lines 1 to 9, starting at the bottom. And label the grid's horizontal lines 1 through 9, starting on the left.
During play, the dots are drawn at these grid-line intersections:
(1,6), (2,7), (3,8), (4,4), (5,5), (6,9), (7,1), (8,3), (9,2).

Player 1 gets 5 points. Player 2 gets 5 points. A tie.

Note that the permutation-line segments connecting (1,6) to (2,7) to (3,8) count as one edge line. Also note that the permutation line (6,9) to (7,1) and the permutation line (4,4) to (8,3) intersect within a grid-square, and that we have 4 edge-lines (2 edge lines to player 1's score, 2 edge lines to player 2's score) meeting at this intersection.

Thanks,
Leroy Quet