TITLE: Game on a square grid
QUESTION [5 upvotes]: Not research level, comments are welcome.
Consider the following game:

The board is the vertices of an $n$ by $n$ square grid.
Two players take moves in turns.
A move is picking two vertices and drawing a straight line between them.
If the line intersects another line or passes through a third vertex,
the game ends and the player who made the move loses the game. Two or more lines are allowed to end at the same vertex.

Is there winning strategy depending on $n$?
Partial result:
We believe if we take the board to be the vertices of regular polygon,
the first player always wins, even if they don't have any skills
except finding a non-losing move if it exists.

REPLY [11 votes]: Doesn't a symmetry argument make this question rather simple?
If $n$ is odd, the second player (blue) can always mirror the first player's (red) moves (reflected through, or rotated by $\pi$ radians around the origin of the grid, i.e. its center vertex), so the first player loses.

If $n$ is even, the first player can draw one of the diagonals of the central square (orange; this move can't be mirrored) and can then mirror all of the second player's moves, so the second player loses.