Game involving an n-gon & lines

The "board" of this game is a regular n-gon drawn (as well as
possible) on paper, n being whatever the 2 players decide.
Each player uses a differently colored pencil and a straightedge to
take turns drawing 1 straight line per turn
between any 2 NONadjacent vertexes of the n-gon.
Lines must not coincide with any already-drawn lines (of either
color), but vertexes can be used more than once. (I suggest that there
be a limit of 2 or 3 lines per vertex, as to make things more
interesting.)
After a fixed number, m, of moves per player, scoring is as follows:
Player gets a point for each of her/his lines where the total number
of lines (of both players) crossing that particular line is an EVEN
integer.
(A shared vertex is not considered a line-interesection.)
The winner is the player with the most points, as you would guess.
If anyone decides to actually play this game, what values do you
suggest for m and n??
(Possible m's are, of course, bounded above by a function of n and the
maximum number of lines which may share one vertex.)
Thanks,
Leroy Quet

Update:

A few suggestions:
I think it might be best that, if n(k) = number of segments (drawn by
either player) crossing the k_th segment (drawn by the player in
question), then that player would get a point if n(k) is ODD, instead
of even.
I also think, for a more interesting game, that the number of sides,
m, of the polygon should be higher, at least 12. (m is limited by the
practical limitations of drawing all of this by hand, however.)
I also suggest that there be a maximum of 2 segments (total,
irrespective of which players drew the segments) that may be connected
to any particular vertex.
And I think that, in some ways, it would be best to have the number of
moves per player (before ending play) equal to floor((m-1)/2)
(ie. floor((m-1)/2)*2 total moves in a game), given a limit of 2
segments per vertex.
This leaves one or two vertex-pairs unconnected at the game's end,
which might have an effect on strategy (since the players cannot be
absolutely certain that there will be, at the game's end, 2 segments
connecting to each vertex that has not yet had 2 segments connected to
it at some point during play).

Please forgive my confusing lack of continuation with the uses of the
variable m (in 'm-gon' in the bottom) and the variable n (in 'n-gon'
in the first quoted post).
And then I use 'n(k)' to further confuse you all...

Thanks,
Leroy Quet

Update 2:

I think a interesting variation might be to, instead of each player
trying to get as many of their segments as possible having an even (or
odd) number of crossings, just have one player win if there are an
even number of *total* crossings, and the other player win if their
are an odd number of crossings.
(A 'crossing' is where 2 lines come together. For our purposes, if n
{n = 3 or more} lines come together at a single point, this counts as
n(n-1)/2 crossings.)
One advantage of this new variation is that we can play this game with
just one color of pencil/pen...
(which seems to actually be an issue when playing the game
in-practice...)
In this variation there are no scores, just a winner and loser, which
may be a disadvantage or an advantage, depending on your own opinion.

Thanks again, Leroy Quet