Polygons-&-Dots Game

Here is another game.
For 2 players.
First, players alternatingly take turns placing the same
number of dots on a piece of paper.
Each dot can be anywhere on the paper as long as no 3 or
more dots are lined up (along any imaginary straight lines),
and each dot is placed where no dot existed previously.
Second, the players take turns connecting the dots by drawing
straight line-segments (with a straight-edge, if necessary).
Each segment is drawn from the last connected-to dot
(connected to by the opposing player) to any unconnected-to dot.
The first player can connect any 2 dots.
The last dot to be connected-to and the first dot are connected
when the game is done, so the line-segments form a closed path
which visits every dot on the paper exactly once.
Scoring:
Player 1 gets a point for every polygon with an odd number of sides,
player 2 gets a point for every polygon with an even number of sides.
By "polygon", I mean those polygons formed by intersecting
line-segments and by dots as vertexes, but with no internal
line-segments.
For example, let us say that we had 5 dots (in reality, we would
only have an even number of dots) arranged at the vertexes of a
regular pentagon. The dots are connected to make a 5-pointed star,
ie: each vertex is connected to the 2 vertexes not adjacent to it.
We would then have 6 polygons, 5 triangles and one smaller pentagon.
The 5 larger triangles, for example, would not count.
So, player 1 would get 6 points, player 2 would get 0 points.
I suggest, especially if there are a large number of polygons to
consider, that the polygons be shaded in after their sides are
counted.
Shading them in with different colors might be aesthetically pleasing.
:)

The no-colinear-points rule is somewhat difficult to enforce in
practice if this game is not played on a computer.
If neither player notices that a player is drawing a point so that 3
points are colinear, this is no big deal, really.

thanks,
Leroy Quet