Subdivide, The Game

The game consists of any number of rounds, but the
number is a multiple of the number of players playing
the game.
Players take turns being the offensive player (snicker
snicker) for each round.
On each round players take turns drawing a predetermined
number of dots on a blank piece of paper.
After the dots are drawn, players take turns (offensive
player first) drawing straight line segments (preferably
with a straight-edge) that connect pairs of dots, one
line segment connecting two dots per move.
The segments can connect any two dots provided that:
*The segments don't cross over any intermediate dots
and don't cross other previously drawn line-segments.
*There is a maximum of 3 line-segments connected to
every internal dot. But as for the dots making up the
perimeter of the convex-hull of the set of dots, any
number of line-segments can connect to these dots.
The round is over when the perimeter of the convex-hull
of the set of dots is completed.
The number of points the offensive player gets is equal
to the maximum number of dots making up the perimeter
of any one polygon drawn during the game.
Here by 'polygon', it is meant that the scoring polygon
has no internal line-segments subdividing it into
smaller polygons. But it may have internally drawn
line-segments, as long as these segments don't
subdivide the polygon.
(As to what to do if the winning polygon has a smaller
polygon within it, touching the permimeter of the
winning polygon at at most 1 dot, I leave that up to
the players to decide.)
Here is a sample round:
There are 2 players, player 1 (offense) drawing the
odd line-segments, player 2 drawing the even segments.
The dots happen to have been drawn as a 4-by-4 array
of dots.
The dots each are, for purpose of illustrating the game,
labeled a,b,c, or d for the row they occur in, and
1,2,3 or 4 for the column they occur in.
Segment 1: a1 - b2. Segment 2: a1 - a2. Segment 3: b1 - c2.
Segment 4: b1 - a1. Segment 5: c2 - b4. Segment 6: b3 - b4.
Segment 7: b2 - a4. Segment 8: a3 - a4. Segment 9: c2 - d3.
Segment 10: a4 - b4. Segment 11: c3 - b4. Segment 12: b4 - c4.
Segment 13: c1 - d2. Segment 14: c1 - d1. Segment 15: d3 - c4.
Segment 16: c4 - d4. Segment 17: b1 - d2. Segment 18: b1 - c1.
Segment 19: d2 - d3. Segment 20: a2 - a3. Segment 21:d3 - d4.
Segment 22: d1 - d2.
The winning polygon has 6 dots making up its perimeter:
a1, b2, a4, b4, c2, b1.
So player 1 gets 6 points this round.
The line-segment within the winning polygon, segment b3 - b4,
does not affect the score.
Thanks,
Leroy Quet

PS
I was thinking that, even though I like the convex-hull rules,
this game would be a bit more accessible if there is a slight
rule modification.
First, allow a maximum of 3 line-segments to be drawn to any
dot, no matter if the dot is part of the perimeter of the dots'
convex-hull or not.
Second, the game is over whenever no more line-segments can
be drawn, either because all dots have 3 segments drawn to them,
or because the dots with fewer segments aren't accessible to
any other dots with fewer than 3 segments.
Maybe with this rule-change the game will be more accessible to
children and non-math people.
Yet, as I said before, I like the convex-hull rules, even if
they may be too mathy.
Thanks,
Leroy Quet