Cross, Don't Cross Circle Game

Here is a game for 2 players.

As in many of my games, the players play an even number of rounds, half of the rounds where one player is offense and the other player is defense, and the other half of the rounds with the players switching who is defense and offense, and then the players adding up their scores for their grand total scores.


A round starts with a carefully drawn circle on paper. (No grids this time. Sorry.)

The defense player starts the game by drawing m (m is fixed number for all rounds) straight line segments from anywhere on the circumference of the circle to anywhere else on the circumference of the circle. (I suggest an m of 4 to 6 for beginning players.) The defense player's line segments may cross each other (but don't have to cross).

(I suggest that the bigger m is, the bigger the circle is drawn.)

After the defense player has drawn the m line segments, then it is the offense player's turn to make his/her moves for the round. Clarification: After the defense player draws her/his m line segments, he/she does not move any more during the round.
(So, in a round, first the defense player draws all his/her line segments, then the offense player draws all his/her line-segment-- see below.)

On a move the offense player draws a straight line-segment from an intersection to another intersection*.
*An intersection is either where any line segment (drawn by the defense player) touches the circle, or is where any pair of previously-drawn line segments (drawn by either player) cross.
On every odd-numbered move (the first move, the third move, the fifth move, etc) the offense player's line segment must not cross any other previously-drawn line segments.
On every even-numbered move the offense player's line segment MUST cross exactly one previously-drawn line segment (crossing no fewer, no more than one segment).

And, oh yeah, neither the defense player's nor the offense player's line segments may coincide (coincide along more than one point) with any other previously-drawn line-segment.

The offense player moves until he/she can't move anywhere, otherwise she/he MUST move.
By the way, the defense player may find possible moves for the offense player if the offense player wrongly claims that he/she can't move any more. (It is advantageous for the defense player if the offense player keeps drawing line segments.)

As the offense player draws line segments, the number of these line segments drawn is tabulated.

After playing all the rounds, the winner of this game is the player who, during all rounds that they were the offense player, drew the FEWEST line-segments all together.

--

Question:

I wonder, is there a pattern of line segments the defense player can draw that will guarantee a larger number of moves by the offense player than with any other pattern of line segments drawn by the defense player?
(By "pattern" I mean, as an example, lines drawn parallel, all lines crossing at a center point, the lines forming the perimeter of an m- gon, etc.)

Thanks,
Leroy Quet

PS: After I post this game to my blog, the list of 66 or so games I posted in September will be hidden. Just click on the triangle next to the September link to get that list of games back.