A game for two players:
First, draw a circle on a piece of paper.
Players start by each drawing a different straight line-segment at any angle they choose from the center of the circle to the circumference.
Players thereafter move like so: (Player 2, Player 1), (Player 1, Player 2), (Player 2, Player 1), (Pl 1, Pl 2), (Pl 2, Pl 1), etc.
So, we have "whole moves", consisting of two moves, with a move by each player. And who moves first in the whole moves alternates.
The first player to move in a whole-move decides if the next line-segment will bounce left or bounce right. This player then draws his straight line-segment in the proper direction (relative to the direction his own last line-segment was traveling) from where his own last line segment ended to where the new line-segment comes up against a pre-existing line-segment (drawn by either player) or up against the circumference of the circle. A player's line-segment may pass through a pre-existing line-segment. But each time a player crosses a line-segment with another line-segment, his score is halved. No line-segments may pass outside of the circle.
The second player to move in a full-move then must bounce the same direction, left or right, as the other player did, but relative to the direction this player's own last segment was traveling. And he draws his segment from where his own last line-segment ended to where his new line-segment comes up against another pre-existing segment or up against the circle's circumference. Again, his segment may pass through a pre-existing line-segment (but not pass through the circle's circumference), but doing so halves his score each time he does it.
After a predetermined number of full-moves (such as 10), each player's score = the length of that player's final line-segment divided by 2^(the number of lines crossed by that player).
Largest score wins.
Note: To be clear, there will be two "pathways" within the circle: One pathway belonging to each player, and each pathway made up of the series of connected line-segments drawn by that player.
Also, line-segments may not coincide, except at the points where they intersect.