Needed: blank paper, pencil and compass (the circle-drawing kind), maybe a protractor.
First, draw a circle on a piece of paper, relatively large.
Let m be a multiple of the number of players playing this game. m should be >= 8, at least, I suggest.
Draw m pencil-marks EVENLY SPACED along the circumference of the circle. (This is why you might need a protractor, if you can't just use the compass and straightedge to accomplish this by geometric construction.)
There will be m moves in the game (or in the round).
Players alternately take turns drawing arcs, drawing one arc with the compass on each move. Each arc must be drawn within the circle's interior from the circle's edge back to the circle's edge. On a move, a player uses the appropriate mark as the center of the circular arc that passes through two other marks. The two marks (which the arc intersects the main circle at) must be equidistant from the arc's center mark, and that distant must be nonzero.
The first player starts at any mark to make the center of his arc; and on each move after the first, the moving player uses for her arc's center the mark immediately clockwise from the mark used for the previous player's arc-center.
Count the number of preexisting arcs (not including the "arc" of the main circle) that the moving player's arc intersects or touches. Each player has a running-total of the number of arcs his/her arcs passed through.
The winner has the FEWEST total number of earlier-drawn arcs (drawn by any player) intersected or touched by their arcs.
Notes: An arc may touch/intersect other single arcs more than once each, but each such incident only counts once towards the number of arcs intersected or touched.
Also, multiple arcs touching at a point are counted as touching all other arcs coming together at that point, no matter which arcs are "behind" or "ahead of" which other arcs.
There seems to be an advantage for who moves first, and a disadvantage for who moves last. So, maybe multiple rounds should be played, a different round where each player is the first player to move. (Draw a new circle with the same number of marks each round.) Then the total scores for the rounds are added up for the grand score for each player. Players try to minimize their grand scores, of course.