(I have posted other games before where you draw lines, then shade in sections bordered by the lines. But I can't think of a better name for this game.)
This game is for any plural number of players.
Start with an n-by-n grid lightly drawn on paper.
Players take turns drawing horizontal and vertical line segments, each segment being one grid-square side in length, from grid-vertex to adjacent vertex along the lightly drawn lines of the grid.
Player 1 starts the game by drawing a line segment from any vertex to adjacent vertex. Players each, thereafter, draw a line segment from where the last line segment left off. Darker line segments must not be drawn where other darker line segments were previously drawn. And the continuous path of line-segments must not cross itself. Yet, the path may be drawn to any vertex more than once.
This part of the game continues until the path cannot be drawn anymore.
(If all players want an interesting game, they should probably try not to lead the path into a situation where it prematurely ends.)
After the darker path of line-segments is complete, the players then take turns filling in un-shaded-in sections of the grid, one section per move. By "section", I mean a polygon bounded by the darker line-segments, or by vertexes where different parts of the path come together, or by the perimeter of the grid. (You can draw darker line segments along the border of the grid. But as far as the sections and the border of the grid are concerned, whether a particular segment of the border of the grid was darkened in or not does not matter.)
As soon as a player is forced to fill in a section bordering (along a line) another filled in section, or accidently does so, then that player is removed from play.
(Two filled in sections may border at a vertex without removing a player.)
Play continues until there is one player left, who then is the winner.
By the way, I suggest that when two parts of the path come together at a single vertex, then the path should be drawn so it is clear that there is no gap between the parts of the path. Otherwise, players may think that two of the sections that meet at that vertex are only one section (with a choke-point).
PS: Don't get confused between "sections" and "segments".