"Maze" Of Polygonal Sections, Game

This game has elements in common with an earlier game of mine, Slice And Fill.
See:
http://gamesconceived.blogspot.com/2008/09/slice-and-fill.html

This game will work with any number of players.

Start with an n-by-n grid lightly and carefully drawn on paper.

Darken in the 4 grid-lines that form the square boundary of the grid. (All of the vertexes along the grid's edge are thereafter each considered to be drawn-to by a line-segment.)

Players take turns drawing straight line-segments, one segment per move, each segment drawn from any vertex of the grid that has a line segment passing through it or terminating at it, to any vertex that touches no line-segments, such that the line segments don't cross any others or coincide with any others.
(Any number of segments may be drawn FROM any single vertex. Line-segments may be diagonal and of any slope. Each line-segment may pass through any number of vertices. But, I repeat, line segments must each be drawn from a vertex of the grid to another vertex of the grid, not from an intersection of a line-segment and a grid-line if that intersection is not a vertex of the grid.)
The first line-segment (after the perimeter of the grid is filled in) is drawn from a vertex along the edge of the grid, of course.

When all vertexes of the grid are touching line-segments, we have a maze (without an entrance or exit), and then the next phase of the game begins.

One player starts the second phase by filling in any "section" of the subdivided grid. A section is defined by* the lines of the grid and/or by straight line-segments drawn by players (a section may be a square, or it may be a polygon which is a subset of a square).
*[By "where the section is defined by...", I mean "where the section is BORDERED by" the lines of the grid and/or by straight line-segments drawn by players. There are no internal line-segments within any given "section".]

Then the players take turns filling in, if possible, any UNFILLED section that is immediately adjacent to the previously filled in section (filled in by another player) and that is not separated from the previously filled in section by a line-segment drawn in the earlier phase of the game. (So, consecutively filled sections must not only be adjacent, but must be in the same "corridor" of the maze.)
If a section can be filled in under the rules, then a section must be filled in.

If, however, a section cannot be filled in by a player (either because it is surrounded by already filled in sections, or it is at one of the maze's dead-ends), then the previous player to move gets a point. The player who cannot fill in a section under the rules above then fills in any unfilled section (so as to start a new string of filled in sections).

The game continues until all sections are filled in.

The player with the greatest number of points wins.

Thanks,
Leroy Quet