Rectangles Of Distinction

This is a game for any plural number of players.

First, draw a (2n)-by-(2n) array of dots (where the dots correspond to the vertices of a grid of (2n-1)-by-(2n-1) squares), where 2n is at least 6, I suggest.

Players take turns drawing a rectangle each move, each rectangle using 4 of the dots as corners. Each rectangle must have a unique non-zero area and have unique corners (see below).

The sides of the rectangles may overlap those of previously drawn rectangles, but no corner should be the corner of a previously drawn rectangle (drawn by any player).
After drawing a rectangle, mark the 4 dots which are its corners with x's so that it is known which dots have been used already.

Also, after drawing a rectangle, write down in a (growing) list the area of this triangle (the area in terms of the "squares" of the original array of dots).
No rectangle may have the same area as any previously drawn rectangle (drawn by any player).

The last player able to successfully draw a rectangle using 4 previously-unused corners and having a unique area is the winner.
In other words, the player wins who moved just before the first player who THINKS he or she cannot move.

Thanks,
Leroy Quet