Please excuse my poor drawing skills...
Essentially is this what you mean?
game_moves_1
-----------------------------------------------------
|..move_id....|....r937....|....blindman....|.... game_id...|
-----------------------------------------------------
|.......1........|...a2a4...|.......c7c5.......|.. ......1........|
-----------------------------------------------------
|.......2........|...a4a5...|.......h7h5.......|.. ......1........|
-----------------------------------------------------
|.......3........|...d2d3...|.......h5h4.......|.. ......1........|
-----------------------------------------------------
And for each new game between two users, a new table would be made? (i.e. game_moves_2, game_moves_3, ...)