I wouldn't expect that the addition of noise would EVER completely
eliminate the first-turn advantage, just make it less significant.
Assuming that players strictly alternate full turns, and that nothing other
than the positions of pieces affects the game (e.g. there's no time
limit), and general chesslike properties such as perfect information, the
only way I can see to have NO first-turn advantage is if the first player
has literally nothing useful to do with his turn. No possible piece
development, no moving forward to claim extra territory, no starting to
launch an attack or race for a promotion, NOTHING. And I question whether
that would even be desirable.
