I think it starts like this:

1) at most nb of stone increase by one each time one player plays

2) nb of stone cannot decrease unless the board is full of stones

3) no one can lose unless the board has been full of stone at least once; this is because until the board has been full of stones it is possible to increase the nb of stone by adding one and this is a new position because of 2)

4) as soon as there are n - 1 stones, the player who puts the last stone wins; because there are n - 1 different positions left with n - 1 stones and n - 1 is even