Permutation Sorting Game

Alice and Bob have a permutation of size $N$, so they started playing a game on it. **They both have their own copies** of an initial permutation on which they perform moves in turns. Note that although Bob's and Alice's permutations are initially similar in values, **they are the not the same permutation**; any move Alice makes only changes her permutation and not Bob's, and vice versa. Alice go