from functools import cache
def permutationGame(arr):
    def is_increasing(remain):
        return all(remain[i] < remain[i+1] for i in range(len(remain)-1))
    def new_perm(remain, removed):
        return tuple(e - 1 if e > removed else e for e in remain if e != removed)
    @cache
    def first_wins(remain):
        if is_increasing(remain):
            return False
        return any(not first_wins(new_perm(remain, e)) for e in remain)
    return 'Alice' if first_wins(tuple(arr)) else 'Bob'