Label the numbers $ (1), \ldots, (m)$. If $ m$ is even, then player B pairs up the labels $ (1,2),(3,4) \ldots$, and mirrors each action that player A does on the other number of the pair. Then B wins. If $ m$ is odd and $ n$ is odd, then player A pairs up $ (2,3),(4,5) \ldots$, and subtracts one from $ (1)$. If B plays on any of the numbers greater than the value of $ (1)$, than A mirrors the action on the paired number. If B subtracts one from a number equal to the value of $ (1)$ (or $ (1)$ itself), then A again subracts one from that number, and swaps that number's label with $ (1)$. At any given move before player B's turn then there will be an odd number of smallest numbers (equal in value with $ (1)$), and a symmetric position left. Then A wins. If $ m$ is odd and $ n$ is even, then player B uses A's strategy from above.