XX OBM

Two players play a game as follows. There are $n>1$ rounds and $d\ge 1$ is fixed. In the first round A picks a positive integer $m_1$, then B picks a positive integer $n_1\ne m_1$. In round $k$ (for $k=2,\dots ,n$), A picks an integer $m_k$ such that $m_{k-1}<m_k\le m_{k-1}+d$. Then B picks an integer $n_k$ such that $n_{k-1}<n_k\le n_{k-1}+d$. A gets $\gcd (m_k,n_{k-1})$ points and B gets $\gcd (m_k,n_k)$ points. After $n$ rounds, A wins if he has at least as many points as B, otherwise he loses. For each $n,d$ which player has a winning strategy?