Brazilian Mathematical Olympiad

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?