In this note, we consider the on-line Ramsey numbers \(\overline{R}(P_n, P_m)\) for paths. Using a high-performance computing cluster, we calculated the values for off-diagonal numbers for paths of lengths at most \(8\). Also, we were able to check that \(\overline{R}(P_9, P_9) = 17\), thus solving the problem raised in [5].