A lower bound of $\Omega(\sqrt{\log k / \log \log k})$ is proved for the competitive ratio of randomized algorithms for the k-server problem against an oblivious adversary. The bound holds for arbitrary metric spaces (having at least k+1 points) and provides a new lower bound for the metrical task system problem as well. This improves the previous best lower bound of $\Omega(\log \log k)$ for arbitrary metric spaces [H. J. Karloff, Y. Rabani, and Y. Ravid, SIAM J. Comput., 23 (1994), pp. 293--312] and more closely approaches the conjectured lower bound of $\Omega(\log k)$. For the server problem on k+1 equally spaced points on a line, which corresponds to a natural motion-planning problem, a lower bound of $\Omega(\frac{\log k}{\log \log k})$ is obtained. The results are deduced from a general decomposition theorem for a simpler version of both the k-server and the metrical task system problems, called the pursuit-evasion game. It is shown that if a metric space $\cal M$ can be decomposed into two spaces $\cal M_L$ and $\cal M_R$ such that the distance between them is sufficiently large compared to their diameter, then the competitive ratio for this game on $\cal M$ can be expressed nearly exactly in terms of the ratios on each of the two subspaces. This yields a divide-and-conquer approach to bounding the competitive ratio of a space.