  There exists a completely metrizable bounded metrizable space $X$ with
compatible metrics $d,d'$ so that the hyperspace $CL(X)$ of nonempty closed
subsets of $X$ endowed with the Hausdorff metric $H_d$, $H_{d'}$, resp. is
$\alpha$-favorable, $\beta$-favorable, resp. in the strong Choquet game. In
particular, there exists a completely metrizable bounded metric space $(X,d)$
such that $(CL(X),H_d)$ is not completely metrizable.

General Topology