There has been considerable effort in exploring completeness properties of the Wijsman topology tWd on the hyperspace CL(X) of a metrizable space (X,d); in particular, results of Beer and Costantini showed that Polishness of tWd is equivalent to Polishness of X. In this respect Beer asked, if complete metrizablity of X alone (without separability) is equivalent to some completeness property of the Wijsman hyperspace. It was Costantini who produced the first example of a complete metric space with a non-Čech-complete Wijsman hyperspace. More recently, Chaber and R. Pol have constructed a complete metric space with a non-hereditarily Baire Wijsman hyperspace, making all closed-hereditary completeness properties irrelevant for answering Beer's question.On the other side, I have shown that a completely metrizable X guarantees strong completeness-type properties on the hyperspace, namely (CL(X), tWd) is a-favorable in the strong Choquet game (which is equivalent to complete metrizability in a metrizable setting) and hence is a Baire space. It is the purpose of the talk to discuss the above and related results and show that strong a-favorability of (CL(X), tWd) is not equivalent to complete metrizability of X.