A topological space $X$ is called (strong) Choquet if the player II has a winning strategy in the (strong) Choquet game.
It is known that a metrizable space $X$ is
$\bullet$ Choquet if and only if it contains a dense complete-metrizable subspace;
$\bullet$ strong Choquet if and only if it is complete-metrizable.
Since a topological group is complete-metrizable if and only if it contains a dense complete-metrizable subspace, we obtain the following characterization.
Theorem. A metrizable topological group is Choquet if and only if it is strong Choquet.
Corollary. A cosmic topological group is Choquet if and only if it is strong Choquet.
Let us recall that a regular topological space is cosmic if it has a countable network of the topology. It is easy to show that a topological group is second-countable if it is cosmic and Baire. That is why Corollary follows from Theorem.
Theorem and Corollary suggest the following intriguing
Problem. Is each Choquet topological group strong Choquet?