Is $\kappa$-distributive the same as $\kappa$-strategically closed?

Question

References for the definitions are Jech's Set Theory Definition 15.5, and Cummings paper in the Handbook of set theory Definition 5.15.

Answer 1

Forcing with a Souslin tree is $\omega$-distributive, but it is not countably strategically closed. No $\omega_1$-tree can be countably strategically closed, since if it were, we could construct a subtree of order type $2^{\lt\omega}$, which had all limits. This would give rise to an uncountable level in the tree, contrary to the assumption that it was an $\omega_1$-tree.

Another example (in ZFC rather than merely relatively consistent) would be the forcing to shoot a club through a stationary co-stationary subset $S\subset \omega_1$, with conditions consisting of closed bounded subsets of $S$. This is $\omega$-distributive when $S$ is stationary, by a reflection argument reflecting $\omega_1$ to a point in $S$, but it is not countably strategically closed when $S$ is co-stationary, since there would have to be ordinals not in $S$ which were in a sense closed under the strategy (go to a countable elementary substructure), and one could find a play according to the strategy which would force this ordinal to appear in the club, even though it is not in $S$.