TITLE: absorption of strategically closed posets
QUESTION [7 upvotes]: It is a (folklore?) fact that if $\kappa$ is a regular cardinal, and $\mathbb{P}$ is a $\kappa$-closed poset such that $\Vdash_\mathbb{P} |\mathbb{P}| = \kappa$, then $\mathbb{P}$ is equivalent to $Col(\kappa,\mathbb{P})$, the collection of partial functions from $\kappa$ to $\mathbb{P}$ of size $<\kappa$, ordered by inclusion.  Hence if $\mathbb{P}$ is any $\kappa$-closed forcing, then $\mathbb{P} \times Col(\kappa,\mathbb{P}) \sim Col(\kappa,\mathbb{P})$, so $\mathbb{P}$ is completely embeddable into $\mathcal{B}(Col(\kappa,\mathbb{P}))$.
If $\mathbb{P}$ is $\kappa$-strategically closed (following the notation from Cummings' chapter of the Handbook of Set Theory, not Jech's book), then one can show that $\mathbb{P}$ is completely embeddable into $\mathcal{B}(Col(\kappa,\mathcal{P}(\mathbb{P})))$, since after forcing with this collapse we can use the strategy to build a filter on $\mathbb{P}$ that is generic over the ground model.  It is not clear to me whether we can always embed such $\mathbb{P}$ into the smaller forcing $Col(\kappa,\mathbb{P})$.
Question:  Are there counterexamples to the statement, "If $\kappa$ is regular and $\mathbb{P}$ is $\kappa$-strategically closed, then there is a complete embedding of $\mathbb{P}$ into $\mathcal{B}(Col(\kappa,\mathbb{P}))$?

REPLY [5 votes]: Meanwhile, let's observe that if you weaken strategic $\kappa$-closure to the requirement that the forcing is strategically ${\lt}\kappa$-closed, that is, strategically $\beta$-closed for all $\beta<\kappa$, then the conclusion does not necessarily hold. The difference is between player II being able to continue playing for $\kappa$ many moves in a single game, versus being able to play for $\beta$ many moves in the length $\beta$ games for arbitrarily large $\beta$ below $\kappa$. 
Suppose $\kappa$ is a Mahlo cardinal, and let $\mathbb{P}$ be the forcing to kill the Mahloness of $\kappa$. Conditions in $\mathbb{P}$ consist of closed bounded subsets of $\kappa$, containing no regular cardinals, ordered by end-extension. This forcing has size $\kappa$, and it adds a club in $\kappa$ containing no regular cardinals. It is strategically $\lt\kappa$-closed, since for any $\beta<\kappa$, the collection of conditions mentioning an ordinal above $\beta$ is $\leq\beta$-closed and dense in $\mathbb{P}$. So player II can win the games of length $\beta+1$ simply by playing into that dense set at the first opportunity. 
Meanwhile, $\mathbb{P}$ does not completely embed into $\text{Coll}(\kappa,\kappa)$, because that forcing is ${\lt}\kappa$ closed and hence preserves every ground model stationary subset of $\kappa$ (one can see this by a pseudo-generic argument, since for any name for a club avoiding a given set, one can produce a club in the ground model avoiding that set), while $\mathbb{P}$ destroys the stationarity of the set of regular cardinals below $\kappa$. So the collapse forcing cannot add a generic filter for $\mathbb{P}$.