The completeness game $G\_{\gamma}(P)$ for a partial order $P$ has players COM and INC play alternatingly and descendingly elements of $P$ with player INC playing first and player COM playing at limit steps. INC wins the game if at some point $\alpha<\gamma$ (necessarily a limit ordinal) there are no legal moves left. Otherwise, COM wins.


By a theorem of Foreman, for a successor cardinal $\kappa=\lambda^+$, $P$ is $<\kappa$-distributive (i.e. every sequence of $<\kappa$ ordinals is in $V$) iff INC does not have a winning strategy in $G\_{\lambda+1}(P)$.


Let $\kappa=\lambda^+$ be a successor cardinal.


* Assume INC has a winning strategy in $G\_{\kappa}(P)$. Does INC have a winning strategy in $G\_{\lambda+1}(P)$? I.e. if INC always wins eventually, can they uniformly bound the time it takes for them to win?
* If $P$ is $<\kappa$-closed (every sequence of $<\kappa$ conditions in $P$ has a lower bound) and $Q$ is $\kappa$-cc., $P$ is $<\kappa$-distributive in $V[G]$, where $G$ is $Q$-generic. Can INC have a winning strategy in $G\_{\kappa}(P)$ in $V[G]$? What if $Q$ is $\kappa$-Knaster?




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**Edit:** The answer to the first question is "no". If $(T,\leq)$ is an $\omega\_1$-Suslin-tree, $(T,\geq)$ is $<\omega\_1$-distributive (see e.g. Lemma 15.28 in Jech, noting the different definition of distributivity). However, INC has an easy winning strategy in $G\_{\omega\_1}((T,\leq))$, by simply playing any node strictly above COMs last play, since a play in which COM wins corresponds to a cofinal chain in $(T,\leq)$.


It still remains if such an object can be added by a tame forcing notion.



Are the completeness Games $G_{\lambda+1}(P)$ and $G_{\lambda^+}(P)$ equivalent for INC?
The answer to the second question is no as well.


Suppose $\dot\tau$ is a $Q$-name for a strategy for the player INC in the game $G\_\kappa(P)$. Let us pretend that COM opens the game instead of INC and note that this is unproblematic. We will define by induction a descending sequence $\langle p\_\alpha\mid\alpha<\kappa\rangle$ of conditions in $P$ and $Q$-names $\langle \dot p\_\alpha\mid\alpha<\kappa\rangle$ so that


* $\Vdash\_Q\check p\_\beta\leq \dot p\_\alpha\leq\check p\_\alpha$ for all $\beta\leq\alpha$
* $\Vdash\_Q$"The sequence $\langle \dot p\_\beta\mid\beta\leq\alpha\rangle$ make up the fist $\alpha+1$-many moves of player COM in a game of $G\_{\check\kappa}(\check P)$ in which INC plays according to $\dot\tau$ and the final response of $\dot \tau$ to this is a condition $\leq\check p\_\alpha$."


for all $\alpha<\kappa$. It follows that if $G$ is $Q$-generic then COM wins against $\dot\tau^G$ by playing the conditions $\langle \dot p\_\alpha^G\mid\alpha<\kappa\rangle$, so $\dot\tau$ is not a winning strategy for INC.


Let us turn to the construction. Assume $\alpha<\kappa$ and that $\dot p\_\beta, p\_\beta$ are defined for all $\beta<\alpha$. Let $s\_0$ be some lower bound of $\langle p\_\beta\mid\beta<\alpha\rangle$ which exists as $P$ is ${<}\kappa$-closed. Build descending sequences of maximal possible length
$$\langle s\_\gamma\mid\gamma\leq\xi\rangle,\ \langle t\_\gamma\mid\gamma<\xi\rangle$$
and an antichain $A\_\xi:=\{q\_\gamma\mid \gamma<\xi\}\subseteq Q$ so that


* $s\_\gamma\leq t\_\gamma\leq s\_{\gamma+1}$ for all $\gamma<\xi$ and
* $q\_\gamma\Vdash\_Q$"if COM plays $\langle \dot p\_\beta\mid\beta<\alpha\rangle^\frown\check s\_\gamma$ and INC follows $\dot\tau$, INC's final play is $t\_\gamma$".


The construction is straightforward and can only break down for two reasons: Either there is no lower bound of $\langle t\_\gamma\mid\gamma<\xi\rangle$ or $A\_\xi$ is a maximal antichain in $Q$. The latter must happen first since $P$ is ${<}\kappa$-closed and $Q$ is $\kappa$-cc. We set $p\_\alpha$ to be the final $s\_\xi$ and by mixing of names, we can find a $Q$-name $\dot p\_\alpha$ so that
$$q\_\gamma\Vdash\_Q\dot p\_\alpha=\check s\_\gamma$$
for all $\gamma<\xi$. This completes the construction.


Finally, let me remark that this is sensitive to the order of play in $G\_\kappa(P)$: If $G\_\kappa'(P)$ is the game where INC plays at limit steps instead of COM then it is possible that INC has a winning strategy for $G\_\kappa'(P)$ in $V^Q$:


Let $Q$ be the poset of finite partial functions $q:\omega\_1\rightarrow 2$ ordered by inclusion and let $P=\mathrm{Add}(\omega\_1, 1)$. $Q$ is ccc and $P$ is ${<}\omega\_1$-closed. Let $G$ be $Q$-generic and consider the strategy for INC for the game $G\_\kappa'(P)$ where INC adds a bit of information which agrees with the generic function $\bigcup G$ at the least ordinal which is not yet in the domain of any condition played so far. Assume toward a contradiction that some game in which INC follows this strategy lasts for $\omega\_1$-many rounds. The result is a function $f\colon\omega\_1\rightarrow 2$ which agrees with $\bigcup G$ on a club $C$. Now $Q$ is ccc, so there is a club $C'\subseteq C$ with $C'\in V$. It follows that $f\notin V$, but we must have $f\upharpoonright\alpha\in V$ for all $\alpha<\omega\_1$. This is impossible by a result of Spencer Unger: As $Q\times Q$ is ccc, $Q$ has the $\omega\_1$-approximation property, see Lemma 1.2 in [Fragility and indestructibility II](https://www.math.toronto.edu/sunger/Indestructible2.pdf), Annals of Pure and Applied Logic 166 (2015) 1110-1122.


