This question is about the argument for Lemma 3.7 in [Forcing axioms and stationary sets](https://doi.org/10.1016/0001-8708(92)90038-M) ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=MR1174395)) by Boban Veličković.


He defines a game $G\_\alpha$ between two players, playing objects in $H\_\kappa$, depending on a function $F : [\kappa]^{<\omega} \to \kappa$. At stage $n$, Player I plays an interval of ordinals $I\_n$ below $\kappa$ and an ordinal $\xi\_n \in I\_n$. Player II plays an ordinal $\mu\_n$. $\min(I\_{n+1})$ must be above $\mu\_n$. Player I wins if the closure $M$ of $\{ \xi\_n : n < \omega \} \cup \alpha$ under $F$ is contained in $\bigcup\_n I\_n$, and $M \cap \omega\_1 = \alpha$.


Now I understand the argument for why player I has a winning strategy when we ignore the requirement on $\alpha$. For the $\alpha$-indexed games, he says: Let $A\_F$ be the set of $\alpha<\omega\_1$ such that player II has a winning strategy $\sigma\_\alpha$. "Since $A\_F$ has cardinality $\leq \aleph\_1$, there is a strategy $\sigma$ which dominates all the $\sigma\_\alpha$. It follows that $\sigma$ is a winning strategy for II in $G\_\alpha$ for every $\alpha \in A\_F$."


What does it mean to say that a strategy "dominates" a lot of other ones (for the same player), and why does such $\sigma$ exist?



Veličković's model game
What constitutes a partial play doesn't depend on $\alpha$. And $\kappa$ is assumed to have cofinality strictly larger than $\aleph\_1$. So he means: define $\sigma$ applied to a partial play to just be some ordinal below $\kappa$ that is above all the outputs of the $\aleph\_1$-many strategies $\sigma\_\alpha$.


(And it is easy to see that for any fixed $\alpha$, if $\sigma\_\alpha$ is a winning strategy for II in the game $G\_\alpha$, then so is any function from partial runs into $\kappa$ that is $\ge \sigma\_\alpha$.)


