Suppose that $G( \omega, A, X)$ denotes a sequential game of perfect information in which player I and player II play an element in $A$ in each turn with a total number of $\omega$ moves. $X$, the winning set, is a subset of $A ^{\omega}$. If the outcome is in $X$, then player I wins. Otherwise, player II wins.
$G(\omega, A, X_1)$ and $G(\omega, A, X_2)$ are two games in which player I has a winning strategy. Clearly in the game $G(\omega, A, X_1\cup X_2)$, player I also has a winning strategy. Let $W_{X_1}$, $W_{X_2}$ and $W_{X_1 \cup X_2}$ be the sets of player I's winning strategies respectively. I'm interested to know their relationship given their non-emptiness in general.
In particular, is it true that $W_{X_1} \cap W_{X_2} =\varnothing$, then $W_{X_1 \cup X_2} = W_{X_1} \cup W_{X_2}$?