set theory - Determinacy interchanging the roles of both players

Let me refer to Jech's "Set Theory" Chap. 33 Determinacy:

"With each subset A of $omega^omega$ we associate the following game $G_A$, played by two players I and II. First I chooses a natural number $a_0$, then II chooses a natural number $b_0$, then I chooses a1, then II chooses b1, and so on. The game ends after $omega$ steps; if the resulting sequence $<a_0, b_0, a_1, b_1, ...>$ is in A, then I wins, otherwise II wins.

A strategy (for I or II) is a rule that tells the player what move to make depending on the previous moves of both players. A strategy is a winning strategy if the player who follows it always wins. The game $G_A$ is determined if one of the players has a winning strategy.

The Axiom of Determinacy (AD) states that for every subset A of $omega^omega$, the game $G_A$ is determined."

Now, there is some apparent lack of symmetry in the definition of the $G_A$ game: the player who plays first (I) attempts for a sequence in A.

What happens if we interchange the roles of both players? I. e. if we let the player who plays first attempt for a sequence not in A? Let us call this game $G'_A$

Is it the case that for every subset A, A is determined wrt $G_A$ iff A is determined wrt $G'_A$?