The game \({\mathcal{G}}_A\) is determined if there is a winning strategy for one of the players. A winning strategy for one of the players, say for player II, is a function \(\sigma\) from the set of finite sequences of natural numbers into \(\mathbb{N}\), such that if the player plays according to this function, i.e., she plays \(\sigma (n_0,\ldots ,n_{2k})\) at the \(k\)-th turn, she will always win the game, no matter what the other player does.