Given \(A\subseteq \mathcal{N}\), the game associated to \(A\), denoted by \({\mathcal G}_A\), has two players, I and II, who play alternatively \(n_i\in \mathbb{N}\): I plays \(n_0\), then II plays \(n_1\), then I plays \(n_2\), and so on. So, at stage \(2k\), player I plays \(n_{2k}\) and at stage \(2k+1\), player II plays \(n_{2k+1} \). We may visualize a run of the game as follows: