[[Ishiu, Yoshinobu - Directive trees and games on posets#Startegic closure]] **Definition.** For a poset $\mathbb{P}$ and an ordinal $α > ω$, we define two-player games: - $G_{\alpha}^{I}(\mathbb{P})$: Players take turns choosing a descending sequence of conditions. **Player I** chooses limits if there are any. - $G_{\alpha}^{II}(\mathbb{P})$: Players take turns choosing a descending sequence of conditions. Player II chooses limits if there are any. - $G_{\alpha}^{V}(\mathbb{P})$: Player I plays all successor stages, player II all limit stages Player II wins if she could move α times, otherwise player I wins. $\mathbb{P}$ is *$\alpha$-startegically closed* if player II has a winning strategy in $G_{\alpha}^{II}(\mathbb{P})$ $\mathbb{P}$ is *strongly $\alpha$-startegically closed* if player II has a winning strategy in $G_{\alpha}^{I}(\mathbb{P})$ # Properties ![[Ishiu, Yoshinobu - Directive trees and games on posets#^6bac72]] - Preserving stationary subsets of $\mathcal{P}_{\omega _{1}}(X)$. # Relations ## Implied by - [[Directed closed]] - [[Closed forcing]] ## Implies - [[Distributivity]] - [[Cardinal preservation]] up to $\kappa$.