We will also show that it is consistent with ZFC that for all infinite cardinals $\kappa$, for each infinite regular carinal $\nu \le \kappa$ there is a $\kappa^+$-Suslin algebra containing a $<\nu$-closed dense subset in which many games of length $\ge \nu$ are all undetermined. To do this, we use $\square_{\kappa}$ and $\diamond_{\kappa^+}(S)$ for all stationary sets $S$ contained in $\kappa^+$. This improves on an earlier result of Dobrinen (03) which showed that for regular cardinals $\kappa$, there is consistently a large gap between the strengths of "$B$ satisfies the $(\kappa,\infty)$-d.l." and "player II has a winning strategy in the game $\mathcal{G}^{\kappa}_1(\infty)$.