The games $\mathcal G_2$ and $\mathcal G_3$ are played on a complete Boolean algebra $\mathbb B$ in $\omega$-many moves. At the beginning White picks a non-zero element $p$ of $\mathbb B$ and, in the $n$-th move, White picks a positive $p_n<p$ and Black chooses an in $i_n\in\{0,1\}$. White wins $\mathcal G_2\operatorname{iff}\lim\inf p^{i_n}_n=0$ and wins $\mathcal G_3\operatorname{iff}\bigvee_{a\in[\omega]^\omega}\bigwedge_{n\in A}p^{i_n}_n=0$. It is shown that White has a winning strategy in the game $\mathcal G_2\operatorname{iff}$ White has a winning strategy in the cut-and-choose game $\mathcal G_{c\&c}$ introduced by Jech. Also, White has a winning strategy in the game $\mathcal G_3\operatorname{iff}$ forcing by $\mathbb B$ produces a subset $R$ of the tree ${}^{<\omega}2$ containing either $\varphi^\frown0$ or $\varphi^\frown1$, for each $\varphi\in{}^{<\omega}2$, and having unsupported intersection with each branch of the tree ${}^{<\omega}2$ belonging to $V$. On the other hand, if forcing by $\mathbb B$ produces independent (splitting) reals then White has a winning strategy in the game $\mathcal G_3$ played on $\mathbb B$. It is shown that $\lozenge$ implies the existence of an algebra on which these games are undetermined.