Four games on Boolean algebras
The games G2 and G3 are played on a complete Boolean algebra B in ?-many moves. At the beginning White picks a non-zero element p of B and, in the n-th move, White picks a positive pn < p and Black chooses an in ? {0,1}. White wins G2 iff lim inf pin,n = 0 and wins G3 iff W A?[?]? ? n?A pin,n = 0. It is shown that White has a winning strategy in the game G2 iff White has a winning strategy in the cut-and-choose game Gc&c introduced by Jech. Also, White has a winning strategy in the game G3 iff forcing by B produces a subset R of the tree <?2 containing either ??0 or ??1, for each ? ? <?2, and having unsupported intersection with each branch of the tree <?2 belonging to V. On the other hand, if forcing by B produces independent (splitting) reals then White has a winning strategy in the game G3 played on B. It is shown that ? implies the existence of an algebra on which these games are undetermined.