I was reading a cake-cutting problem here (not really related, so I won't link to it), and for some reason, this variation occurred to me. I have no idea whether this problem is even well-formed:
Alice and Bob have bought a cube-shaped cake, $10$ cm on a side. Alice makes one ordinary linear cut dividing the cake into two ordinary parts (she does not separate the parts, however), and Bob selects a piece* that is a subset (not necessarily proper) of one of the two parts. Continuing with Alice, they then take turns selecting pieces from the whole cake until the cake is entirely divvied up.
Each person's objective is to obtain pieces sufficient to be reassembled into a cube-shaped cake, $10$ cm on a side. Can either person "win"? Can both? Note that I do not require the actual algorithm either person would follow (though I would be happy with one); I merely want to know if such an algorithm must exist.
*Assume that by "piece," we mean the usual definition of piece with respect to the Banach-Tarski paradox.