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.

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    $\begingroup$ If I were Alice and I were to go first, I would simply take the entire cake, thereby giving me a cube-shaped cake, 10cm each side, and leaving Bob without any. $\endgroup$ – JMoravitz Aug 3 '15 at 20:13
  • $\begingroup$ How is this game theory? They both play towards the same goal. It's like if we play chess in order to make let me win. $\endgroup$ – Asaf Karagila Aug 3 '15 at 20:15
  • $\begingroup$ @JMoravitz: Sorry, yes, I had thought of that and made up an additional condition. I then promptly (as is my wont) forgot about it. I'll add that now. $\endgroup$ – Brian Tung Aug 3 '15 at 20:16
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    $\begingroup$ I'm not sure I really understand the rules of the game you're proposing. On turn 1, Alice makes a linear cut to divide the cake into pieces $A$ and $B$, and then Bob chooses a subset of either $A$ or $B$ for himself. What happens on turn 2? Does Bob choose one of the remaining pieces and cut it again? Does Alice choose a subset of one of the remaining pieces? $\endgroup$ – Eric Wofsey Aug 3 '15 at 20:23
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    $\begingroup$ So Alice cuts a line on the cake, and Bob takes a piece? How can you hope, with such Borel pieces to reconstruct the Banach-Tarski Banach-Tarski theorem? It's impossible. $\endgroup$ – Asaf Karagila Aug 3 '15 at 20:35

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