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David Gale's subset take-away game is a game where two players A and B play with a finite set $S$. Players alternately choose proper nonempty subsets of $S$ such that if a subset is chosen, then none of its subsets may be chosen after that. Gale conjectured that the second player always wins; this has been checked for $|S|\leq 6$.

Make the following modification to the game: specify a topology $\tau$ on $S$. Two players A and B play with $(S,\tau)$. Players alternately choose proper nonempty open subsets of $S$ such that if an open subset is chosen, then none of its subsets may be chosen after that.

Question: For what nontrivial topologies on $S$ does the second player always win?

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  • $\begingroup$ I don't see why, in Gale's original game, the same player wins for both $|S|=1$ and $|S|=2$. $\endgroup$
    – S. Carnahan ♦
    Commented Aug 12, 2015 at 7:24
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    $\begingroup$ This is in turn just a special case of en.wikipedia.org/wiki/Poset_game. Apparently determining the winner of such a game is PSPACE-complete, so there seems to be no reason to expect a simple answer. $\endgroup$
    – Qiaochu Yuan
    Commented Aug 12, 2015 at 7:52
  • $\begingroup$ @QiaochuYuan I agree that for this game there is no obvious reason to expect a simple answer, but there are many classes of poset game that do admit simple descriptions of the outcome classes. $\endgroup$
    – Gabriel C. Drummond-Cole
    Commented Aug 12, 2015 at 7:57
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    $\begingroup$ I think everybody will realize, but it might be a good idea to say explicitly that the loser is the first player without a legal move. $\endgroup$
    – Jeremy Rickard
    Commented Aug 12, 2015 at 18:16

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