TITLE: Boolean Prime Ideal Theorem versus the Axiom of Determinacy
QUESTION [7 upvotes]: I'm assuming someone must have scooped me on this simple argument.  Where does it (first) appear in the literature?
Fix an ultrafilter $\mu$ on $\omega$, the natural numbers. 
Alice and Bob play a nim-like game.  At the start each player "holds" the empty set and the
the starting "position" consists of $\omega$.  Beginning with Alice, each player in turn will remove a non-empty finite initial segment from the current position (leaving some final segment of $\omega$) and deposit the removed segment into his or her holdings.  Play proceeds for $\omega$ rounds.
Now the object of the game: to finish with holdings that belong to the ultrafilter $\mu$.
Strategy stealing obviates the possibility of either player possessing a winning strategy; the existence of $\mu$ thus contradicts AD.  In more detail, if either player has a winning strategy, the game must admit infinitely many winning positions, but that would allow the other player to possibility of moving to a winning position on his or her very first move.

Many sources work much harder than this to prove the weaker results that AC contradicts
AD.  (Afterthought: I'd love to see a big-list question collecting theorems where unnecessarily complicated proofs permeate the literature despite the availability of simpler treatments...MO appropriate?)

REPLY [2 votes]: I "discovered" this argument while I was a grad student (probably in 1969), but, alas, it was already known to Gene Kleinberg (if I remember correctly).  My first guess for the original discoverer would be Fred Galvin; my second guess would be folklore.