Also, I know which player in the integer-choosing game (if any) has a winning strategy, and can prove it. It's kind of interesting.
Anyone else up for the challenge?
First, two remarks. (1) It's a bit weird to have the winner be the first player with no legal move; the usual convention in combinatorial game theory is the reverse. (2) I concede that "parity of sum of coefficients in Cantor normal form" is a *simple* rule but have trouble seeing why I should consider it *natural*. [Next two paragraphs serve mostly as filler to enable people who don't want to read the answer to avoid doing so.] Now. It looks at first as if there should be a strategy-stealing argument showing that the game can't be a second-player win: if second player wins then first player plays 0 and then copies second player's strategy but with all numbers increased by 1 and wins, contradiction. But once we reach turn w the players' roles are no longer swapped so this doesn't work. It also looks at first as if there should be a simple strategy enabling the second player to win with the normal play convention (i.e., reverse of what's stipulated here): always play first player's move xor 1. But again this doesn't work with the CNF rule at limit ordinals because at w the second player has to play without an immediately preceding move to look at. [OK, now here's the actual answer.] However, it looks as if the following works with the misere convention Dan stipulated: the second player always plays the smallest available number. Because then at w there are no numbers left. (Proof: For n finite, after n full turns all numbers <n are taken, by induction on n; hence at w all numbers are taken.) And at w it's the second player's turn, boom. With the normal play convention, the *first* player adopts that strategy instead and wins. (Lightly unified version of the discussion above: either player can arrange that the game ends at turn w with no numbers left to play; the game cannot end before turn w; therefore, the winner is whoever wins when the game ends at turn w.) -- g