Round 1: Defunct Magazine Smackdown
Here we had Baen's Galaxy: The Best of My Years vs. Cholfin's The Best of Crank! Looking at the polls, slightly more people thought I'd like the Cholfin (9 to 7).The Baen here suffers from not having enough stories (only seven!) and a preoccupation with the relativistic effects of spaceships traveling under constant one-g accelerations. Also, problematic depictions of women, which I would have said almost goes without saying if I hadn't later read the two Women of Wonder anthologies ... Nontheless, on the upside, I like pretty much all of the stories.
The Crank! collection had some stories I disliked, but there were more stories and more variety. Also, it contained the single best piece in any of the volumes I read: Ursula K. Le Guin's The Matter of Seggri, which I see has three pluses and an exclamation mark next to it on my page of notes. (The Baen also has a Le Guin story, The Day Before The Revolution, but I didn't like it as much.)
Winner: The Best of Crank!
Round 2: Sargent Battles Herself
Here I pitted two anthologies edited by Pamela Sargent: Women of Wonder: The Classic Years vs. Women of Wonder: The Contemporary Years. Opinion on which I'd prefer was split right down the middle, with seven votes each.
As 
mmcirvin observed, the covers of the first book has a pretty pulp-era sci-fi depiction of scantily-clad women wearing strange helmets, which seems like kind of a disservice to the content. The second volume's cover features a female cyborg and is also kind of questionable.
No complaints about the contents, though. There's a variety of stuff here in both volumes, which keeps things interesting. Volume 1 ends with Le Guin's The Day Before The Revolution, which made me wonder if there would be a story by her in the other anthologies until I remembered that she isn't Japanese.
Volume 1 ultimately had more stuff that I outright enjoyed, while looking at my notes I see a lot of question marks next to stories in volume 2. (I see that the story about Hitler's wife helping a Gypsy woman fleeing from the United States has a big question mark next to it, for instance.) So:
Winner: Women of Wonder: The Classic Years
Round 3: The Odd Couple
Here we've got Apostoluo and Greenberg's The Best Japanese Science Fiction Stories vs. Fadiman's Fantasia Mathematica. People really thought I'd like the book of Japanese stories by a margin of 11 to 5.I hadn't remembered liking the Japanese tales that much the first time I read them, but I think my memory was defective, because almost all of them were excellent; I see that Triceratops by Tensei Kono is my pick for second-best piece in any of these anthologies (although the Le Guin mentioned earlier is still about fifty times better than even Triceratops). On the downside, I didn't care for the final novella, The Legend of the Paper Spaceship at all. The stories were also mostly really short, which I like for some reason.
Against this we have Fantasia Mathematica, which is split into three parts. The first has stories and excerpts from what I guess would be called literature -- Huxley, Plato, etc. The second is mostly science fictional. The last is mostly poetry.
The first section had its ups and downs, but the second section was pretty consistently entertaining and has some Martin Gardner stories that I don't think you see show up many other places. And I liked the poetry, which was a little unexpected! Taking all of this together, I pick:
Winner: The Best Japanese Science Fiction Stories
Championship Bout: Crank! vs. Classic Years vs. Japanese Science Fiction Stories
Looking at the poll, I see two people thought I'd like Crank! the best, two thought I'd like Women of Wonder: The Classic Years, and three thought I'd like The Best Japanese Short Stories. Plus, two thought that Crank! would be the best overall, three thought Classic Years would be, and four thought the Japanese collection would be. So what is it?Champion: The Best Japanese Science Fiction Stories edited by John L. Apostoluo and Martin H. Greenberg(Seven people thought I'd pick Fantasia Mathematica overall.)
Runner-up: Fantasia Mathematica edited by Clifton Fadiman