I don't think you can use the birthday paradox here - since the expected values of go moves are best treated as being surreal numbers:
http://en.wikipedia.org/wiki/Surreal_number
Surreal numbers were actually originally developed to handle go move values:
But the only special value mentioned on those pages, = { 0 | 0 }, is not a surreal number. It's a combinatorial game, and every surreal number is a combinatorial game, but 0 ≤ 0, making non-numeric.
Also, while values of fragments of Go games are best treated as combinatorial games, the final value of a Go game is always simply an integer (or even an element of the set {WIN, DRAW, LOSS}), and therefore so will the maximin.
Tyler Cowen argues in a TED talk (~15 min) that stories pervade our mental lives. He thinks they are a major source of cognitive biases and, on the margin, we should be more suspicious of them - especially simple stories. Here's an interesting quote about the meta-level: