[gjm11]

I agree. In particular, I wasn't arguing for anything special to happen at p=2/3. I was hopeful, though, that p=2/3 might be a tolerable approximation over a reasonable range of skill levels.

Having played a bit with some toy models, I've changed my mind a bit; my guess is that p=2/3 is a reasonable approximation for few-dan and few-kyu amateurs, but that outside, say, the 5k-5d range it's far enough off to make a substantial difference.

So, what does this do to those (anyway fairly bogus) "depth" figures? My crappy toy model suggests that for a 2/3 win probability you need a 3-rank difference around 24k, a 2-rank difference around 12k, a 1-rank difference around 2d, a 0.5-rank difference around 8d. And I estimate God at 15 amateur dan (if Cho Chikun is 9p and needs 4 stones from God then God is 21p; if, handwavily, 9d=3p and one p-step is 1/3 the size of one d-step, then God is 21p = (3+18)p = (9+6)d = 15d). So we need maybe 20 steps from God to 5d, then maybe 10 from there to 5k, then maybe 5 from there to 15k, then maybe 5 from there to 30k. That's 40 steps -- not so very different from what we get just by pretending one rank = one "2/3 win probability" step, as it happens.