[stromgo]

> In go, beating someone 2/3 of the time corresponds to being about one kyu/dan rank stronger

This isn't true. One kyu/dan rank stronger means being 1 stone stronger (so winning 50% of the time when playing White with reverse komi). In practice this may correspond to winning 2/3 of the time with normal komi for high dan players, but that doesn't hold for low kyu players. A 29k has maybe a 51% chance of winning against a 30k because both will make huge mistakes. So although the 29k can score on average 13-15 more points than the 30k in a given game, this advantage is swamped by the large standard deviation of scores in beginner games, turning the win/loss outcome into essentially a coin flip.

[gjm11]

Fair comment about very weak players. My impression is that the element of chance goes way down well before you get to high dan level, though. How sure are you that I'm wrong?

[stromgo]

Quite sure. 17k players still make many huge mistakes, and at the other extreme, God (say 13d) would win 100% of the time against a 12d player. Given that the win ratio for a one rank difference starts at 50% for extreme beginners and ends at 100% for God, maybe you should be the one explaining why you'd expect a significant plateau at 2/3 instead of a smooth increase.

[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.