So if you want a straighter curve over time just increase your effort exponentially. Players are disappointed because they are insufficiently lazy when they start out.

Or there is another option: stop caring about the curve and just have fun.

The article is not as precise as it could be, but R0 is not claimed to be the observed strength of the teacher. It is a parameter that is designed to fit the other data. EDIT: wineandgolover understands this. I am making the point to others.

It is an interesting article, and the author is quite honest in the sense that he hints at the severe limitations of his study more than once.

Note that R0, t0, and tau are all fitted parameters NOT data. Understanding this distinction is important. They are not direct observations. Furthermore, the discussion in the article seems to suggest that these parameters are individualized to each player. If that is indeed the case, then it would be shocking if the author was not able to generate such suggestive pictures that fit these functional forms nicely. There are enough degrees of freedom there to fit almost any steadily declining rate of growth.

Furthermore, it is unclear whether t, which is an observed variable in the study, is correlated with other factors that may affect growth rates.

Conclusion: We observe that rates of growth slow with time. We do not know if time is the cause of this deceleration or if time is simply correlated with the real causes.

The exercise is interesting and intellectually stimulating, but we should treat it more like engaging parlor conversation rather than anything resembling a serious model. As far as I can see, the author seems to feel that way as well.

PS: This is perhaps a trivial thing to say, but go learning has to be asymptotic in the sense that you cannot exceed a 100% win rate. The presence or absence of an asymptote can be merely a byproduct of how ratings are measured. The rulers we use are important.


A Dr. Strangelove reference? Yes, having fun is the best. I agree.

This may sound trivial but in fact it's not. If you e. g. assume that each rank improvement represents a certain reduction factor in mistakes-per-move, then a 100% win rate (-> 0% mistakes) would be represented by an infinite rating. In the hypothetic case of two players with very high ratings, a limiting factor may be also the board size since very low error rates might be not very well measureable with a finite number of moves.

Regarding the curve fits in the publication it is clear that there are many candidate functions which yield good fits. However, it seems that e. g. a logarithmic increase in rank can be excluded, since it is very difficult to fit a function like a*(1-exp(-t/tau)) against log(t):


Thus we can conclude at least that after a certain time the rank increase for typical players is slower than logarithmic.

I don't think that that last point is trivial. As discussed in another thread, you can describe a certain increase in strength as reducing your losses against a fixed set of opponents by a factor of ten (eg, .99 win rate becomes .999), or as maintaining your loss rate against that set while increasing the handicap by four stones, or as maintaining your old loss rate against a new set of opponents who are four stones stronger than the old set.

Now, the first description of getting strongeris "trivially" asymptotic, while the second one is ambiguous and the third one doesn't admit of any trivial limit. Which just goes to show you can parametrize these things however you damn well please. With the right kind of scaling, even the first description could be graphed in a way that didn't show an asymptote.

Is this time shorter or longer than the lifespan of an average human?

This theoretical bound definitely exists, I just have trouble concluding from there that "my asymptote" (as in the thread title) exists. I'm 3k - it's unlikely that I'll ever get near the top ranks of players in my lifetime, so I will always have a wealth of learning material and stronger players available to me to learn from, so the biggest factor that decides how fast I improve isn't some theoretical upper bound I'll never get near, it's my brain (along with how much time I make it spend on go). If I eventually stop improving, I probably haven't "trailed off" like people usually think of asymptotes doing, e.g. due to my brain reaching full capacity - I've probably died or given up go. And this holds as true for me as it does for all but the very strongest amateurs.

(Analogy: when you were in primary school learning about fractions, did how fast you learnt depend fundamentally on whether mathematics was finite or infinite in scope, or on how well educated your teacher was in degree-level or research-level mathematics? Probably not - you learnt at your own speed, regardless of those things, which were way out of your grasp. It's that internal speed of learning that interests me.)

However if your teacher taught you the wrong way to do fractions, you might have some problems. There's a 10k I see teaching on KGS a bit, and it can make me cringe.

All other things being equal, if I had a million years to study go how strong would I get?

I'm not convinced I could beat the current pros. I might learn everything there is to learn about go but in terms of raw reading strength, would my brain be able to adapt itself in the same way a younger brain can?

Not that it matters anyway, I just play a few moves a day on DGS when my work gets too boring, I'm not going to come close to my potential limits that way.

This isn't different from what I said, which is that existence of asymptotes is sensitive to the rating system used. Perhaps we mean different things by triviality. I meant that my statement is essentially a tautology. I wasn't making any statement about its importance.

After a million years you have a sporting chance that your brains are in better condition than the ones of some current pros.

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I think I am so I think I am.

There is that. Most people equally well reach an "asymptote" in their mathematical education, though all I really mean by that is that they have a panic attack every time they see numbers, and run far away as quickly as they can.

To be fair, that mostly originates from the typical teaching style employed in public schools:

Student: "I don't understand this."
Teacher: "You're not trying hard enough."
Student: "I've tried, but I still don't understand."
Teacher: "Do your worksheets."
Student: "The repetition doesn't help; it doesn't make sense to me."
Teacher: "I can't help you if you won't help yourself."

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Sincerely, Chad R. Sobodash

not to be conceited, but i don't feel like i have one

later on the things to improve will get finer and finer grained such that it might require more effort, but most people at that rank have that same problem, so it's not like i'll be the only one. and improving at that point is still improving

if we're talking rank graphs, then of course it will look asymptotic due to less people to play and also how the lines of strength seem to blur.

improvement at the dan level is the same as improvement at the kyu level. the "leaps and bounds" you make in the beginning are the same "leaps and bounds" you make as a dan, it just looks more dramatic when you're at the kyu level because it's something everyone is able to easily identify.

in the very end, your improvement in go is only limited by how much work you put into it, not by age

Says the youngster! (and maybe someday 9d?)

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- Brady
Want to see videos of low-dan mistakes and what to learn from them? Brady's Blunders